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The work reported in this paper deals with the study of a coupled system for fractional terminal value problems involving ψ-Hilfer fractional derivative. The existence and uniqueness theorems to the problem at hand are investigated. Besides, the stability analysis in the Ulam–Hyers sense of a given system is studied. Our discussion is based upon known fixed point theorems of Banach and Krasnoselskii. Examples are also provided to demonstrate the applicability of our results. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of positive solutions for a class of a fractional differential equation system of Riemann–Liouville type on infinite intervals with infinite-point boundary conditions. First, the higher-order equation is reduced to the lower-order equation, and then it is transformed into the equivalent integral equation. Secondly, we obtain the existence and uniqueness of positive solutions for each fixed parameter λ > 0 by using the mixed monotone operators fixed-point theorem. The results obtained in this paper show that the unique positive solution has good properties: continuity, monotonicity, iteration, and approximation. Finally, an example is given to demonstrate the application of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we study fractional p 1 (x , ⋅) & p 2 (x , ⋅) -Laplacian Schrödinger-type equations for Robin boundary conditions. Under some suitable assumptions, we show that two solutions exist using the mountain pass lemma and Ekeland's variational principle. Then, the existence of infinitely many solutions is derived by applying the fountain theorem and the Krasnoselskii genus theory, respectively. Different from previous results, the topic of this paper is the Robin boundary conditions in R N ∖ Ω ‾ for fractional order p 1 (x , ⋅) & p 2 (x , ⋅) -Laplacian Schrödinger-type equations, including concave-convex nonlinearities, which has not been studied before. In addition, two examples are given to illustrate our results. [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]

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]

In this paper, we presented a new analytical method for one of the rapidly emerging branches of fractional calculus, the distributed order fractional differential equations (DFDE). Due to its significant applications in modeling complex physical systems, researchers have shown profound interest in developing various analytical and numerical methods to study DFDEs. With this motivation, we proposed an easy computational technique with the help of graph theoretic polynomials from algebraic graph theory for nonlinear distributed order fractional ordinary differential equations (NDFODE). In the method, we used clique polynomials of the cocktail party graph as an approximation solution. With operational integration and fractional differentiation in the Caputo sense, the NDFODEs transformed into a system of algebraic equations and then solved by Newton–Raphson's method to determine the unknowns in the Clique polynomial approximation. The proficiency of the proposed Clique polynomial collocation method (CCM) is illustrated with four numerical examples. The convergence and error analysis are discussed in tabular and graphical depictions by comparing the CCM results with the results of existing numerical methods. [ABSTRACT FROM AUTHOR]

We study in this paper the monotonicity properties of the numerical solutions to Volterra integral equations with nonincreasing completely positive kernels on nonuniform meshes. There is a duality between the complete positivity and the properties of the complementary kernel being nonnegative and nonincreasing. Based on this, we propose the “complementary monotonicity” to describe the nonincreasing completely positive kernels, and the “right complementary monotone” (R-CMM) kernels as the analogue for nonuniform meshes. We then establish the monotonicity properties of the numerical solutions inherited from the continuous equation if the discretization has the R-CMM property. Such a property seems weaker than log-convexity and there is no restriction on the step size ratio of the discretization for the R-CMM property to hold. [ABSTRACT FROM AUTHOR]

In the present paper, the regularity and finite difference methods for Caputo-Hadamard fractional differential equations with initial value singularity are taken into consideration. To overcome the weak singularity and enhance convergence precision, a fitted scheme on nonuniform meshes is applied to such problems. Firstly, based on L log , 2 - 1 σ approximation, the temporal convergence accuracy of fitted scheme for the sub-diffusion equations is O (N - min { 2 r α , 2 }) , where N denotes the number of time steps, α is the fractional order and r is the mesh grading parameter. It is indicated that the performance of the fitted scheme is better than that of the standard L log , 2 - 1 σ scheme on (i) exponent meshes (i.e., r = 1 ) and (ii) graded meshes with the optimal choice of the mesh grading. Secondly, a second-order fitted scheme on exponent meshes for the diffusion-wave equations is obtained. Furthermore, for the sake of improving the computational efficiency and demonstrating the effectiveness of the decomposition of the solution, the fast algorithm and further decomposition of the solution for the sub-diffusion equations are investigated. Ultimately, some examples are presented to verify the availability of our theoretical results. [ABSTRACT FROM AUTHOR]

This paper aims to study the existence and stability results concerning a fractional partial differential equation with variable exponent source functions. The local existence result for α ∈ (0 , 1) is established with the help of the α -resolvent kernel and the Schauder-fixed point theorem. The non-continuation theorem is proved by the fixed point technique and accordingly the global existence of solution is achieved. The uniqueness of the solution is obtained using the contraction principle and the stability results are discussed by means of Ulam-Hyers and generalized Ulam-Hyers-Rassias stability concepts via the Picard operator. Examples are provided to illustrate the results. [ABSTRACT FROM AUTHOR]

In his first paper on the generalization of the tautochrone problem, that was published in 1823, Niels Henrik Abel presented a complete framework for fractional-order calculus, and used the clear and appropriate notation for fractional-order integration and differentiation. [ABSTRACT FROM AUTHOR]

In this paper, we study a new class of coupled systems of nonlinear generalized fractional differential equations complemented with coupled nonlocal Riemann–Stieltjes and generalized fractional integral boundary conditions. The nonlinearities also include the lower order generalized fractional derivatives of the unknown functions. We apply the Banach contraction mapping principle and Leray–Schauder alternative to derive the desired results. An illustrative example is also discussed. The results presented in this work are novel in the given configuration and yield some new results as special cases (for details, see the Conclusion section). [ABSTRACT FROM AUTHOR]

In this paper, the coupled system of sequential ψ-Hilfer fractional boundary value problems with non-instantaneous impulses is investigated. The existence results of the system are proved by means of topological degree theory. An example is constructed to demonstrate our results. Additionally, a graphical analysis is performed to verify our results. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying a class of fuzzy fractional switched implicit differential equations (FFSIDEs) with non-instantaneous impulses that there are few papers considering this issue. Considering switching law and the memory property of fractional calculus, we first present a formula of solution for FFSIDEs with non-instantaneous impulses. Subsequently, based on a sequence of Picard functions, we explore the existence of solutions for the addressed equations by successive approximation. Furthermore, Ulam–Hyers (U–H) stability for this considered equations is derived. The main results are obtained using fuzzy-valued fractional calculus and nonlinear analysis. Finally, two numerical examples illustrating the theoretical result are given. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study a generalized Riemann–Liouville fractional differential equation and system with nonlocal boundary conditions. Firstly, some properties of the Green function are presented and then Lyapunov-type inequalities for a sequential ψ-Riemann–Liouville fractional boundary value problem are established. Also, the existence and uniqueness of solutions are proved by using Banach and Schauder fixed-point theorems. Furthermore, the existence and uniqueness of solutions to a sequential nonlinear differential system is established by means of Schauder's and Perov's fixed-point theorems. Examples are given to validate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper focuses on exploring the existence of solutions for a specific class of FDEs by leveraging fixed point theorem. The equation in question features the Caputo fractional derivative of order 3 < u ˆ ≤ 4 and includes a term Θ (β , Z (β)) alongside boundary conditions. Through the application of a fixed point theorem in appropriate function spaces, we consider nonlocal conditions along with necessary assumptions under which solutions to the given FDE exist. Furthermore, we offer an example to illustrate the results. [ABSTRACT FROM AUTHOR]

The notion of A -condensing operators via the measure of noncompactness is proposed, which retains the existing classes of condensing operators. Results concerning the existence of the best proximity point (pair) of cyclic (noncyclic) A -condensing operators along with the coupled best proximity-point theorem for cyclic A -condensing operators have been formulated. An application to a (k , ℷ) -Hilfer fractional differential equation of order 2 < p < 3 , type q ∈ [ 0 , 1 ] satistfying some boundary conditions is presented. The paper is the first to investigate the optimum solution of such a generalized fractional differential equation. The hypothesis involved in the investigation is independent of the incorporated measure of noncompactness, thereby making our result better in application than that present in the literature. Moreover, added numerical examples validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we apply the fractional homotopy perturbation transform method (FHPTM) to deliver an effective semi-analytical technique for determining fractional-order Kuramoto–Sivashinsky equations. The project technique combines the Laplace transform with the Caputo–Fabrizio fractional derivative of order α where α ∈ (0 , 1 ] . Fractional-order Kuramoto–Sivashinsky equation is indeed important in the field of nonlinear physics and mathematics. It is a fractional partial differential equation that describes the behaviour of waves in certain dissipative media, such as flames and chemicals. The FHPTM is described to be fast and accurate. Illustrative examples are included to demonstrate the efficiency and reliability of the presented techniques. [ABSTRACT FROM AUTHOR]

In this paper we consider the numerical solution of fractional differential equations. In particular, we study a step-by-step procedure, defined over a graded mesh, which is based on a truncated expansion of the vector field along the orthonormal Jacobi polynomial basis. Under mild hypotheses, the proposed procedure is capable of getting spectral accuracy. A few numerical examples are reported to confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

The paper investigates an inverse problem for finding the order of the fractional derivative in the sense of Gerasimov–Caputo in the wave equation with an arbitrary positive self-adjoint operator having a discrete spectrum. By means of the classical Fourier method, it is proved that the value of the projection of the solution onto some eigenfunction at a fixed time uniquely restores the order of the derivative. Several examples of the operator are discussed, including a linear system of fractional differential equations, fractional Sturm–Liouville operators, and many others. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a spectral collocation method for the numerical solutions of fractional multipantograph delay differential equations. The fractional derivatives are described in the Caputo sense. We present that some suitable variable transformations can convert the equations to a Volterra integral equation defined on the standard interval [−1, 1]. Then the Jacobi–Gauss points are used as collocation nodes, and the Jacobi–Gauss quadrature formula is used to approximate the integral equation. Later, the convergence analysis of the proposed method is investigated in the infinity norm and weighted L2 norm. To perform the numerical simulations, some test examples are investigated, and numerical results are presented. Further, we provide the comparative study of the proposed method with some existing numerical methods. [ABSTRACT FROM AUTHOR]

This paper primarily investigates uncertain barrier swaption pricing problem based on the fractional differential equation in Caputo sense and analyzes the corresponding efficiency index (validity index and survival index). To a certain extent, barrier swaption can control the gains or losses of swaption investors within a certain range. The existing barrier swaption pricing model cannot fully reflect the hereditability and memorability of the real financial market, so this paper aims to solve such difficulties and further measure the exercise ability of the barrier swaption pricing model. Firstly, the floating interest rate is regarded as an uncertain process because there exists difficult to obtain historical data for real financial model. Then the Caputo type fractional differential operator is introduced into the original barrier swaption pricing model, and a new uncertain barrier swaption model of floating interest rate is established. Secondly, based on the first hitting time, the pricing formulas and the corresponding efficiency index of four kinds of barrier swaptions under the floating rate model are derived, respectively. Finally, the rationality of the model is verified by numerical examples and corresponding methods, and gives the monotonicity of four numerical examples. [ABSTRACT FROM AUTHOR]

A new lumped capacitance model that employs fractional order operators is proposed for use on transient heat conduction problems. Details and implications of the fractional lumped capacitance model's development and application are discussed. The model is shown to agree with observed heating and cooling temperature profiles of laser aiming paper being heated by a laser under various conditions. [ABSTRACT FROM AUTHOR]

This paper deals with some existence and uniqueness results for a class of deformable fractional differential equations. These problems encompassed nonlinear implicit fractional differential equations involving boundary conditions and various types of delays, including finite, infinite, and state-dependent delays. Our approach to proving the existence and uniqueness of solutions relied on the application of the Banach contraction principle and Schauder's fixed-point theorem. In the last section, we provide different examples to illustrate our obtained results. [ABSTRACT FROM AUTHOR]

This paper solves fractional differential equations using the Shehu transform in combination with the q-homotopy analysis transform method (q-HATM). As the Shehu transform is only applicable to linear equations, q-HATM is an efficient technique for approximating solutions to nonlinear differential equations. In nonlinear systems that explain the emergence of stripes in 2D systems, the Newell–Whitehead–Segel equation plays a significant role. The findings indicate that the outcomes derived from the tables yield superior results compared to the existing LTDM in the literature. Maple is utilized to depict three-dimensional surfaces and find numerical values that are displayed in a table. [ABSTRACT FROM AUTHOR]

Over the years, the research of backward stochastic differential equations (BSDEs) has come a long way. As a extension of the BSDEs, the BSDEs with time delay have played a major role in the stochastic optimal control, financial risk, insurance management, pricing, and hedging. In this paper, we study a class of BSDEs with time-delay generators driven by Caputo fractional derivatives. In contrast to conventional BSDEs, in this class of equations, the generator is also affected by the past values of solutions. Under the Lipschitz condition and some new assumptions, we present a theorem on the existence and uniqueness of solutions. [ABSTRACT FROM AUTHOR]

In this paper we provide a way to handle some symmetric fractional boundary-value problems. Indeed, first, we consider some system of fractional equations. We introduce the existence and uniqueness of solutions of the systems of equations and we show that they are entire functions of the spectral parameter. In particular, we show that the solutions are at most of order 1/2. Moreover we share the integration by parts rule for vector-valued functions that enables us to obtain some symmetric equations. These symmetries allow us to handle 2 - sequential and 4 - sequential fractional boundary-value problems. We provide some expansion formulas for the bilinear forms of the solutions of 2 - sequential and 4 - sequential fractional equations which admit us to impose some unusual boundary conditions for the solutions of fractional differential equations. We show that the systems of eigenfunctions of 2 - sequential and 4 - sequential fractional boundary value problems are complete in both energy and mean. Furthermore, we study on the zeros of solutions of 2 - sequential fractional differential equations. At the end of the paper we show that 6 - sequential fractional differential equation can also be handled as a system of equations and hence almost all the results obtained in the paper can be carried for such boundary-value problems. [ABSTRACT FROM AUTHOR]

In this paper, we apply two methods for solving nonlinear system of fractional differential equations (FDEs); these two methods are Picard and Adomian decomposition methods (ADM). The type of fractional derivative in this system will be the Atangana–Baleanu derivative. The existence and uniqueness of the solution will be proved. In addition, the convergence of ADM series solution and the maximum expected error will be discussed. Some numerical examples will be solved by using these two method and a comparison between their solutions will be done. There exist an important application to these types of systems, this application is the fractional-order rabies model and it will be solved here. From the obtained results, it is noticed that the obtained results from using these two methods are coincide with each other, and also these results are coincide with the obtained results from the classical fractional derivatives such as Caputo sense. [ABSTRACT FROM AUTHOR]

The study of the complex model associated with chaotic models is always the most complicated and fundamental in the current scientific environment. The primary goal of the current paper is to provide an illustration of the fundamental theory while analysing dynamical systems and validating the chaotic behaviour of the Lorenz–Haken (LH) equations, a system of fractional differential equations. The LH equations are used to describe the 4D chaotic laser. The Adams–Bashforth numerical method is used to extract the numerical solutions projected model. The classical model introduces the bifurcation of the parameter linked with the system. The system's uniqueness and existence are confirmed using the fixed-point hypothesis with the Caputo–Fabrizio fractional operator, followed by boundedness and dynamical analysis. Furthermore, the chaotic character of the numerical solutions with different orders obtained at different beginning circumstances is presented. [ABSTRACT FROM AUTHOR]

We prove existence of solutions, and particularly positive solutions, of initial value problems (IVPs) for nonlinear fractional differential equations involving the Caputo differential operator of order α ∈ (0 , 1) . One novelty in this paper is that it is not assumed that f is continuous but that it satisfies an L p -Carathéodory condition for some p > 1 α (detailed definitions are given in the paper). We prove existence on an interval [0, T ] in cases where T can be arbitrarily large, called global solutions. The necessary a priori bounds are found using a new version of the Bihari inequality that we prove here. We show that global solutions exist when f (t , u) grows at most linearly in u , and also in some cases when the growth is faster than linear. We give examples of the new results for some fractional differential equations with nonlinearities related to some that occur in combustion theory. We also discuss in detail the often used alternative definition of Caputo fractional derivative and we show that it has severe disadvantages which restricts its use. In particular we prove that there is a necessary condition in order that solutions of the IVP can exist with this definition, which has often been overlooked in the literature. [ABSTRACT FROM AUTHOR]

In a previous paper Guo et al. (Fractional integration on mixed norm spaces. I. Preprint, 2022), we characterized the boundedness of fractional integration operators between mixed norm spaces over the unit disk. In this paper, we characterize the boundedness between X and Y, where X , Y ∈ { H p (0 < p < ∞) , H ∞ , BMOA , B , H (p , q , α) }. As in Guo et al. (Fractional integration on mixed norm spaces. I. Preprint, 2022), we cover three types of fractional integration: Flett, Hadamard, and Riemann-Liouville, and we consider complex orders t ∈ C instead of mere t ∈ R . Our findings provide, in particular, a complete answer to a problem started by Flett in 1972 (Theorem C). [ABSTRACT FROM AUTHOR]

In this paper, we propose the notion of extended neutrosophic rectangular metric space and prove some fixed point results under contraction mapping. Finally, as an application of the obtained results, we prove the existence and uniqueness of the Caputo fractional differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and uniqueness of solutions for the following nonlinear multi-order fractional differential equation with integral boundary conditions { (C D 0 + α u) (t) + ∑ i = 1 m λ i (t) (C D 0 + α i u) (t) + ∑ j = 1 n μ j (t) (C D 0 + β j u) (t) + ∑ k = 1 p ξ k (t) (C D 0 + γ k u) (t) + ∑ l = 1 q ω l (t) (C D 0 + δ l u) (t) + σ (t) u (t) + f (t , u (t)) = 0 , t ∈ [ 0 , 1 ] , u ″ (0) = u ‴ (0) = 0 , u ′ (0) = η 1 ∫ 0 1 u (s) d s , u (1) = η 2 ∫ 0 1 u (s) d s , where 0 < δ 1 < δ 2 < ⋯ < δ q < 1 < γ 1 < γ 2 < ⋯ < γ p < 2 < β 1 < β 2 < ⋯ < β n < 3 < α 1 < α 2 < ⋯ < α m < α < 4 and η 1 + 2 (1 − η 2) ≠ 0 . Using a fixed point theorem and Banach contractive mapping principle, we obtain some existence and uniqueness results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method of using the theory of functional connections (TFC) to approximate the solution of fractional differential equations. For functions with one constraint at one point, several constraints at one point, distinct points, and relative constraints, the theoretical approach of the suggested method is investigated. The choice of the basis function is described, and the issue of using monomials is discussed. For the first time in the literature, the suggested method is used to solve fractional differential initial value problems, boundary value problems, and higher-order problems. Wherever the exact solution exists, the numerical results are compared. The numerical findings for the fractional order scenario are compared with the predictor-corrector method and polynomial least squares method. The error plot for the integer order case with the exact solution is also provided. The proposed approach is also used to solve a corneal shape model. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of solutions for a boundary value problem of singular two-term fractional differential equation with delay and sign-changing nonlinearity. By means of the Guo–Krasnosel'skii fixed point theorem and the Leray–Schauder nonlinear alternative theorem, we obtain some results on the existence and multiplicity of solutions, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear fractional p-Laplacian boundary value problem containing both left Riemann–Liouville and right Caputo fractional derivatives with initial and integral conditions. Some new results on the existence and uniqueness of a solution for the model are obtained as well as the Ulam stability of the solutions. Two examples are provided to show the applicability of our results. [ABSTRACT FROM AUTHOR]

This paper presents a new way to solve numerically multiterm variable-order fractional differential equations (MTVOFDEs) with initial conditions by using a class of modified shifted Jacobi polynomials (MSJPs). As their defining feature, MSJPs satisfy the given initial conditions. A key aspect of our methodology involves the construction of operational matrices (OMs) for ordinary derivatives (ODs) and variable-order fractional derivatives (VOFDs) of MSJPs and the application of the spectral collocation method (SCM). These constructions enable efficient and accurate numerical computation. We establish the error analysis and the convergence of the proposed algorithm, providing theoretical guarantees for its effectiveness. To demonstrate the applicability and accuracy of our method, we present five numerical examples. Through these examples, we compare the results obtained with other published results, confirming the superiority of our method in terms of accuracy and efficiency. The suggested algorithm yields very accurate agreement between the approximate and exact solutions, which are shown in tables and graphs. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some user-friendly versions of integral-type fixed-point results and give some modifications of the classical Banach contraction principle by constructing a special type of contractive restrictions of integral forms for weak contraction mappings defined on 1 E -complete multiplicative partial cone metric spaces over Banach algebras and formulate some existence and uniqueness results regarding the fixed-point theorems using some integrative conditions. Moreover, we validate the significance our results and exploit them to find the unique solution of a fractional nonlinear differential equation of Caputo type, which complements some previously well-known generalizations found in the literature. [ABSTRACT FROM AUTHOR]

This paper aims to propose a complete superconvergence analysis for a postprocessing technique based on collocation methods for fractional differential equations (FDEs). We start with the simple linear FDEs with Caputo derivative of order 0 < α < 1 . The problem is reformulated as a weakly singular Volterra integral equation (VIE), and based on the resolvent theory of VIEs, the existence, uniqueness and regularity for the exact solution for the original FDE are obtained. Then the piecewise polynomial collocation method is adopted to solve the reformulated VIE, and based on the regularity of the original FDE, the convergence for the collocation method and the superconvergence for the iterated collocation method are investigated in detail, respectively. Further, based on the obtained collocation solution, the interpolation postprocessing approximation of higher accuracy is constructed on graded mesh, and the superconvergence is obtained. Compared with classical iterated collocation method, the cost on computation of interpolation postprocessing technique is less. Numerical experiments are given to illustrate the theoretical results, and it is also shown that the proposed postprocessing technique can be extended to certain nonlinear and systems of FDEs. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new multiple-parameters (multi-index) extension of the Wright function that arises from an eigenvalue problem for a case of hyper-Bessel operator involving Caputo fractional derivatives. We show that by giving particular values to the parameters involved in this special function, this leads to some known special functions (as the classical Wright function, the α-Mittag-Leffler function, the Tricomi function, etc.) that on their turn appear as cases of the so-called multi-index Mittag-Leffler functions. As an application, we mention that this new generalization Wright function nis an isochronous solution of a nonlinear fractional partial differential equation. [ABSTRACT FROM AUTHOR]

This paper concerns with the existence, uniqueness, Ulam's Hyer (UH) stability and total controllability results for the Hilfer fractional switched impulsive systems in finite-dimensional spaces. Mainly, this paper can be divided into three parts. In the first part, we examine the existence of a unique solution. In the second part, we establish the UH stability results, and in the third part, we study the total controllability results. The main outcomes are acquired by utilising the nonlinear analysis, fractional calculus, Mittag–Leffler function and Banach contraction principle. Finally, we have given two examples to validate the obtained analytical results. [ABSTRACT FROM AUTHOR]

This paper presents a dynamic system for estimating the spreading profile of COVID-19 in Thailand, taking into account the effects of vaccination and social distancing. For this purpose, a compartmental network is built in which the population is divided into nine mutually exclusive nodes, including susceptible, insusceptible, exposed, infected, vaccinated, recovered, quarantined, hospitalized, and dead. The weight of edges denotes the interaction between the nodes, modeled by a series of conversion rates. Next, the compartmental network and corresponding rates are incorporated into a system of fractional partial differential equations to define the model governing the problem concerned. The fractional degree corresponding to each compartment is considered the node weight in the proposed network. Next, a Monte Carlo-based optimization method is proposed to fit the fractional compartmental network to the actual COVID-19 data of Thailand collected from the World Health Organization. Further, a sensitivity analysis is conducted on the node weights, i.e., fractional orders, to reveal their effect on the accuracy of the fit and model predictions. The results show that the flexibility of the model to adapt to the observed data is markedly improved by lowering the order of the differential equations from unity to a fractional order. The final results show that, assuming the current pandemic situation, the number of infected, recovered, and dead cases in Thailand will, respectively, reach 4300, 4.5 × 10 6 , and 36,000 by the end of 2021. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

In this paper, we focus on the tempered subdiffusive Black–Scholes model. The main part of our work consists of the finite difference method as a numerical approach to option pricing in the considered model. We derive the governing fractional differential equation and the related weighted numerical scheme. The proposed method has an accuracy order 2 - α with respect to time, where α ∈ (0 , 1) is the subdiffusion parameter and 2 with respect to space. Furthermore, we provide stability and convergence analysis. Finally, we present some numerical results. [ABSTRACT FROM AUTHOR]

Nonlinear fractional evolution equations are important for determining various complex nonlinear problems that occur in various scientific fields, such as nonlinear optics, molecular biology, quantum mechanics, plasma physics, nonlinear dynamics, water surface waves, elastic media and others. The space-time fractional modified equal width (MEW) equation is investigated in this paper utilizing a variety of solitary wave solutions, with a particular emphasis on their implications for wave propagation characteristics in plasma and optical fibre systems. The fractional-order problem is transformed into an ordinary differential equation using a fractional wave transformation approach. In this article, the polynomial expansion approach and the sardar sub-equation method are successfully used to evaluate the exact solutions of space-time fractional MEW equation. Additionally, in order to graphically represent the physical significance of created solutions, the acquired solutions are shown on contour, 3D and 2D graphs. Based on the results, the employed methods show their efficacy in solving diverse fractional nonlinear evolution equations generated across applied and natural sciences. The findings obtained demonstrate that the two approaches are more effective and suited for resolving various nonlinear fractional differential equations. [ABSTRACT FROM AUTHOR]

Recently, fractional partial differential equations have been widely applied in option pricing problems, which better explains many important empirical facts of financial markets, but rare paper considers the multi-state options pricing problem based on fractional diffusion models. Thus, multi-state European option pricing problem under regime-switching tempered fractional partial differential equation is considered in this paper. Due to the expensive computational cost caused by the implicit finite difference scheme, a novel implicit-explicit finite difference scheme has been developed with consistency, stability, and convergence guarantee. Since the resulting coefficient matrix equals to the direct sum of several Toeplitz matrices, a preconditioned direct method has been proposed with O (S ̄ N log N + S ̄ 2 N) operation cost on each time level with adaptability analysis, where S ̄ is the number of states and N is the number of grid points. Related numerical experiments including an empirical example have been presented to demonstrate the effectiveness and accuracy of the proposed numerical method. [ABSTRACT FROM AUTHOR]

The theory of averaging is a classical component of applied mathematics and has been applied to solve some engineering problems, such as in the filed of control engineering. In this paper, we develop a theory of averaging on both finite and infinite time intervals for fractional non-autonomous differential equations. The closeness of the solutions of fractional no-autonomous differential equations and the averaged equations has been proved. The main results of the paper are applied to the switched capacitor voltage inverter modeling problem which is described by the fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, the time-fractional coupled viscous Burgers' equation (CVBE) and Drinfeld-Sokolov-Wilson equation (DSWE) are solved by the Sawi transform coupled homotopy perturbation method (HPM). The approximate series solutions to these two equations are obtained. Meanwhile, the absolute error between the approximate solution given in this paper and the exact solution given in the literature is analyzed. By comparison of the graphs of the function when the fractional order α takes different values, the properties of the equations are given as a conclusion. [ABSTRACT FROM AUTHOR]