12,537 results on '"FRACTIONAL differential equations"'
Search Results
2. Bifurcation analysis and soliton structures of the truncated [formula omitted]-fractional Kuralay-II equation with two analytical techniques.
- Author
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Arafat, S M Yiasir and Islam, S M Rayhanul
- Subjects
PARTIAL differential equations ,FRACTIONAL differential equations ,NONLINEAR waves ,HYPERBOLIC functions ,WAVE analysis - Abstract
The truncated M-fractional Kuralay (TMFK)-II equation is prevalent in the exploration of specific complex nonlinear wave phenomena. Such types of wave phenomena are more applicable in science and engineering. These equations could potentially provide insights into understanding the intricate dynamics of optical phenomena, encompassing solitons, nonlinear effects, and wave interactions. This study aims to uncover a diverse range of soliton solutions to the model, spanning trigonometric, hyperbolic, exponential, and rational expressions. These solutions are unveiled through the application of extended hyperbolic functions and improved F-expansion techniques, representing the primary objective of this research. The three-dimensional (3D) and two-dimensional (2D) combined charts are plotted for some of these solutions. The impact of the fractional parameters and time variations is also illustrated. Moreover, the models are converted into a planar dynamical system using a Galilean transformation, and the analysis of bifurcation is examined. This research underscores the versatility of the aforementioned techniques for exploring complex nonlinear phenomena across various engineering and scientific disciplines. Finally, the findings of this study hold significant implications for advancing our understanding and analysis of nonlinear wave dynamics in diverse physical systems. [ABSTRACT FROM AUTHOR]
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- 2024
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3. Generalized Fractional (ρ, k, φ)‐Proportional Hilfer Derivatives and Some Properties.
- Author
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Wang, Haihua and Zafer, Agacik
- Subjects
FRACTIONAL calculus ,MATHEMATICS ,FRACTIONAL differential equations ,LAPLACE transformation ,INTEGRALS - Abstract
Building on previous work in fractional calculus, this paper introduces new definitions for the (ρ, k, φ)‐proportional integral and (ρ, k, φ)‐proportional H fractional derivative. This new approach retains the semigroup properties of traditional fractional integrals. A significant advantage of this fractional calculus is its compatibility with the majority of existing studies on fractional differential equations. Furthermore, we delve into the properties of the generalized fractional integrals and derivatives. We discuss, for instance, the mapping properties of the (ρ, k, φ)‐proportional integral. To elucidate these concepts, we introduce a set of new weighted spaces. Additionally, we explore the generalized Laplace transform of both the (ρ, k, φ)‐proportional integrals and (ρ, k, φ)‐proportional H fractional derivatives. Also, examples concerning the linear (ρ, k, φ)‐proportional H fractional equations are given to illustrate the main results. [ABSTRACT FROM AUTHOR]
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- 2024
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4. Fractional differential equation on the whole axis involving Liouville derivative.
- Author
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Matychyn, Ivan and Onyshchenko, Viktoriia
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FRACTIONAL differential equations , *PARTIAL differential equations , *ORDINARY differential equations , *FRACTIONAL calculus , *INTEGRAL transforms - Abstract
The paper investigates fractional differential equations involving the Liouville derivative. Solution to these equations under a boundary condition inside the time interval are derived in explicit form, their uniqueness is established using integral transforms technique. [ABSTRACT FROM AUTHOR]
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- 2024
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5. Attractors of Caputo semi-dynamical systems.
- Author
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Doan, T. S. and Kloeden, P. E.
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FRACTIONAL differential equations , *VOLTERRA equations , *VECTOR fields , *CONTINUOUS functions , *VECTOR valued functions - Abstract
The Volterra integral equation associated with autonomous Caputo fractional differential equation (FDE) of order α ∈ (0 , 1) in R d was shown by the authors [4] to generate a semi-group on the space C of continuous functions f : R + → R d with the topology uniform convergence on compact subsets. It serves as a semi-dynamical system for the Caputo FDE when restricted to initial functions f(t) ≡ i d x 0 for x 0 ∈ R d . Here it is shown that this semi-dynamical system has a global Caputo attractor in C , which is closed, bounded, invariant and attracts constant initial functions, when the vector field function in the Caputo FDE satisfies a dissipativity condition as well as a local Lipschitz condition. [ABSTRACT FROM AUTHOR]
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- 2024
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6. Averaging principle for stochastic Caputo fractional differential equations with non-Lipschitz condition.
- Author
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Guo, Zhongkai, Han, Xiaoying, and Hu, Junhao
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STOCHASTIC differential equations , *FRACTIONAL differential equations , *NONLINEAR differential equations , *SINGULAR integrals , *GRONWALL inequalities - Abstract
In this paper, the averaging principle for stochastic Caputo fractional differential equations with the nonlinear terms satisfying the non-Lipschitz condition is considered. The work in the article is roughly divided into three parts. Firstly, we establish a generalized Gronwall inequality with singular integral kernel which is a key part in our analysis. Secondly, we discuss the existence and uniqueness of solution. And finally, the averaging principle is considered. [ABSTRACT FROM AUTHOR]
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- 2024
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7. A high order predictor-corrector method with non-uniform meshes for fractional differential equations.
- Author
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Mokhtarnezhadazar, Farzaneh
- Subjects
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VOLTERRA equations , *FRACTIONAL differential equations , *INTEGRAL equations - Abstract
This article proposes a predictor-corrector scheme for solving the fractional differential equations 0 C D t α y (t) = f (t , y (t)) , α > 0 with non-uniform meshes. We reduce the fractional differential equation into the Volterra integral equation. Detailed error analysis and stability analysis are investigated. The convergent order of this method on non-uniform meshes is still 3 though 0 C D t α y (t) is not smooth at t = 0 . Numerical examples are carried out to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]
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- 2024
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8. Mittag-Leffler stability and Lyapunov stability for a problem arising in porous media.
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Hassan, Jamilu Hashim, Tatar, Nasser-eddine, and Al-Homidan, Banan
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LYAPUNOV stability , *FRACTIONAL differential equations , *PARTIAL differential equations , *POROUS materials , *FRACTIONAL calculus - Abstract
A fractional order problem arising in porous media is considered. Well-posedness as well as stability are discussed. Mittag-Leffler stability is proved in case of a strong fractional damping in the displacement component and a fractional frictional one in the volume fraction component. This extends an existing result from the integer-order (second-order) case to the non-integer case. In the absence of the fractional damping in the volume fraction component, it is shown a convergence to zero and a Lyapunov uniform stability. [ABSTRACT FROM AUTHOR]
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- 2024
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9. Lie symmetries, exact solutions and conservation laws of (2+1)-dimensional time fractional cubic Schrödinger equation.
- Author
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Yu, Jicheng and Feng, Yuqiang
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FRACTIONAL differential equations , *PARTIAL differential equations , *SCHRODINGER equation , *CUBIC equations , *CONSERVATION laws (Physics) - Abstract
In this paper, Lie symmetry analysis method is applied to ( 2 + 1 ) {(2+1)} -dimensional time fractional cubic Schrödinger equation. We obtain all the Lie symmetries and reduce the ( 2 + 1 ) {(2+1)} -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to ( 1 + 1 ) (1+1) -dimensional counterparts with Erdélyi–Kober fractional derivative. Then we obtain the power series solutions of the reduced equations and prove their convergence. In addition, the conservation laws for the governing model are constructed by the new conservation theorem and the generalization of Noether operators. [ABSTRACT FROM AUTHOR]
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- 2024
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10. Fractional weak adversarial networks for the stationary fractional advection dispersion equations.
- Author
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Feng, Dian, Yang, Zhiwei, and Zou, Sen
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FRACTIONAL differential equations , *SPECIAL functions , *ADVECTION , *EQUATIONS , *DISPERSION (Chemistry) - Abstract
In this article, we propose the fractional weak adversarial networks (f-WANs) for the stationary fractional advection dispersion equations based on their weak formulas. This enables us to handle less regular solutions for the fractional equations. To handle the non-local property of the fractional derivatives, convolutional layers and special loss functions are introduced in this neural network. Numerical experiments for both smooth and less regular solutions show the validity of f-WANs. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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11. An effective computational approach and sensitivity analysis to pseudo-parabolic-type equations.
- Author
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Kaplan, Melike, Butt, Asma Rashid, Thabet, Hayman, Akbulut, Arzu, Raza, Nauman, and Kumar, Dipankar
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ORDINARY differential equations , *NONLINEAR differential equations , *FRACTIONAL differential equations , *PARTIAL differential equations , *EXPONENTIAL functions - Abstract
The researchers have developed numerous analytical and numerical techniques for solving fractional partial differential equations most of which provide approximate solutions. Exact solutions, however, are vitally important in a convenient conception of the qualitative properties of the concerned phenomena and processes. In this paper, the pseudo-parabolic-type equations with conformable fractional derivatives are reduced to conformable fractional nonlinear ordinary differential equations by implementing a simple wave transformation. An important benefit of the proposed transformation is that it yields analytical solutions of the conformable pseudo-parabolic type equations by applying the exponential rational function strategy. The sensitivity behaviour of the model has been mentioned thoroughly. [ABSTRACT FROM AUTHOR]
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- 2024
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12. Exploring the stochastic patterns of hyperchaotic Lorenz systems with variable fractional order and radial basis function networks.
- Author
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Awais, Muhammad, Khan, Muhammad Adnan, and Bashir, Zia
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DIFFERENTIABLE dynamical systems , *FRACTIONAL differential equations , *FRACTIONAL calculus , *LYAPUNOV exponents , *CHAOS theory , *LORENZ equations - Abstract
This research explores the incorporation of variable order (VO) fractional calculus into the hyperchaotic Lorenz system and studies various chaotic features and attractors. Initially, we propose a variable fractional order hyperchaotic Lorenz system and numerically solve it. The solutions are obtained for multiple choices of control parameters, and these results serve as reference solutions for exploring chaos with the artificial intelligence tool radial basis function network (RBFN). We rebuild phase spaces and trajectories of system states to exhibit chaotic behavior at various levels. To further assess the sensitivity of chaotic attractors, Lyapunov exponents are calculated. The efficacy of the designed computational RBFN is validated through the RMSE and extensive error analysis. The proposed research on AI capabilities aims to introduce an innovative methodology for modeling and analyzing hyperchaotic dynamical systems with variable orders. [ABSTRACT FROM AUTHOR]
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- 2024
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13. A novel hybrid variation iteration method and eigenvalues of fractional order singular eigenvalue problems.
- Author
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Kumari, Sarika, Kannaujiya, Lok Nath, Kumar, Narendra, Verma, Amit K., and Agarwal, Ravi P.
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NONLINEAR boundary value problems , *CAPUTO fractional derivatives , *ENERGY levels (Quantum mechanics) , *FRACTIONAL differential equations , *EIGENFUNCTIONS , *LAGRANGE multiplier - Abstract
In response to the challenges posed by complex boundary conditions and singularities in molecular systems and quantum chemistry, accurately determining energy levels (eigenvalues) and corresponding wavefunctions (eigenfunctions) is crucial for understanding molecular behavior and interactions. Mathematically, eigenvalues and normalized eigenfunctions play crucial role in proving the existence and uniqueness of solutions for nonlinear boundary value problems (BVPs). In this paper, we present an iterative procedure for computing the eigenvalues (μ ) and normalized eigenfunctions of novel fractional singular eigenvalue problems, D 2 α y (t) + k t α D α y (t) + μ y (t) = 0 , 0 < t < 1 , 0 < α ≤ 1 , with boundary condition, y ′ (0) = 0 , y (1) = 0 , where D α , D 2 α represents the Caputo fractional derivative, k ≥ 1 . We propose a novel method for computing Lagrange multipliers, which enhances the variational iteration method to yield convergent solutions. Numerical findings suggest that this strategy is simple yet powerful and effective. [ABSTRACT FROM AUTHOR]
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- 2024
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14. A new hybrid special function class and numerical technique for multi-order fractional differential equations.
- Author
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Ghanim, F., Khan, Fareeha Sami, Al-Janaby, Hiba F., and Ali, Ali Hasan
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FRACTIONAL calculus ,NUMERICAL functions ,FRACTIONAL differential equations ,INTEGRAL calculus ,HYPERGEOMETRIC functions - Abstract
This study aims to investigate the properties of fractional calculus theory (FCT) in the complex domain. We focus on the relationship between the theories of special functions (SFT) and FCT, which have seen recent advancements and have led to various successful applications in fields such as engineering, mathematics, physics, biology, and other allied disciplines. Our main contribution is the development of a special function, specifically the confluent hypergeometric function (CHF) on the complex domain. By deriving various implementations of fractional order derivatives and integral operators using this function, we present a new class of special functions combining certain cases of Mittag-Leffler and confluent hypergeometric functions. Moreover, a new numerical technique for solving linear and nonlinear multi-order fractional differential equations has been developed using the proposed class of functions and the point collocation method. Graphical results are shown to demonstrate the efficacy of this proposed technique and its applicability. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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15. Tempered fractional differential equations on hyperbolic space.
- Author
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Garra, Roberto and Orsingher, Enzo
- Subjects
FRACTIONAL differential equations ,HYPERBOLIC spaces ,CAPUTO fractional derivatives ,WIENER processes ,HYPERBOLIC geometry - Abstract
In this paper we study linear fractional differential equations involving tempered Caputo-type derivatives in the hyperbolic space. We consider in detail the three-dimensional case for its simple and useful structure. We also discuss the probabilistic meaning of our results in relation to the distribution of an hyperbolic Brownian motion time-changed with the inverse of a tempered stable subordinator. The generalization to an arbitrary dimension n can be easily obtained. We also show that it is possible to construct a particular solution for the non-linear porous-medium type tempered equation by using elementary functions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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16. Approximation Approach to the Fractional BVP with the Dirichlet Type Boundary Conditions.
- Author
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Marynets, Kateryna and Pantova, Dona
- Abstract
We use a numerical-analytic technique to construct a sequence of successive approximations to the solution of a system of fractional differential equations, subject to Dirichlet boundary conditions. We prove the uniform convergence of the sequence of approximations to a limit function, which is the unique solution to the boundary value problem under consideration, and give necessary and sufficient conditions for the existence of solutions. The obtained theoretical results are confirmed by a model example. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
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17. A Collocation Technique via Pell-Lucas Polynomials to Solve Fractional Differential Equation Model for HIV/AIDS with Treatment Compartment.
- Author
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Yıldırım, Gamze and Yüzbaşı, Şuayip
- Subjects
FRACTIONAL differential equations ,AIDS treatment ,NONLINEAR equations ,AIDS patients ,COLLOCATION methods - Abstract
In this study, a numerical method based on the Pell-Lucas polynomials (PLPs) is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment. The HIV/AIDS mathematical model with a treatment compartment is divided into five classes, namely, susceptible patients (S), HIV-positive individuals (I), individuals with full-blown AIDS but not receiving ARV treatment (A), individuals being treated (T), and individuals who have changed their sexual habits sufficiently (R). According to the method, by utilizing the PLPs and the collocation points, we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations. Also, the error analysis is presented for the Pell-Lucas approximation method. The aim of this study is to observe the behavior of five populations after days when drug treatment is applied to HIV-infectious and full-blown AIDS people. To demonstrate the usefulness of this method, the applications are made on the numerical example with the help of MATLAB. In addition, four cases of the fractional order derivative () are examined in the range. Owing to applications, we figured out that the outcomes have quite decent errors. Also, we understand that the errors decrease when the value of N increases. The figures in this study are created in MATLAB. The outcomes indicate that the presented method is reasonably sufficient and correct. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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18. Separate families of fuzzy dominated nonlinear operators with applications.
- Author
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Rasham, Tahair
- Abstract
This paper presents novel fixed-point results for two distinct families of fuzzy-dominated operators satisfying a generalized nonlinear contraction condition on a closed ball in a complete strong b-metric-like space. Our research introduces innovative fixed-point theorems for separate families of ordered fuzzy-dominated mappings in ordered complete strong b-metric-like spaces. Two different kinds of mappings are used in our methodology: a class of fuzzy-dominated mappings and a class of strictly non-decreasing mappings. Furthermore, we establish new fixed-point results for fuzzy-graph-dominated contractions. To substantiate our findings, we provide both rigorous and illustrative examples. We demonstrate the uniqueness of our results by applying them to obtain common solutions for fractional differential equations and fuzzy Volterra integral equations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
19. An innovative computational approach for fuzzy space-time fractional Telegraph equation via the new iterative transform method.
- Author
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Kshirsagar, Kishor Ashok, Nikam, Vasant R., Gaikwad, Shrikisan B., and Tarate, Shivaji Ashok
- Subjects
FRACTIONAL differential equations ,ITERATIVE methods (Mathematics) ,FRACTIONAL calculus ,REACTION-diffusion equations ,APPROXIMATION theory - Abstract
In this paper, the Fuzzy Sumudu Transform Iterative method (FSTIM) was applied to find the exact fuzzy solution of the fuzzy space-time fractional telegraph equations using the Fuzzy Caputo Fractional Derivative operator. The Telegraph partial differential equation is a hyperbolic equation representing the reaction-diffusion process in various fields. It has applications in engineering, biology, and physics. The FSTIM provides a reliable and efficient approach for obtaining approximate solutions to these complex equations improving accuracy and allowing for fine-tuning and optimization for better approximation results. The work introduces a fuzzy logic-based approach to Sumudu transform iterative methods, offering flexibility and adaptability in handling complex equations. This innovative methodology considers uncertainty and imprecision, providing comprehensive and accurate solutions, and advancing numerical methods. Solving the fuzzy space-time fractional telegraph equation used a fusion of the Fuzzy Sumudu transform and iterative approach. Solution of fuzzy fractional telegraph equation finding analytically and interpreting its results graphically. Throughout the article, whenever we draw graphs, we use Mathematica Software. We successfully employed FSTIM, which is elegant and fast to convergence. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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20. Existence and uniqueness of positive solutions for a Hadamard fractional integral boundary value problem.
- Author
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Ahmadkhanlu, Asghar and Jamshidzadeh, Shabnam
- Subjects
UNIQUENESS (Mathematics) ,EXISTENCE theorems ,BOUNDARY value problems ,FRACTIONAL differential equations ,FIXED point theory - Abstract
The main aim of this paper is to study a kind of boundary value problem with an integral boundary condition including Hadamard-type fractional differential equations. To do this, upper and lower solutions are used to guarantee their existence, and Schauder’s fixed point theorem is used to prove the uniqueness of the positive solutions to this problem. An illustrated example is presented to explain the theorems that have been proved. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
21. Existence of solutions for a class of asymptotically linear fractional Schrödinger equations.
- Author
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Abid, Imed, Baraket, Sami, and Mahmoudi, Fethi
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PARTIAL differential equations , *FRACTIONAL differential equations , *SCHRODINGER equation , *POTENTIAL energy - Abstract
In this paper, we focus on studying a fractional Schrödinger equation of the form { (− Δ) s u + V (x) u = f (x , u) in Ω , u > 0 in Ω , u = 0 in R n ∖ Ω , where 0 < s < 1 , n > 2 s , Ω is a smooth bounded domain in R n , (− Δ) s denotes the fractional Laplacian of order s, f (x , t) is a function in C (Ω ‾ × R) , and f (x , t) / t is nondecreasing in t and converges uniformly to an L ∞ function q (x) as t approaches infinity. The potential energy V satisfies appropriate assumptions. In the first part of our study, we analyze the asymptotic linearity of the nonlinearity and investigate the occurrence of the bifurcation phenomenon. We employ variational techniques and a "mountain pass" approach in our proof, notable for not assuming the Ambrosetti–Rabinowitz condition or any replacement condition on the nonlinearity. Additionally, we extend our methods to handle cases where the function f (x , t) exhibits superlinearity in t at infinity, represented by q (x) ≡ + ∞ . [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
22. Similarity reduction, group analysis, conservation laws, and explicit solutions for the time-fractional deformed KdV equation of fifth order.
- Author
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Al-Denari, Rasha B., Ahmed, Engy. A., Seadawy, Aly R., Moawad, S. M., and EL-Kalaawy, O. H.
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LIE groups , *ORDINARY differential equations , *FRACTIONAL differential equations , *CONSERVATION laws (Physics) , *ANALYTICAL solutions - Abstract
Through this paper, we consider the time-fractional deformed fifth-order Korteweg–de Vries (KdV) equation. First of all, we detect its symmetries by Lie group analysis with the help of Riemann–Liouville (R-L) fractional derivatives. These symmetries are employed to convert the considered equation into a fractional ordinary differential (FOD) equation in the sense of Erdélyi-Kober (E-K) fractional operator. Also, a set of new analytical solutions for the equation under study are obtained via the power series method. We test the accuracy and effectiveness of this method by providing a numerical simulation of the obtained solution and studying the effect of α which is represented graphically in 2D and 3D plots. Added to that, we prove the convergence of the power series solutions. Finally, the computation of the conservation laws is introduced in detail. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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23. Strong Convergence of Euler-Type Methods for Nonlinear Fractional Stochastic Differential Equations without Singular Kernel.
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Ali, Zakaria, Abebe, Minyahil Abera, and Nazir, Talat
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STOCHASTIC differential equations , *FRACTIONAL differential equations , *WHITE noise , *NOISE , *EULER method - Abstract
In this paper, we first prove the existence and uniqueness of the solution to a variable-order Caputo–Fabrizio fractional stochastic differential equation driven by a multiplicative white noise, which describes random phenomena with non-local effects and non-singular kernels. The Euler–Maruyama scheme is extended to develop the Euler–Maruyama method, and the strong convergence of the proposed method is demonstrated. The main difference between our work and the existing literature is the fact that our assumptions on the nonlinear external forces are those of one-sided Lipschitz conditions on both the drift and the nonlinear intensity of the noise as well as the proofs of the higher integrability of the solution and the approximating sequence. Finally, to validate the numerical approach, current results from the numerical implementation are presented to test the efficiency of the scheme used in order to substantiate the theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
24. Unique Solutions for Caputo Fractional Differential Equations with Several Delays Using Progressive Contractions.
- Author
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Tunç, Cemil and Akyildiz, Fahir Talay
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FRACTIONAL differential equations , *CAPUTO fractional derivatives , *DIFFERENTIAL equations , *MATHEMATICAL models , *DELAY differential equations - Abstract
We take into account a nonlinear Caputo fractional-order differential equation including several variable delays. We examine whether the solutions to the Caputo fractional-order differential equation taken under consideration, which has numerous variable delays, are unique. In the present study, first, we will apply the method of progressive contractions, which belongs to T.A. Burton, to Caputo fractional-order differential equation, including multiple variable delays, which has not yet appeared in the relevant literature by this time. The significant point of the method of progressive contractions consists of a very flexible idea to discuss the uniqueness of solutions for various mathematical models. Lastly, we provide two examples to demonstrate how this paper's primary outcome can be applied. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
25. A fractional differential quadrature method for fractional differential equations and fractional eigenvalue problems.
- Author
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Mohamed, Salwa A.
- Subjects
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FRACTIONAL differential equations , *DIFFERENTIAL quadrature method , *DIFFERENTIAL calculus , *PARTIAL differential equations , *DIFFERENTIAL equations - Abstract
In this paper, based on the differential quadrature method (DQM), matrix operators are derived for fractional integration and Caputo differentiation. These operators generalize the efficient DQM to fractional calculus. The proposed fractional differential/integral quadrature method (FDIQM) is used to solve various types of fractional ordinary and partial differential equations. FDIQM unifies the solution of multi‐integer fractional‐order differential equations leading to significant simplification in the implementation. Numerous examples are presented to demonstrate the accuracy of the operators. Other examples are presented to solve various fractional differential equations including time‐fractional sub‐diffusion equation, linear/nonlinear, and multiorder fractional differential equations. In addition, numerous boundary conditions are considered including mixed fractional derivatives. Further, a nonlinear fractional eigenvalue problem is solved efficiently, and its bifurcation diagrams are obtained. Comparisons between the proposed method and the existing ones are included, showing the ease of implementation, efficiency, and applicability of FDIQM. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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26. Delayed analogue of three‐parameter Mittag‐Leffler functions and their applications to Caputo‐type fractional time delay differential equations.
- Author
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Huseynov, Ismail T. and Mahmudov, Nazim I.
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FRACTIONAL differential equations , *CAPUTO fractional derivatives , *TIME delay systems , *MATRIX functions , *INTEGRAL transforms - Abstract
In this paper, we consider a Cauchy problem for a Caputo‐type time delay linear system of fractional differential equations with permutable matrices. First, we provide a new representation of solutions to linear homogeneous fractional differential equations using the Laplace integral transform and variation of constants formula via a newly defined delayed Mittag‐Leffler type matrix function introduced through a three‐parameter Mittag‐Leffler function. Second, with the help of a delayed perturbation of a Mittag‐Leffler type matrix function, we attain an explicit formula for solutions to a linear nonhomogeneous time delay fractional order system using the superposition principle. Furthermore, we prove the existence and uniqueness of solutions to nonlinear fractional delay differential equations using the contraction mapping principle. Finally, we present an example to illustrate the applicability of our results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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27. Rotational periodic boundary value problem for a fractional nonlinear differential equation.
- Author
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Cheng, Yi, Gao, Shanshan, and Agarwal, Ravi P.
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NONLINEAR differential equations , *FRACTIONAL differential equations , *BOUNDARY value problems , *DIFFERENTIAL inclusions , *DIFFERENTIAL equations - Abstract
This paper is devoted to study the rotational periodic boundary value problem for a fractional‐order nonlinear differential equation. Applying topology‐degree theory and the Leray‐Schauder fixed‐point theorem, we prove the existence and uniqueness of solution for the fractional‐order differential system. Furthermore, the existence of solution for a nonlinear differential system with a multivalued perturbation term is investigated by using set‐valued theory and techniques of functional analysis. Two examples of applications are given at the end. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
28. Fractional differential equation pertaining to an integral operator involving incomplete H‐function in the kernel.
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Bansal, Manish Kumar, Lal, Shiv, Kumar, Devendra, Kumar, Sunil, and Singh, Jagdev
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FRACTIONAL differential equations , *FREE electron lasers , *INTEGRAL operators , *INTEGRAL equations , *MATHEMATICAL physics - Abstract
Fractional differential equations (FDEs) involving a family of special functions and their solutions represent different physical phenomena. FDEs are characterizing and solving many problems of mathematical physics, chemistry, biology, and engineering. In this article, we establish an integral operator involving the family of incomplete H‐function (IHF) in its kernel. First, we derive the solutions for FDEs involving the generalized composite fractional derivative (GCFD) and integral operator associated with the incomplete H‐function. Several important special cases are revealed and analyzed. The main result derived in this study contains first‐order Volterra‐type integro‐differential equation describing the unsaturated nature of the free electron laser as a special case. Further, we give the graphical interpretation of the solution of FDEs. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
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29. On the solution of a boundary value problem associated with a fractional differential equation.
- Author
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Sevinik Adigüzel, Rezan, Aksoy, Ümit, Karapinar, Erdal, and Erhan, İnci M.
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GREEN'S functions , *FRACTIONAL differential equations , *BOUNDARY value problems , *NONLINEAR differential equations , *EXISTENCE theorems - Abstract
The problem of the existence and uniqueness of solutions of boundary value problems (BVPs) for a nonlinear fractional differential equation of order 2<α ≤ 3 is studied. The BVP is transformed into an integral equation and discussed by means of a fixed point problem for an integral operator. Conditions for the existence and uniqueness of a fixed point for the integral operator are derived via b‐comparison functions on complete b‐metric spaces. In addition, estimates for the convergence of the Picard iteration sequence are given. An estimate for the Green's function related with the problem is provided and employed in the proof of the existence and uniqueness theorem for the solution of the given problem. Illustrative examples are presented to support the theoretical results. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
- View/download PDF
30. A coupled system of Langevin differential equations of fractional order and associated to antiperiodic boundary conditions.
- Author
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Baghani, Hamid, Alzabut, Jehad, and Nieto, Juan J.
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BOUNDARY value problems , *CAPUTO fractional derivatives , *LANGEVIN equations , *FRACTIONAL differential equations - Abstract
In this paper, we provide an extension result for the existence and uniqueness of solutions for a coupled system of fractional Langevin differential equations with antiperiodic boundary conditions. The system involves two different Caputo fractional derivatives defined on different intervals and associated with boundary conditions described by sequential fractional derivatives. As a conclusion for our main result, we deduce the results of H. Fazli, J.J. Nieto, Fractional Langevin equation with anti‐periodic boundary conditions, Chaos Soliton Fract, 114 (2018),332–337 under less restrictive conditions. The consistency of the main results is demonstrated by two numerical examples. For the sake of completeness, we end the paper by a concluding discussion. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
- View/download PDF
31. Degree theory and existence of positive solutions to coupled system involving proportional delay with fractional integral boundary conditions.
- Author
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Ali, Anwar, Sarwar, Muhammad, Zada, Mian Bahadur, and Shah, Kamal
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CAPUTO fractional derivatives , *TOPOLOGICAL degree , *BOUNDARY value problems , *INTEGRAL equations , *FRACTIONAL integrals - Abstract
The purpose of this paper is to obtain the existence of at least one solution to the following coupled system of nonlinear fractional order differential equations under the integral type boundary conditions by using topological degree theory 0.1Dδμ(ℓ)=F1(ℓ,μ(λℓ),ν(λℓ)),ℓ∈[0,1],Dϱν(ℓ)=F2(ℓ,μ(λℓ),ν(λℓ)),ℓ∈[0,1],μ(0)=r(μ),μ(1)=1Γ(δ)∫01(1−η)δ−1φ(η,μ(η))dη,ν(0)=h(ν),ν(1)=1Γ(ϱ)∫01(1−η)ϱ−1ψ(η,ν(η))dη, where δ,ϱ∈(1,2], 0<λ<1, D denotes the standard Caputo fractional derivative, F1,F2:[0,1]×ℜ×ℜ→ℜ, φ,ψ:[0,1]×ℜ→ℜ and r,h:[0,1]→ℜ are continuous functions. For this intention, some results for the existence of at least one solution are constructed. For the validity of our results, an appropriate example is presented. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
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32. Eigenvalue superposition for Toeplitz matrix-sequences with matrix order dependent symbols.
- Author
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Bogoya, M., Grudsky, S.M., and Serra-Capizzano, S.
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TOEPLITZ matrices , *EIGENVALUES , *FINITE difference method , *FRACTIONAL differential equations , *DIFFERENTIAL operators - Abstract
The eigenvalues of Toeplitz matrices T n (f) with a real-valued generating function f , satisfying some conditions and tracing out a simple loop over the interval [ − π , π ] , are known to admit an asymptotic expansion with the form λ j (T n (f)) = f (σ j , n) + c 1 (σ j , n) h + c 2 (σ j , n) h 2 + O (h 3) , where h = 1 / (n + 1) , σ j , n = π j h , and c k are some bounded coefficients depending only on f. The numerical results presented in the literature suggest that the effective conditions for the expansion to hold are weaker and reduce to a fixed smoothness and to having only two intervals of monotonicity over [ − π , π ]. In this article we investigate the superposition caused over this expansion, when considering the following linear combination λ j (T n (f 0) + β n , 1 T n (f 1) + β n , 2 T n (f 2)) , where β n , 1 , β n , 2 are certain constants depending on n and the generating functions f 0 , f 1 , f 2 are either simple loop or satisfy the weaker conditions mentioned before. We formally obtain an asymptotic expansion in this setting under simple-loop related assumptions, and we show numerically that there is much more to investigate, opening the door to linear in time algorithms for the computation of eigenvalues of large matrices of this type including a multilevel setting. The problem is of concrete interest, considering spectral features of matrices stemming from the numerical approximation of standard differential operators and distributed order fractional differential equations, via local methods such as Finite Differences, Finite Elements, and Isogeometric Analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
33. GLT sequences and automatic computation of the symbol.
- Author
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Sarathkumar, N.S. and Serra-Capizzano, S.
- Subjects
- *
FRACTIONAL differential equations , *PARTIAL differential equations , *DIFFERENTIAL operators , *FINITE differences , *ISOGEOMETRIC analysis , *PSEUDODIFFERENTIAL operators - Abstract
Spectral and singular value symbols are valuable tools to analyse the eigenvalue or singular value distributions of matrix-sequences in the Weyl sense. More recently, Generalized Locally Toeplitz (GLT) sequences of matrices have been introduced for the spectral/singular value study of the numerical approximations of differential operators in several contexts. As an example, such matrix-sequences stem from the large linear systems approximating Partial Differential Equations (PDEs), Fractional Differential Equations (FDEs), Integro Differential Equations (IDEs), using any discretization on reasonable grids via local methods, such as Finite Differences, Finite Elements, Finite Volumes, Isogeometric Analysis, Discontinuous Galerkin etc. Studying the asymptotic spectral behaviour of GLT sequences is useful in analysing classical techniques for the solution of the corresponding PDEs/FDEs/IDEs and in designing novel fast and efficient methods for the corresponding large linear systems or related large eigenvalue problems. The theory of GLT sequences, in combination with the results concerning the asymptotic spectral distribution of perturbed sequences of matrices, is one of the most powerful and successful tools for computing the spectral symbol f. In this regard, it would be beneficial to design an automatic procedure to compute the spectral symbols of such matrix-sequences and Ahmed Ratnani partially pursued it. Here, in the case of one-dimensional and two-dimensional differential problems, we continue in this direction by proposing an automatic procedure for computing the symbol of the underlying sequences of matrices, assuming that it is a GLT sequence satisfying mild conditions. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
- View/download PDF
34. A comprehensive mathematical analysis of fractal–fractional order nonlinear re-infection model.
- Author
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Eiman, Shah, Kamal, Sarwar, Muhammad, and Abdeljawad, Thabet
- Subjects
DIFFERENTIAL operators ,MATHEMATICAL analysis ,NUMERICAL analysis ,FRACTIONAL differential equations ,VIRUS diseases - Abstract
In recent times, various real word problems including infectious diseases, engineering problems, and chemical processes, etc are modelled by using differential equations with fractals and fractional orders. The mentioned area provides a powerful tool to investigate the aforesaid problems from different perspectives like analysis, numerical investigation, and qualitative theory. Similarly, one of the global issue is devoted to re-infection of viral diseases like COVID-19. The said behaviour has produced very saviour impact on human life worldwide. Also, this is very critical for health system of a country as well as for its economics situation. To study the aforesaid phenomenon of re-infection, a hybrid type fractal–fractional three compartments model is formulated. The basic results related to boundedness, positivity, equilibrium points, basic reproductive number, and sensitivity analysis of the basic threshold are included in the theoretical aspect. Additionally, the Volterra–Lyapunov approach is used to establish both local and global stability analysis under fractals and fractional derivative. Furthermore, utilizing techniques from numerical functional analysis and fixed point theory, the existence and Hyers–Ulam stability of the solutions are also examined. A numerical algorithm based on Lagrange's interpolation polynomials is extended to simulate the results of various compartments. Graphical interpretations corresponding to different fractals–fractional order values are presented. For verification of our numerical results, a comparison is provided between the real and simulated data. Here it should be kept in mind that the differential operator of fractals–fractional order is considered with Mittag-Leffler kernel. The mentioned operator is more general among the available fractals fractional differential operators in literature. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
- View/download PDF
35. Utilizing fractional derivatives and sensitivity analysis in a random framework: a model-based approach to the investigation of random dynamics of malware spread.
- Author
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Bekiryazici, Zafer
- Subjects
- *
FRACTIONAL differential equations , *CAPUTO fractional derivatives , *BASIC reproduction number , *EULER method , *RANDOM variables - Abstract
In this study, an ordinary-deterministic equation system modeling the spread dynamics of malware under mutation is analyzed with fractional derivatives and random variables. The original model is transformed into a system of fractional-random differential equations (FRDEs) using Caputo fractional derivatives. Normally distributed random variables are defined for the parameters of the original system that are related to the mutations and infections of the nodes in the network. The resulting system of FRDEs is simulated using the predictor-corrector method based fde12 algorithm and the forward fractional Euler method (ffEm) for various values of the model components such as the standard deviations, orders of derivation, and repetition numbers. Additionally, the sensitivity analysis of the original model is investigated in relation to the random nature of the components and the basic reproduction number ( R 0 ) to underline the correspondence of random dynamics and sensitivity indices. Both the normalized forward sensitivity indices (NFSI) and the standard deviation of R 0 with random components give matching results for analyzing the changes in the spread rate. Theoretical results are backed by the simulation outputs on the numerical characteristics of the fractional-random model for the expected number of infections and mutations, expected timing of the removal of mutations from the network, and measurement of the variability in the results such as the coefficients of variation. Comparison of the results from the original model and the fractional-random model shows that the fractional-random analysis provides a more generalized perspective while facilitating a versatile investigation with ease and can be used on different models as well. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
- View/download PDF
36. Symmetry Analysis and Wave Solutions of Time Fractional Kupershmidt Equation.
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Saini, Shalu, Kumar, Rajeev, Kumar, Kamal, and Francomano, Elisa
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- *
ORDINARY differential equations , *FRACTIONAL differential equations , *NONLINEAR differential equations , *PARTIAL differential equations , *WAVE analysis - Abstract
This study employs the Lie symmetry technique to explore the symmetry features of the time fractional Kupershmidt equation. Specifically, we use the Lie symmetry technique to derive the symmetry generators for this equation, which incorporates a conformal fractional derivative. We use the symmetry generators to transform the fractional partial differential equation into a fractional ordinary differential equation, thereby simplifying the analysis. The obtained reduced equation is of fourth order nonlinear ordinary differential equation. To find the wave solutions, F/G‐expansion process has been used to obatin different types of solutions of the time‐fractional Kuperschmidt equation. The obtained wave solutions are hyperbolic and trigonometric in nature. We then use Maple software to visually depict these wave solutions for specific parameter values, providing insights into the behaviour of the system under investigation. Peak and kink wave solutions are achieved for the given problem. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
- View/download PDF
37. Fractional Parabolic Systems of Vector Order.
- Author
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Ashurov, R. and Sulaymonov, I.
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- *
PARTIAL differential equations , *FRACTIONAL differential equations , *DIFFERENTIAL operators , *MATRICES (Mathematics) , *EQUATIONS , *CAUCHY problem - Abstract
We consider the Cauchy problem for a system of partial differential equations of fractional order D t B U(t, x) + A (D)U(t, x) = H(t, x), where U and H are vector-valued functions and the m × m-matrix of differential operators A (D) is triangular with elliptic operators on the diagonal. The main feature of this system is that the vector order B has different components βj ∈ (0, 1] which are not necessarily rational. We find sufficient (and necessary in some cases) conditions on the initial function and right-hand side of the equation that guarantee the existence of the classical solution. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
- View/download PDF
38. Epidemic transmission modeling with fractional derivatives and environmental pathogens.
- Author
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Khalighi, Moein, Ndaïrou, Faïçal, and Lahti, Leo
- Subjects
- *
FRACTIONAL differential equations , *INFECTIOUS disease transmission , *DISEASE management , *COMMUNICABLE diseases , *FRACTIONAL calculus , *BASIC reproduction number - Abstract
This research presents an advanced fractional-order compartmental model designed to delve into the complexities of COVID-19 transmission dynamics, specifically accounting for the influence of environmental pathogens on disease spread. By enhancing the classical compartmental framework, our model distinctively incorporates the effects of order derivatives and environmental shedding mechanisms on the basic reproduction numbers, thus offering a holistic perspective on transmission dynamics. Leveraging fractional calculus, the model adeptly captures the memory effect associated with disease spread, providing an authentic depiction of the virus’s real-world propagation patterns. A thorough mathematical analysis confirming the existence, uniqueness and stability of the model’s solutions emphasizes its robustness. Furthermore, the numerical simulations, meticulously calibrated with real COVID-19 case data, affirm the model’s capacity to emulate observed transmission trends, demonstrating the pivotal role of environmental transmission vectors in shaping public health strategies. The study highlights the critical role of environmental sanitation and targeted interventions in controlling the pandemic’s spread, suggesting new insights for research and policy-making in infectious disease management. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
39. Inverse-Initial Problem for Time-Degenerate PDE Involving the Bi-Ordinal Hilfer Derivative.
- Author
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Karimov, E. T., Tokmagambetov, N. E., and Usmonov, D. A.
- Subjects
- *
DEGENERATE differential equations , *SEPARATION of variables , *FRACTIONAL differential equations , *PARTIAL differential equations , *INFINITE series (Mathematics) - Abstract
The authors have proved a unique solvability of an inverse-initial problem for a time-fractional degenerate partial differential equation. Using a method of separation of variables, they obtained the Cauchy problem for fractional differential equation involving the bi-ordinal Hilfer derivative in time-variable. The authors present the solution to this Cauchy problem in an explicit form via the Kilbas–Saigo function. Further, using the upper and lower bounds of the function, they prove uniform convergence of infinite series corresponding to the solution of the formulated inverse-initial problem. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
- View/download PDF
40. On a Generic Fractional Derivative Associated with the Riemann–Liouville Fractional Integral.
- Author
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Luchko, Yuri
- Subjects
- *
FRACTIONAL differential equations , *FRACTIONAL integrals , *LINEAR operators , *OPERATOR theory , *PROJECTORS - Abstract
In this paper, a generic fractional derivative is defined as a set of the linear operators left-inverse to the Riemann–Liouville fractional integral. Then, the theory of the left-invertible operators developed by Przeworska-Rolewicz is applied to deduce its properties. In particular, we characterize its domain, null-space, and projector operator; establish the interrelations between its different realizations; and present a generalized fractional Taylor formula involving the generic fractional derivative. Then, we consider the fractional relaxation equation containing the generic fractional derivative, derive a closed-form formula for its unique solution, and study its complete monotonicity. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. Non-Local Problems for the Fractional Order Diffusion Equation and the Degenerate Hyperbolic Equation.
- Author
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Ruziev, Menglibay, Parovik, Roman, Zunnunov, Rakhimjon, and Yuldasheva, Nargiza
- Subjects
- *
FRACTIONAL differential equations , *VOLTERRA equations , *BOUNDARY value problems , *HEAT equation , *GAUSSIAN function - Abstract
This research explores nonlocal problems associated with fractional diffusion equations and degenerate hyperbolic equations featuring singular coefficients in their lower-order terms. The uniqueness of the solution is established using the energy integral method, while the existence of the solution is equivalently reduced to solving Volterra integral equations of the second kind and a fractional differential equation. The study focuses on a mixed domain where the parabolic section aligns with the upper half-plane, and the hyperbolic section is bounded by two characteristics of the equation under consideration and a segment of the x-axis. By utilizing the solution representation of the fractional-order diffusion equation, a primary functional relationship is derived between the traces of the sought function on the x-axis segment from the parabolic part of the mixed domain. An explicit solution form for the modified Cauchy problem in the hyperbolic section of the mixed domain is presented. This solution, combined with the problem's boundary condition, yields a fundamental functional relationship between the traces of the unknown function, mapped to the interval of the equation's degeneration line. Through the conjugation condition of the problem, an equation with fractional derivatives is obtained by eliminating one unknown function from two functional relationships. The solution to this equation is explicitly formulated. For a specific solution of the proposed problem, visualizations are provided for various orders of the fractional derivative. The analysis demonstrates that the derivative order influences both the intensity of the diffusion (or subdiffusion) process and the shape of the wave front. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
42. Approximate Solutions of Fractional Differential Equations Using Optimal q-Homotopy Analysis Method: A Case Study of Abel Differential Equations.
- Author
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Şengül, Süleyman, Bekiryazici, Zafer, and Merdan, Mehmet
- Subjects
- *
FRACTIONAL differential equations , *CAPUTO fractional derivatives , *BERNOULLI equation , *RICCATI equation , *DIFFERENTIAL equations - Abstract
In this study, the optimal q-Homotopy Analysis Method (optimal q-HAM) has been used to investigate fractional Abel differential equations. This article is designed as a case study, where several forms of Abel equations, containing Bernoulli and Riccati equations, are given with ordinary derivatives and fractional derivatives in the Caputo sense to present the application of the method. The optimal q-HAM is an improved version of the Homotopy Analysis Method (HAM) and its modification q-HAM and focuses on finding the optimal value of the convergence parameters for a better approximation. Numerical applications are given where optimal values of the convergence control parameters are found. Additionally, the correspondence of the approximate solutions obtained for these optimal values and the exact or numerical solutions are shown with figures and tables. The results show that the optimal q-HAM improves the convergence of the approximate solutions obtained with the q-HAM. Approximate solutions obtained with the fractional Differential Transform Method, q-HAM and predictor–corrector method are also used to highlight the superiority of the optimal q-HAM. Analysis of the results from various methods points out that optimal q-HAM is a strong tool for the analysis of the approximate analytical solution in Abel-type differential equations. This approach can be used to analyze other fractional differential equations arising in mathematical investigations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
43. Neural Fractional Differential Equations: Optimising the Order of the Fractional Derivative.
- Author
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Coelho, Cecília, Costa, M. Fernanda P., and Ferrás, Luís L.
- Subjects
- *
FRACTIONAL differential equations , *INITIAL value problems , *NUMERICAL analysis , *ARCHITECTURAL design - Abstract
Neural Fractional Differential Equations (Neural FDEs) represent a neural network architecture specifically designed to fit the solution of a fractional differential equation to given data. This architecture combines an analytical component, represented by a fractional derivative, with a neural network component, forming an initial value problem. During the learning process, both the order of the derivative and the parameters of the neural network must be optimised. In this work, we investigate the non-uniqueness of the optimal order of the derivative and its interaction with the neural network component. Based on our findings, we perform a numerical analysis to examine how different initialisations and values of the order of the derivative (in the optimisation process) impact its final optimal value. Results show that the neural network on the right-hand side of the Neural FDE struggles to adjust its parameters to fit the FDE to the data dynamics for any given order of the fractional derivative. Consequently, Neural FDEs do not require a unique α value; instead, they can use a wide range of α values to fit data. This flexibility is beneficial when fitting to given data is required and the underlying physics is not known. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
44. On the Numerical Investigations of a Fractional-Order Mathematical Model for Middle East Respiratory Syndrome Outbreak.
- Author
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Abd Alaal, Faisal E., Hadhoud, Adel R., Abdelaziz, Ayman A., and Radwan, Taha
- Subjects
- *
MIDDLE East respiratory syndrome , *NONLINEAR differential equations , *FRACTIONAL differential equations , *PUBLIC health , *CORONAVIRUSES - Abstract
Middle East Respiratory Syndrome (MERS) is a human coronavirus subtype that poses a significant public health concern due to its ability to spread between individuals. This research aims to develop a fractional-order mathematical model to investigate the MERS pandemic and to subsequently develop two numerical methods to solve this model numerically to evaluate and comprehend the analysis results. The fixed-point theorem has been used to demonstrate the existence and uniqueness of the solution to the suggested model. We approximate the solutions of the proposed model using two numerical methods: the mean value theorem and the implicit trapezoidal method. The stability of these numerical methods is studied using various results and primary lemmas. Finally, we compare the results of our methods to demonstrate their efficiency and conduct a numerical simulation of the obtained results. A comparative study based on real data from Riyadh, Saudi Arabia is provided. The study's conclusions demonstrate the computational efficiency of our approaches in studying nonlinear fractional differential equations that arise in daily life problems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
45. Interaction Solutions for the Fractional KdVSKR Equations in (1+1)-Dimension and (2+1)-Dimension.
- Author
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Zhang, Lihua, Zheng, Zitong, Shen, Bo, Wang, Gangwei, and Wang, Zhenli
- Subjects
- *
FINITE groups , *SYMMETRY groups , *FRACTIONAL differential equations , *SOCIAL interaction , *NONLINEAR systems - Abstract
We extend two KdVSKR models to fractional KdVSKR models with the Caputo derivative. The KdVSKR equation in (2+1)-dimension, which is a recent extension of the KdVSKR equation in (1+1)-dimension, can model the soliton resonances in shallow water. Applying the Hirota bilinear method, finite symmetry group method, and consistent Riccati expansion method, many new interaction solutions have been derived. Soliton and elliptical function interplaying solution for the fractional KdVSKR model in (1+1)-dimension has been derived for the first time. For the fractional KdVSKR model in (2+1)-dimension, two-wave interaction solutions and three-wave interaction solutions, including dark-soliton-sine interaction solution, bright-soliton-elliptic interaction solution, and lump-hyperbolic-sine interaction solution, have been derived. The effect of the order γ on the dynamical behaviors of the solutions has been illustrated by figures. The three-wave interaction solution has not been studied in the current references. The novelty of this paper is that the finite symmetry group method is adopted to construct interaction solutions of fractional nonlinear systems. This research idea can be applied to other fractional differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
46. Fractional Differential Equations with Impulsive Effects.
- Author
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Fečkan, Michal, Danca, Marius-F., and Chen, Guanrong
- Subjects
- *
FRACTIONAL differential equations , *IMPULSIVE differential equations , *CAPUTO fractional derivatives , *DYNAMICAL systems - Abstract
This paper discusses impulsive effects on fractional differential equations. Two approaches are taken to obtain our results: either with fixed or changing lower limits in Caputo fractional derivatives. First, we derive an existence result for periodic solutions of fractional differential equations with periodically changing lower limits. Then, the impulsive effects are modeled for fractional differential equations regarding the nonlinearities rather than the initial value conditions. The proposed impulsive model differs from common discontinuous and nonsmooth dynamical systems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
47. Approximate Controllability of Hilfer Fractional Stochastic Evolution Inclusions of Order 1 < q < 2.
- Author
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Shukla, Anurag, Panda, Sumati Kumari, Vijayakumar, Velusamy, Kumar, Kamalendra, and Thilagavathi, Kothandabani
- Subjects
- *
DIFFERENTIAL inclusions , *FRACTIONAL differential equations , *STOCHASTIC differential equations , *FRACTIONAL calculus , *SET-valued maps - Abstract
This paper addresses the approximate controllability results for Hilfer fractional stochastic differential inclusions of order 1 < q < 2 . Stochastic analysis, cosine families, fixed point theory, and fractional calculus provide the foundation of the main results. First, we explored the prospects of finding mild solutions for the Hilfer fractional stochastic differential equation. Subsequently, we determined that the specified system is approximately controllable. Finally, an example displays the theoretical application of the results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
48. Conformable Double Laplace Transform Method (CDLTM) and Homotopy Perturbation Method (HPM) for Solving Conformable Fractional Partial Differential Equations.
- Author
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GadAllah, Musa Rahamh and Gadain, Hassan Eltayeb
- Subjects
- *
PARTIAL differential equations , *FRACTIONAL differential equations , *DECOMPOSITION method - Abstract
In the present article, the method which was obtained from a combination of the conformable fractional double Laplace transform method (CFDLTM) and the homotopy perturbation method (HPM) was successfully applied to solve linear and nonlinear conformable fractional partial differential equations (CFPDEs). We included three examples to help our presented technique. Moreover, the results show that the proposed method is efficient, dependable, and easy to use for certain problems in PDEs compared with existing methods. The solution graphs show close contact between the exact and CFDLTM solutions. The outcome obtained by the conformable fractional double Laplace transform method is symmetrical to the gain using the double Laplace transform. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
49. Lattice Boltzmann Simulation of Spatial Fractional Convection–Diffusion Equation.
- Author
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Bi, Xiaohua and Wang, Huimin
- Subjects
- *
LATTICE Boltzmann methods , *FRACTIONAL differential equations , *PARTIAL differential equations , *ADVECTION-diffusion equations , *COMPUTER simulation , *EQUATIONS , *TRANSPORT equation - Abstract
The space fractional advection–diffusion equation is a crucial type of fractional partial differential equation, widely used for its ability to more accurately describe natural phenomena. Due to the complexity of analytical approaches, this paper focuses on its numerical investigation. A lattice Boltzmann model for the spatial fractional convection–diffusion equation is developed, and an error analysis is carried out. The spatial fractional convection–diffusion equation is solved for several examples. The validity of the model is confirmed by comparing its numerical solutions with those obtained from other methods The results demonstrate that the lattice Boltzmann method is an effective tool for solving the space fractional convection–diffusion equation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
50. Euler–Maruyama methods for Caputo tempered fractional stochastic differential equations.
- Author
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Huang, Jianfei, Shao, Linxin, and Liu, Jiahui
- Subjects
- *
STOCHASTIC differential equations , *FRACTIONAL differential equations , *INITIAL value problems , *PROBLEM solving - Abstract
In this paper, we introduce the initial value problem of Caputo tempered fractional stochastic differential equations and then study the well-posedness of its solution. Further, a Euler–Maruyama (EM) method is derived for solving the considered problem. The strong convergence order of the derived EM method is proved to be $ \alpha -\frac {1}{2} $ α − 1 2 with $ \frac {1}{2} \lt \alpha \lt 1 $ 1 2 < α < 1. Additionally, a fast EM method is also developed which is based on the sum-of-exponentials approximation. Finally, numerical experiments are given to support the theoretical findings of the above two methods and verify computational efficiency of the fast EM method. The fast EM method can greatly improve the computational performance of the original EM method. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
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