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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]

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]

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]

This article's goal is to examine the existence and uniqueness of the solution for a novel class of $-Hilfer fractional differential Langevin equations. The suggested work depends on several basic definitions of topological degree theory and fractional calculus. Employing topological degree method, we were able to obtain the existence result, and we establish the uniqueness result using Banach's fixed-point theorem. We provide an example to illustrate our theoretical finding. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

This study explores the application of advanced mathematical techniques to solve fractional differential equations, focusing particularly on the fractional diffusion equation. The fractional diffusion equation, used to simulate a range of physical and engineering phenomena, poses considerable difficulties when applied to fractional orders. Thus, by utilizing the mighty powers of fractional calculus, we employ the variational iteration method (VIM) with the Elzaki transform to produce highly accurate approximations for these specific differential equations. The VIM provides an iterative framework for refining solutions progressively, while the Elzaki transform simplifies the complex integral transforms involved. By integrating these methodologies, we achieve accurate and efficient solutions to the fractional diffusion equation. Our findings demonstrate the robustness and effectiveness of combining the VIM and the Elzaki transform in handling fractional differential equations, offering explicit functional expressions that are beneficial for theoretical analysis and practical applications. This research contributes to the expanding field of fractional calculus, providing valuable insights and useful tools for solving complex, nonlinear fractional differential equations across various scientific and engineering disciplines. [ABSTRACT FROM AUTHOR]

In this study, we investigate the stability and asymptotic stability properties of Caputo fractional time-dependent systems with delay by employing vector Lyapunov functions. Utilizing the Caputo fractional Dini derivative on Lyapunov-like functions, along with a new comparison theorem and differential inequalities, we derive and prove sufficient conditions for the stability and asymptotic stability of these complex systems. An example is included to showcase the method's practicality and to specifically illustrate its advantages over scalar Lyapunov functions. Our results improves, extends, and generalizes several existing findings in the literature. [ABSTRACT FROM AUTHOR]

This paper presents a novel spatio-temporal meshless method (STMM) for solving the time fractional partial differential equations (TFPDEs) with variable coefficients based on the space-time metric. The main idea of the STMM is to directly approximate the solutions of fractional PDEs by using a multiquadric function with the space-time distance within a space-time scale framework. Compared with the existing methods, the present meshless STMM entirely avoids the difference approximation of fractional temporal derivatives and can be easily applied to complicated irregular geometries. Furthermore, both regular and irregular nodal distribution can be used without loss of accuracy. For these reasons, this new space-time meshless method could be regarded as a competitive alternative to the conventional numerical algorithms based on difference decomposition for solving the TFPDEs with variable coefficients. Numerical experiments confirm the ability and accuracy of the proposed methodology. [ABSTRACT FROM AUTHOR]

This research introduced a new method, the Aboodh Tamimi Ansari transform method ( (A T) 2 method), for solving systems of linear and nonlinear fractional partial differential equations. The method combined the Aboodh transform method and the Tamimi Ansari method, allowing for the simultaneous solution of linear and nonlinear terms without restrictions. The Caputo sense was considered for fractional derivatives. The effectiveness of the proposed method was demonstrated through numerical solutions, graphical representations, and tabular data, showing strong agreement with exact solutions. The approach was deemed precise, easy to apply, and could be extended to address further challenges in fractional-order problems. Computational tasks were carried out using Mathematica 13. [ABSTRACT FROM AUTHOR]

This paper investigated the application of analytical methods, specifically the Mohand transform iterative method (MTIM) and the Mohand residual power series method (MRPSM), to solve the fractional Boussinesq equation. Utilizing the Caputo operator to manage fractional derivatives, these semi-analytical approaches provide accurate solutions to complex fractional differential equations. Through convergence analysis and error estimation, the study validated the efficacy of these methods by comparing numerical solutions to known exact solutions. Graphical and tabular representations illustrated the accuracy of the proposed methods, highlighting their performance for varying fractional orders. The findings demonstrated that both MTIM and MRPSM offer reliable, efficient solutions, making them valuable tools for addressing fractional differential systems in fields such as applied mathematics, engineering, and physics. [ABSTRACT FROM AUTHOR]

In this paper, we introduced the gradient-enhanced fractional physics-informed neural networks (gfPINNs) for solving the forward and inverse problems of the multiterm time-fractional Burger-type equation. The gfPINNs leverage gradient information derived from the residual of the fractional partial differential equation and embed the gradient into the loss function. Since the standard chain rule in integer calculus is invalid in fractional calculus, the automatic differentiation of neural networks does not apply to fractional operators. The automatic differentiation for the integer order operators and the finite difference discretization for the fractional operators were used to construct the residual in the loss function. The numerical results demonstrate the effectiveness of gfPINNs in solving the multiterm time-fractional Burger-type equation. By comparing the experimental results of fractional physics-informed neural networks (fPINNs) and gfPINNs, it can be seen that the training performance of gfPINNs is better than fPINNs. [ABSTRACT FROM AUTHOR]

This paper explores a fractional integro-differential equation with boundary conditions that incorporate the Hilfer-Hadamard fractional derivative. We model the RLC circuit using fractional calculus and define weighted spaces of continuous functions. The existence and uniqueness of solutions are established, along with their Ulam-Hyers and Ulam-Hyers-Rassias stability. Our analysis employs Schaefer's fixed-point theorem and Banach's contraction principle. An illustrative example is presented to validate our findings. [ABSTRACT FROM AUTHOR]

This paper aims to study the existence of solutions for Hadamard fractional nonlocal boundary value problems with mean curvature operator at resonance. Based on the coincidence degree theory, some new results are established. Moreover, an example is given to verify our main results. [ABSTRACT FROM AUTHOR]

This work provided a detailed theoretical analysis of fractional ordinary differential equations with Caputo and the Atangana-Baleanu fractional derivative. The work started with an extension of Tychonoff's fixed point and the Perron principle to prove the global existence with extra conditions due to the properties of the fractional derivatives used. Then, a detailed analysis of the existence of maximal and minimal solutions was presented for both cases. Then, using Chaplygin's approach with extra conditions, we also established the existence and uniqueness of the solutions of these equations. The Abel and the Bernoulli equations were considered as illustrative examples and were solved using the fractional middle point method. [ABSTRACT FROM AUTHOR]

In this study, we proved existence results for nonlinear implicit fractional differential equations with the Caputo version of the Atangana-Baleanu derivative, subject to the boundary and nonlocal initial conditions. The Kuratowski's measure of non-compactness and its associated fixed point theorems–Darbo's fixed point theorem and Mönchh's fixed point theorem, are the foundation for the analysis in this paper. We support our results with examples of nonlinear implicit fractional differential equations involving the Caputo version of the Atangana-Baleanu derivative subject to both boundary and nonlocal initial conditions. In addition, we provide solutions to the problems we considered. [ABSTRACT FROM AUTHOR]

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]

In the present paper, we prove the existence and uniqueness of solutions to impulsive coupled system of nonlinear hybrid fractional differential equations involving Caputo fractional derivative of order α ∊ (0; 1) with linear and nonlinear perturbations. We prove our main results by applying the nonlinear alternative of Leray-Schauder type and Banach's fixed-point theorem. As application, On example is included to show the applicability of our results. [ABSTRACT FROM AUTHOR]

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]

The fractional Lane–Emden model illustrates different phenomena in astrophysics and mathematical physics. This pafa involves the Vieta–Lucas (Vt-L) bases to solve types of variable-order (V-O) fractional Lane–Emden equation (linear and nonlinear). The ofaational matrix of the V-O fractional derivative is obtained for the Vt-L polynomials. In the established approach, these polynomials are applied to transform the main problem into an algebraic equations system. To indicate the faformance and capability of the scheme, a number of examples are presented for various types of V-O fractional Lane–Emden equations. Also, for one example, a comparison is done between the calculated results by our technique and those obtained via the Bernoulli polynomials. Overall, this pafa introduces a new methodology for solving V-O fractional Lane–Emden equations using Vt-L bases. The derived ofaational matrix and the transformation to an algebraic equation system offer practical advantages in solving these equations efficiently. The presented examples and comparative analysis highlight the effectiveness and validity of the proposed technique, contributing to the understanding and advancement of fractional Lane–Emden models in astrophysics and mathematical physics. [ABSTRACT FROM AUTHOR]

The present article introduces an ofaational approach based on modified hat functions to solve the space-time-fractional differential equations in the Caputo sense. In this method, the derivative of the unknown function is considered as a linear combination of modified hat functions. We use the ofaational matrix of the Riemann–Liouville fractional integral of modified hat functions to approximate the Caputo fractional derivative in order to reduce the problem to a system of Sylvester equations. The error of the mentioned method is of the order O(h3). In addition, we examine several  numerical examples to confirm the ability of the proposed approach. [ABSTRACT FROM AUTHOR]

A class of linear and nonlinear fractional differential equations (FDEs) in the Caputo sense is considered and studied through two novel techniques called the Homotopy analysis method (HAM). A reliable approach is proposed for solving fractional order nonlinear ordinary differential equations, and the Haar wavelet technique (HWT) is a numerical approach for both integer and noninteger orders. Perturbation techniques are widely applied to gain analytic approximations of nonlinear equations. However, faturbation methods are essentially based on small physical parameters (called faturbation quantity), but unfortunately, many nonlinear problems have no such kind of small physical parameters at all. HAM overcomes this, and HWT does not require any parameters. Due to this, we opt for HAM and HWT to study FDEs. We have drawn a semi-analytical solution in terms of a series of polynomials and numerical solutions for FDEs. First, we solve the models by HAM by choosing the preferred control parameter. Second, HWT is considered. Through this technique, the ofaational matrix of integration is used to convert the given FDEs into a set of algebraic equation systems. Four problems are discussed using both techniques. Obtained results are expressed in graphs and tables. Results on convergence have been discussed in terms of theorems. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

The aim of this manuscript is to showcase novel distance mappings, specifically gamma distance mappings. Through the utilization of these mappings, we can effectively introduce substantial contractions on a mapping f : B → B. In order to demonstrate the importance and applicability of our work, a significant example and application are presented. [ABSTRACT FROM AUTHOR]

We introduce FdeSolver, an open-source Julia package designed to solve fractional-order differential equations efficiently. The available solutions are based on product-integration rules, predictor–corrector algorithms, and the Newton-Raphson method. The package covers solutions for one-dimensional equations with orders of positive real numbers. For higher-dimensional systems, it supports orders up to one. Incommensurate derivatives are allowed and defined in the Caputo sense. Here, we summarize the implementation for a representative class of problems and compare it with available alternatives in Julia and MATLAB. Moreover, FdeSolver leverages the power and flexibility of the Julia environment to offer enhanced computational performance, and our development emphasizes adherence to the best practices of open research software. To highlight its practical utility, we demonstrate its capability in simulating microbial community dynamics and modeling the spread of COVID-19. This latter application involves fitting the order of derivatives grounded on real-world epidemiological data. Overall, these results highlight the efficiency, reliability, and practicality of the FdeSolver Julia package. [ABSTRACT FROM AUTHOR]

This paper aims to investigate nonlinear implicit differential equation(DEs) using the fractional derivative(FD) of Hilfer-Katugampola(H-K). The presence of the implicit equation's solution is established through the application of the Gronwall inequality and the Banach fixed point theorem, leading to pertinent findings regarding Ulam Hyers stability(UHS). Lastly, an example confirms that the findings are valid. [ABSTRACT FROM AUTHOR]

This paper aims to explore the richer dynamics of the Lazer-McKenna suspension bridge model, which is a asymmetric system of equations that has previously shown chaotic dynamics, here the study will be made by means of fractional order differential equations where the order of the entire system is varied. For the solutions it was used integration schemes previously shown in the literature, which were implemented in the Python programming language, the dynamics were then analyzed in the time domain by means of phase-space, Poincard sections, and bifurcation diagrams, and in the frequency domain by Discrete Fourier Transform (DFT), Continuous Wavelet Transform (CWT) and Hilbert-Huang Transform (HHT). The investigated responses demonstrated a great influence of the fractional order in the system, changing its time dynamics from chaotic to periodic. [ABSTRACT FROM AUTHOR]

This study presents a comprehensive analysis of the dynamics of Mpox viral transmission using a compartmental mathematical model. The model incorporates the impact of immunization, isolation, and hospitalization on disease management, as well as the interaction between humans and rodents. Through numerical simulations, the study highlights the effectiveness of isolation in mitigating disease transmission and emphasizes the significance of mathematical modeling and simulation techniques in understanding disease dynamics. The utilization of Caputo's fractional differential equation in the human dynamical model is shown to be effective in regulating disease in all compartments. Sensitivity analysis is conducted to identify the most influential parameters in virus transmission. The findings contribute valuable insights for public health strategies and provide a foundation for further research in disease control and management. [ABSTRACT FROM AUTHOR]

Different types of fractional calculus have been defined, grouped into categories based on their properties. Two types particularly studied are Hadamard-type fractional calculus and tempered fractional calculus. This paper establishes a connection between these two types by expressing one in terms of the other, using the theory of fuzzy fractional calculus with respect to functions. By naturally extending this connection, a generalization is developed that unifies several existing fractional operators, including Riemann–Liouville, Caputo, ψ -Caputo, classical Hadamard, Hadamard-type, tempered, and all these operators considered with respect to functions via the Hadamard ψ -Caputo tempered fractional derivative (CTFD) using the Mittag-Leffler function. In addition, we explore the Cauchy problem for fuzzy fractional differential equations (FDEs) with this kind of derivative and demonstrate various results of existence and uniqueness using fixed-point theorems of Krasnoselskii, Schaüder's, and Banach. Applications further serve as proof of the utility of our findings, which converge, expand, and generalize several previously reported results in research. [ABSTRACT FROM AUTHOR]

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]