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In this article, we analyse the fractional Cattaneo heat equation for studying the conduction of heat in porous medium. This equation is also used in studying extended irreversible thermodynamics, material, plasma, cosmological model, computational biology, and diffusion theory in crystalline solids. The Sumudu adomian decomposition technique, which is combination of Sumudu transform and a numerical technique, is applied for getting numerical solution. The existence and uniqueness is analysed by using the fixed point theorem and the highest error of the designed technique is also analysed. Finally, the accuracy of the designed numerical method is presented by solving two examples and the findings are compared with the existing method. [ABSTRACT FROM AUTHOR]

Constructing a higher order collocation framework for solving the Caputo multi-term time-fractional advection–diffusion and diffusion-type problems is the primary objective of this work, which has influenced the field of scientific disciplines. Advection–diffusion and reaction–diffusion equations were developed by modeling scientific phenomena in fluid flow issues, solid oxide fuel cells, and solvent diffusion into heavy oils. As a result, numerical solutions to these problems have garnered significant attention. The L 1 - 2 approximation approach approximates the fractional derivatives of orders η , η i ∈ (0 , 1) that are present in the considered problem. This approach provides a higher accuracy of O (k 3 - max { η , η i }) in time direction. Fourth-order convergence in space is achieved by employing a spline collocation technique with trigonometric quintic splines. Results from applying the suggested computational approach to four test examples have demonstrated its superiority and validity. [ABSTRACT FROM AUTHOR]

Anomalous diffusion of particles in fluids is better described by the fractional diffusion models. A robust hybrid numerical algorithm for a two-dimensional time fractional diffusion equation with the source term is presented. The well-known L1 scheme is considered for semi-discretization of the diffusion equation. To interpolate the semi-discretized equation, orthogonal collocation with bi-quintic Hermite splines as the basis is chosen for the smooth solution. Quintic Hermite splines interpolate the solution as well as its first and second order derivatives. The technique reduces the proposed problem to an algebraic system of equations. Stability analysis of the implicit scheme is studied using H ~ 1 m -norm defined in Sobolev space. The optimal order of convergence is found to be of order O (h 4) in spatial direction and is of order O (Δ t) 2 - α in the temporal direction where h is the step size in space direction and Δ t is the step size in time direction and α is the fractional order of the derivative. Numerical illustrations have been presented to discuss the applicability of the proposed hybrid numerical technique to the problems having fractional order derivative. [ABSTRACT FROM AUTHOR]

In this article, we construct the series solution of the time-fractional Korteveg de Vries (K-dV) equation through a computational approach named as Laplace residual power series (LRPS) that combines the Laplace transform with the residual power series method (RPS). Time-fractional K-dV equation is used to modeled various real life phenomena like propagation of waves in elastic rods, dispersion effects in shallow coastal regions, anomalous diffusion observed in financial markets. The Caputo fractional derivative is used in the formulation of time-fractional K-dV equation. LRPS method is characterized by its rapid convergence and easy finding of the unknown coefficients using the concept of limit at infinity without any perturbation, discretization and linearization. To assess the effectiveness of proposed computational strategy, we perform a comparative analysis among the fractional residual power series method, the Adomian decomposition method, and the RPS method. Additionally, we examine the convergence of the fractional series solution across different α values and assess the solution's behavior as the time domain increased. The efficiency and authenticity of the LRPS method is shown by computing the absolute error, relative error and residual error. This work is supported by 2D and 3D graphical representations made in accordance with Maple and MATLAB. [ABSTRACT FROM AUTHOR]

This article introduces an extension of classical fuzzy partial differential equations, known as fuzzy fractional partial differential equations. These equations provide a better explanation for certain phenomena. We focus on solving the fuzzy time diffusion equation with a fractional order of 0 < α ≤ 1, using two explicit compact finite difference schemes that are the compact forward time center space (CFTCS) and compact Saulyev's scheme. The time fractional derivative uses the Caputo definition. The double-parametric form approach is used to transfer the governing equation from an uncertain to a crisp form. To ensure stability, we apply the von Neumann method to show that CFTCS is conditionally stable, while compact Saulyev's is unconditionally stable. A numerical example is provided to demonstrate the practicality of our proposed schemes. [ABSTRACT FROM AUTHOR]

This paper investigates Cauchy problems of nonlinear parabolic equation with a Caputo fractional derivative. When the initial datum is sufficiently small in some appropriate spaces, we demonstrate the existence in global time and uniqueness of a mild solution in fractional Sobolev spaces using some novel techniques. Under some suitable assumptions on the initial datum, we show that the mild solution of the time fractional parabolic equation converges to the mild solution of the classical problem when α → 1 − . Under some appropriate assumptions on the initial datum, we show that the mild solution of the time fractional diffusion equation converges to the mild solution of the classical problem when α → 1 − . Our theoretical results can be widely applied to many different equations such as the Hamilton–Jacobi equation, the Navier–Stokes equation in two cases: the fractional derivative and the classical derivative. Our paper also provides a completely new answer to the related open problem of convergence of solutions to fractional diffusion equations as the order of fractional derivative approaches 1−. [ABSTRACT FROM AUTHOR]

In this work, we suggest a new method for solving linear multi-term time-fractional wave-diffusion equations, which is named the modified fractional reduced differential transform method (m-FRDTM). The importance of this technique is that it suggests a solution for a multi-term time-fractional equation. Very few techniques have been proposed to solve this type of equation, as will be shown in this paper. To show the effectiveness and efficiency of this proposed method, we introduce two different applications in two-term fractional differential equations. The three-dimensional and two-dimensional plots for different values of the fractional derivative are depicted to compare our results with the exact solutions. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to construct an efficient recursive algorithm to solve a time-space fractional Poisson's equation which can be treated as a time-space fractional diffusion equation in two dimensions. The fractional derivatives in both time and space are defined in the Caputo sense. A homotopy perturbation method is introduced to approximate the solution, and a comparison is made between the exact and the approximate solutions. In addition, we present a procedure for solving higher-order fractional Poisson's equations. In this case, the equation is converted to a system of fractional differential equations in which the order of the time derivatives is less than or equal to one. The convergence analysis is carried out, and an apriori bound of the solution is obtained for the present problem. Numerical examples are provided and the experimental evidence proves the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order 0 < α < 1 , and 1 < β < 2 . The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to obtain an approximate solution for the reduced problem; (iii) finally, the Stehfest method is employed to convert the obtained solution from the frequency domain back to the time domain. The use of the Laplace transform eliminates the need for classical time-stepping techniques, which often require very small time steps to achieve accuracy. Additionally, the application of local radial basis functions helps overcome issues related to ill-conditioning and sensitivity to shape parameters typically encountered in global radial basis function methods. To validate the efficiency and accuracy of the proposed method, several test problems in regular and irregular domains with uniform and non-uniform nodes are considered. [ABSTRACT FROM AUTHOR]

The main goal in this study is to determine the analytic solution of onedimensional initial boundary value problem including sequential space-time fractional differential equation with boundary conditions in Neumann sense. The solution of the space-time fractional diffusion problem is accomplished in series form by employing the separation of variables method. To obtain coefficients in the Fourier series is utilized a fractional inner product. The obtained results are supported by an illustrative example. Moreover, it is observed that the implementation of the method is straightforward and smooth. [ABSTRACT FROM AUTHOR]

Various scholars have lately employed a wide range of strategies to resolve specific types of symmetrical fractional differential equations. This paper introduces a new implicit finite difference method with variable-order time-fractional Caputo derivative to solve semi-linear initial boundary value problems. Despite its extensive use in other areas, fractional calculus has only recently been applied to physics. This paper aims to find a solution for the fractional diffusion equation using an implicit finite difference scheme, and the results are displayed graphically using MATLAB and the Fourier technique to assess stability. The findings show the unconditional stability of the implicit time-fractional finite difference method. This method employs a variable-order fractional derivative of time, enabling greater flexibility and the ability to tackle more complicated problems. [ABSTRACT FROM AUTHOR]

In this paper we investigate the Cauchy problem for a fractional diffusion equation and the time‐fractional derivative is taken in the Caputo type sense. We give a representation of solutions under Fourier series and analyze initial value problems for the semi‐linear fractional diffusion equation with a memory term. We also discuss the stability of the fractional derivative order for the time under some assumptions on the input data. Our key idea is to use Mittag‐Leffler functions, the Banach fixed point theorem, and some Sobolev embeddings. [ABSTRACT FROM AUTHOR]

We consider the multi-term time fractional diffusion equation on a bounded convex domain. We analyze the L 1 method on a graded mesh in time to compensate for the weak singularity of the solution and a weak Galerkin finite element method in space discretization. The stability analyses are presented for both semi-discrete and fully-discrete schemes and we prove that two schemes are unconditionally stable. Error estimates in L 2 -norm and a discrete H 1 equivalent norm for both schemes are rigorously derived. Further we discuss the optimal spatial order error estimates in L 2 -norm. Finally, we give some numerical experiments to show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we utilize a time‐fractional anisotropic diffusion equation for image denoising. Theoretical results are provided thanks to Schauder fixed‐point theorem. A discretization scheme by finite difference is also presented. Numerical experiments show a great performance in deleting the noise while preserving important features. The model robustness is tested, which is clear visually and quantitatively. All the obtained results conclude that our model surpasses the competitive models, such as Perona–Malik and Weickert. [ABSTRACT FROM AUTHOR]

The time-fractional fourth-order reaction-diffusion problem, which contains more than one time-fractional derivative of orders lying between 0 and 1, is considered. This problem is the generalized version of the problem discussed by Nikan et al. Appl. Math. Model. 89 (2021), 819–836 that has only one time-fractional derivative. It is widely used in the study of chemical waves and patterns in reaction-diffusion systems. The analysis of non-smooth solutions to this problem is discussed broadly using the Caputo-time fractional derivative. The non-smooth solutions to the problem have a weak singularity close to zero that can be efficiently handled by considering the non-uniform mesh. The method based on the non-uniform time stepping is an efficacious way to regain accuracy. The current study presents the trigonometric quintic B-spline approach to solve this multi-term time-fractional fourth-order problem using graded mesh and effective grading parameters. The stability and convergence results are proved through rigorous analysis, which helps choose the optimal grading parameter. The accuracy and effectiveness of our technique are observed in our numerical experiments that manifest the comparison of uniform and non-uniform meshes. [ABSTRACT FROM AUTHOR]

This paper investigates the temporally loaded time-fractional diffusion equation with initial and Dirichlet-type boundary conditions. To begin with, a solution form is established using the method of eigenfunction expansions, and its existence and uniqueness are examined along with some apriori estimates. Thereafter, a finite difference approximation is performed using the so-called L1 method for the Caputo fractional derivative, resulting in a loaded difference scheme. The superposition property of systems of linear algebraic equations is applied to solve the loaded difference scheme by appointing an appropriate solution representation. The unique solvability of the proposed scheme is set up. The stability and convergence of the proposed difference scheme are analysed by the discrete energy method with an order of accuracy O(τ2-α+h2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}(\tau ^{2-\alpha }+h^2)$$\end{document}. Numerical results via two test problems are presented to validate the theoretical findings of the proposed scheme by observing the errors. [ABSTRACT FROM AUTHOR]

We consider a phase-change problem for a one-dimensional material with a nonlocal flux, expressed in terms of the Caputo derivative, which derives in a space-fractional Stefan problem. We prove existence of a unique solution to a phase-change problem with the fractional Neumann boundary condition at the fixed face x = 0, where the domain, at the initial time, consists of liquid and solid. Then we use this result to prove the existence of a solution to an analogous problem with solid initial domain, when it is not possible to transform the domain into a cylinder. [ABSTRACT FROM AUTHOR]

In the current paper, we are concerned with the existence and uniqueness of mild solutions to a Cauchy problem involving a time-space fractional diffusion equation with an exponential semilinear source. By using the iteration method and some L p − L q -type estimates of fundamental solutions associated with the Mittag-Leffler function, we study the well-posedness of the problem in two different cases corresponding to two assumptions on the Cauchy data. On the one hand, when considering initial data in L p ( R N) ∩ L ∞ ( R N) , the problem possesses a local-in-time solution. On the other hand, we obtain a global existence result for a mild solution with small data in an Orlicz space. [ABSTRACT FROM AUTHOR]

The axisymmetric time-fractional diffusion equation with mass absorption is studied in a circle under the time-harmonic Dirichlet boundary condition. The Caputo derivative of the order 0 < α ≤ 2 is used. The investigated equation can be considered as the time-fractional generalization of the bioheat equation and the Klein–Gordon equation. Different formulations of the problem for integer values of the time-derivatives α = 1 and α = 2 are also discussed. The integral transform technique is employed. The outcomes of numerical calculations are illustrated graphically for different values of the parameters. [ABSTRACT FROM AUTHOR]

This paper is concerned with numerical approach for solving space fractional diffusion equation using shifted Gegenbauer polynomials, where the fractional derivatives are expressed in Caputo sense. The properties of Gegenbauer polynomials are exploited to reduce space fractional diffusion equation to a system of ordinary differential equations, that are then solved using finite difference method. Some selected numerical simulations of space fractional diffusion equations are presented and the results are compared with the exact solution, also with the results obtained via other methods in the literature. The comparison reveals that the proposed method is reliable, effective and accurate. All the computations were carried out using Matlab package. [ABSTRACT FROM AUTHOR]

This paper focuses on development of an efficient numerical technique for approximation of a fractional neutron diffusion equation with delayed neutrons for neutron flux profile in a nuclear reactor. The L 1 method is used to approximate the Caputo time-fractional derivatives in the governing equation. A collocation technique based on quintic B-spline (QBS) basis function is employed for discretization of space derivative. The proposed numerical scheme is applied to solve the considered fractional neutron diffusion equation in order to illustrate its applicability and efficiency. The influences of fractional order derivative (α), radioactive decay constant (λ) and relaxation time (τ) on the neutron flux profile are investigated. The numerical results of fractional neutron diffusion equation are compared with those of classical neutron diffusion equation. Further, the results obtained by the proposed approach are compared with the results obtained by the finite difference method to justify the superiority of our method. The CPU time (the execution time of the computer program) is given in order to demonstrate the method's computational efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we present a high-order approach for solving one- and two-dimensional time-space fractional diffusion equations (FDEs) with Caputo-Riesz derivatives. To design the scheme, the Caputo temporal derivative is approximated using a high-order method, and the spatial Riesz derivative is discretized by the second-order weighted and shifted Grünwald difference (WSGD) method. It is proved that the scheme is unconditionally stable and convergent with the order of O (τ α h 2 + τ 4) , where τ and h are time and space step sizes, respectively. We illustrate the accuracy and effectiveness of the method by providing several numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we study fractional subdiffusion fourth parabolic equations containing Caputo and Caputo-Fabrizio operators. The main results of the paper are presented in two parts. For the first part with the Caputo derivative, we focus on the global and local well-posedness results. We study the global mild solution for biharmonic heat equation with Caputo derivative in the case of globally Lipschitz source term. A new weighted space is used for this case. We then proceed to give the results about the local existence in the case of locally Lipschitz source term. To overcome the intricacies of the proofs, we applied Lp − Lq estimate for biharmonic heat semigroup, Banach fixed point theory, some estimates for Mittag-Lefler functions and Wright functions, and also Sobolev embeddings. For the second result involving the Cahn-Hilliard equation with the Caputo-Fabrizio operator, we first show the local existence result. In addition, we first provide that the connections of the mild solution between the Cahn-Hilliard equation in the case 0 < α < 1 and α = 1. This is the first result of investigating the Cahn-Hilliard equation with this type of derivative. The main key of the proof is based on complex evaluations involving exponential functions, and some embeddings between Lp spaces and Hilbert scales spaces. [ABSTRACT FROM AUTHOR]

In this paper, the properties of Chebyshev polynomials and the Gauss–Legendre quadrature rule are employed to construct a new operational matrix of distributed-order fractional derivative. This operational matrix is applied for solving some problems such as distributed-order fractional differential equations, distributed-order time-fractional diffusion equations and distributed-order time-fractional wave equations. Our approach easily reduces the solution of all these problems to the solution of some set of algebraic equations. We also discuss the error analysis of approximation distributed-order fractional derivative by using this operational matrix. Finally, to illustrate the efficiency and validity of the presented technique five examples are given. Abbreviations: DFDEs: distributed-order fractional differential equations; DTFDEs: distributed-order time-fractional diffusion equations; DTFWEs: distributed-order time-fractional wave equations; OMFD: operational matrix of fractional derivative; SCP: shifted Chebyshev polynomial [ABSTRACT FROM AUTHOR]

In this article, we propose a method to solve the time‐fractional hyperbolic heat equation. We first formulate a boundary value problem for the standard hyperbolic heat equation in a finite domain and provide an analytical solution by means of separation of variables and Fourier series. Then, we consider the same boundary value problem for the fractional hyperbolic heat equation. The fractional problem is solved using three different definitions of the fractional derivative: the Caputo fractional derivative and two recently defined alternative versions of this derivative, the Caputo–Fabrizio and the Atangana–Baleanu. A closed form of the solution is provided for each case. Finally, we compare the solutions of the fractional and the standard problem and show numerically that the solution of the standard hyperbolic heat equation can be retrieved from the solution of the fractional equation in the limit γ→2, where γ represents the exponent of the fractional derivative. [ABSTRACT FROM AUTHOR]

Under a priori boundedness conditions of solutions and coefficients, we prove a Hölder stability estimate for an inverse problem of determining two spatially varying zeroth order non-diagonal elements of a coefficient matrix in a one-dimensional fractional diffusion system of half order in time. The proof relies on the conversion of the fractional diffusion system to a system of order 4 in the space variable and the Carleman estimate. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to present a stable and efficient numerical technique based on modified trigonometric cubic B-spline functions for solving the time-fractional diffusion equation (TFDE). The TFDE has numerous applications to model many real objects and processes. Design/methodology/approach: The time-fractional derivative is used in the Caputo sense. A modification is made in trigonometric cubic B-spline (TCB) functions for handling the Dirichlet boundary conditions. The modified TCB functions have been used to discretize the space derivatives. The stability of the technique is also discussed. Findings: The obtained results are compared with those reported earlier showing that the present technique gives highly accurate results. The stability analysis shows that the method is unconditionally stable. Furthermore, this technique is efficient and requires less storage. Originality/value: The current work is novel for solving TFDE. This technique is unconditionally stable and gives better results than existing results (Ford et al., 2011; Sayevand et al., 2016; Ghanbari and Atangana, 2020). [ABSTRACT FROM AUTHOR]

The linear non-polynomial spline is used here to solve the fractional partial differential equation (FPDE). The fractional derivatives are described in the Caputo sense. The tensor products are given for extending the one-dimensional linear nonpolynomial spline S1 to a two-dimensional spline S1 ⊗ S2 to solve the heat equation. In this paper, the convergence theorem of the method used to the exact solution is proved and the numerical examples show the validity of the method. All computations are implemented by Mathcad15. [ABSTRACT FROM AUTHOR]

In this paper, we study a fast linearized numerical method for solving nonlinear time-fractional diffusion equations. A new weighted method is proposed to construct linearized approximation, which enables the unconditional convergence to be established when the nonlinearity f(u) is only locally Lipschitz continuous. In order to reduce the computational cost, the sum-of-exponentials (SOE) technique is employed to evaluate the kernel function in the Caputo derivative. By using the complementary discrete kernels for the coefficients of the refined fast weighted discretization, the proposed method is shown to be unconditionally convergent with respect to the discrete H1-norm. The fast linearized method can also be extended to nonlinear multi-term and distributed-order time-fractional diffusion equations. Numerical examples with different types of nonlinear functions are provided to demonstrate the behavior of proposed methods for both smooth and weakly singular solutions. [ABSTRACT FROM AUTHOR]

This paper presents an iterative procedure based on an implicit finite difference method to solve a mathematical model of drug delivery from a planar matrix with a moving boundary condition. This model includes the diffusion equation with space–time fractional-order derivatives. We establish the stability and convergence analysis of the method. We compare the numerical results with the scale-invariant and the homotopy perturbation solutions for different space–time-fractional orders and the problem parameters. [ABSTRACT FROM AUTHOR]

In this paper, we develop two fast implicit difference schemes for solving a class of variable‐coefficient time–space fractional diffusion equations with integral fractional Laplacian (IFL). The proposed schemes utilize the graded L1 formula for the Caputo fractional derivative and a special finite difference discretization for IFL, where the graded mesh can capture the model problem with a weak singularity at initial time. The stability and convergence are rigorously proved via the M‐matrix analysis, which is from the spatial discretized matrix of IFL. Moreover, the proposed schemes use the fast sum‐of‐exponential approximation and Toeplitz matrix algorithms to reduce the computational cost for the nonlocal property of time and space fractional derivatives, respectively. The fast schemes greatly reduce the computational work of solving the discretized linear systems from O(MN3+M2N) by a direct solver to O(MN(logN+Nexp)) per preconditioned Krylov subspace iteration and a memory requirement from 𝒪(MN2) to 𝒪(NNexp), where N and (Nexp ≪) M are the number of spatial and temporal grid nodes, respectively. The spectrum of preconditioned matrix is also given for ensuring the acceleration benefit of circulant preconditioners. Finally, numerical results are presented to show the utility of the proposed methods. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to construct and analyze an efficient numerical scheme for solving the fractional neutron diffusion equation with delayed neutrons, which describes neutron transport in a nuclear reactor. The L 1 approximation is used for discretization of time derivative and finite difference method is used for discretization of space derivative. The stability and convergence analysis of the proposed method are studied. The method is shown to be second-order convergent in space and (2 − 2 α) -th order convergent in time, where α is the order of fractional derivative. Numerical experiments are carried out to demonstrate the performance of the method and theoretical analysis. The effects of fractional order derivative, relaxation time and radioactive decay constant on the neutron flux behaviour are investigated. Moreover, the CPU time of the present method is provided. [ABSTRACT FROM AUTHOR]

In this article, an efficient local meshless technique is implemented for the numerical solution of an anomalous mobile-immobile solute transport process. The process is mathematically modeled as a time fractional mobile-immobile diffusion equation in sense of Caputo derivative. An implicit time integration procedure is used to semi-discretize the model in the time direction whereas the space derivatives of the model is discretized by the proposed meshless technique based on inverse multquadric radial basis function. The demand of meshless techniques increment because of its meshless nature and simplicity of usage in higher dimensions. This technique approximate the solution on set of uniform and scattered nodes and it leads to a sparse and well-conditioned coefficient matrices. Numerical examinations on some test problems are performed to exhibit successful applications and accuracy of the local meshless technique on regular and irregular computational domains. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new high-order finite difference method to solve the time-fractional diffusion equation with a source. We first construct a finite difference approximation of the Caputo fractional derivative of order α (0 < α < 1), and show that the convergence rate of our approximation is (4 - α). We then investigate the properties of the fractional differentiation matrix for our new approximations, and introduce an implicit finite difference method which employs such approximations for the time discretization of the fractional diffusion equation, coupled with a Fourier-type expansion in space. By taking advantage of the special structure of our fractional differentiation matrix, each of the linear systems resulted from our new high-order approximations for each mode of time-fractional diffusion equation can be solved in order O(N³). Numerical experiments about the performance of our method in evaluating fractional derivatives, and solving fractional ordinary differential equations and time-fractional diffusion equation are also presented, to demonstrate the efficiency of our method. [ABSTRACT FROM AUTHOR]

In this paper, the effect of the fractional order of the Caputo time-derivative occurring in heat conduction models on the temperature distribution in a finite cylinder consisting of an inner solid cylinder and an outer concentric layer is investigated. The inner cylinder (core) and the cylindrical layer are in perfect thermal contact. The Robin boundary condition on the outer surface and the Neumann conditions on the ends of the cylinder are assumed. An internal heat source is represented in the mathematical model by taking into account in the heat conduction equation of a function which depends on the space and time variable. An analytical solution of the problem is derived in the form of the double series of eigenfunctions. Numerical examples are presented. [ABSTRACT FROM AUTHOR]

In this paper, for the first time, the shifted Legendre operational matrix of distributed order fractional derivative has been derived. Also, this new operational matrix is used together with tau method for approximation of solutions of linear distributed order fractional differential equations and diffusion equations with distributed order in time. Moreover, eight numerical examples are implemented in order to show the validity and reliability of the suggested methods. [ABSTRACT FROM AUTHOR]

We examine the transition region between two solids which state differs from the state of contacting media. Small thickness of the intermediate region allows us to reduce a three-dimensional problem to a two-dimensional one for a median surface endowed with equivalent physical properties. In the present paper, we consider the generalized boundary conditions of nonperfect thermal contact for the time-fractional heat conduction equation with the Caputo derivative and solve the problem for a composite medium consisting of two semi-infinite regions. Numerical results are illustrated graphically. [ABSTRACT FROM AUTHOR]

The time-fractional heat conduction equation with the Caputo derivative is solved for an infinite plane with an external half-infinite crack which surfaces are exposed to the heat flux loading. The solution is obtained using the integral transform technique and is expressed in terms of the Mittag-Leffler function. The stress intensity factor is calculated for different values of the order of fractional derivative. Numerical results are illustrated graphically. [ABSTRACT FROM AUTHOR]

The one-dimensional fractional advection–diffusion equation with Robin-type boundary conditions is studied by using the Laplace and finite sine-cosine Fourier transforms. The mathematical model with memory is developed by employing the generalized Fick's law with time-fractional Caputo derivative. The influence of the fractional parameter (the non-local effects) on the solute concentration is studied. It is found that solute concentration can be minimized by decreasing the memory parameter. Also, it is found that, at small values of time the ordinary model leads to minimum concentration, while at large values of the time the fractional model is recommended. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate the performance of a hybridizable discontinuous Galerkin (HDG) method for approximating the solution of conservative fractional diffusion equations (CFDE). The main attractive feature of these methods is the fact that the only globally coupled unknowns are those at the element boundaries. We first introduce the HDG method for the CFDE and prove the existence and uniqueness of the numerical solution provided that the stabilization parameter is strictly positive. We provide extensive numerical results to test the convergence behavior of the HDG approximation. [ABSTRACT FROM AUTHOR]

Two fractional Stefan problems are considered by using Riemann‐Liouville and Caputo derivatives of order α ∈ (0,1) such that, in the limit case (α = 1), both problems coincide with the same classical Stefan problem. For the one and the other problem, explicit solutions in terms of the Wright functions are presented. We prove that these solutions are different even though they converge, when α↗1, to the same classical solution. This result also shows that some limits are not commutative when fractional derivatives are used. [ABSTRACT FROM AUTHOR]

A reaction-diffusion problem with a Caputo time derivative of order α∈(0,1) is considered. The solution of such a problem has in general a weak singularity near the initial time t=0. Some new pointwise bounds on certain derivatives of this solution are derived. The numerical method of the paper uses the well-known L1 discretisation in time on a graded mesh and a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh. Discrete stability of the computed solution is proved. The error analysis is based on a non-trivial projection into the finite element space, which for the first time extends the analysis of the DDG method to non-periodic boundary conditions. The final convergence result implies how an optimal grading of the temporal mesh should be chosen. Numerical results show that our analysis is sharp. [ABSTRACT FROM AUTHOR]

In this paper, a new numerical technique for solving the fractional order diffusion equation is introduced. This technique basically depends on the Non-Standard finite difference method (NSFD) and Chebyshev collocation method, where the fractional derivatives are described in terms of the Caputo sense. The Chebyshev collocation method with the (NSFD) method is used to convert the problem into a system of algebraic equations. These equations solved numerically using Newton’s iteration method. The applicability, reliability, and efficiency of the presented technique are demonstrated through some given numerical examples. [ABSTRACT FROM AUTHOR]

We compare the model of heat transfer proposed by Cattaneo, Maxwell, and Vernotte with another one, formulated in terms of fractional differential equations, in one and two dimensions. These are only some of the numerous models that have been proposed in the literature over many decades to model heat transport and possibly heat waves, in place of the classical heat equation due to Fourier. These models are characterized by sound as well as by critical properties. In particular, we found that the Cattaneo model does not exhibit necessarily oscillations or negative values of the (absolute) temperature when the relaxation parameter, $\tau$, drops below some value. On the other hand, the fractional derivative model may be affected by oscillations, depending on the specific initial profile. We also estimate the error made when the Cattaneo equation is adopted in place of the heat equation, and show that the approximation error is of order $\tau$. Moreover, the solution of the Cattaneo equation converges uniformly to that of the heat equation as $\tau \to 0+$ in the full closed time interval $[0,T]$ (for any given $T > 0$), while this does not occur for the time derivative, and the higher-order time derivatives blow up. [ABSTRACT FROM AUTHOR]

In this study, approximate solutions of a system of time-fractional coupled Burger equations were obtained by means of a local fractional operator (LFO) in the sense of the Caputo derivative. The LFO technique was built on the basis of the standard differential transform method (DTM). Illustrative examples used in demonstrating the effectiveness and robustness of the proposed method show that the solution method is very efficient and reliable as - unlike the variational iteration method - it does not depend on any process of identifying Lagrange multipliers, even while still maintaining accuracy. [ABSTRACT FROM AUTHOR]

In this paper, a finite difference scheme is proposed to solve time–space fractional diffusion equation which has second-order accuracy in both time and space direction. The time and space fractional derivatives are considered in the senses of Caputo and Riesz, respectively. First, the centered difference approach is used to approximate the Riesz fractional derivative in space. Then, the obtained fractional ordinary differential equations are transformed into equivalent Volterra integral equations. And then, the trapezoidal rule is utilized to approximate the Volterra integral equations. The stability and convergence of our scheme are proved via mathematical induction method. Finally, numerical experiments are performed to confirm the high accuracy and efficiency of our scheme. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the stability and boundedness of solutions of the initial value problem for a class of time-fractional diffusion equations. We first establish a fractional Duhamel principle for the nonhomogeneous time-fractional diffusion equation. Then based on it and the superposition principle, the solution of the above initial value problem is represented. Finally, we obtain the stability and boundedness of the solution and present an illustrative example. [ABSTRACT FROM AUTHOR]

We consider the initial value problem of the time fractional diffusion equation on the whole line and the fractional derivative is described in Caputo sense. A fully discrete Hermite pseudospectral approximation scheme is structured basing Hermite-Gauss points in space and finite difference in time. Unconditionally stability and convergence are proved. Numerical experiments are presented and the results conform to our theoretical analysis. [ABSTRACT FROM AUTHOR]