THE WELL-POSEDNESS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN COMPLEX BANACH SPACES

Let X be a complex Banach space and let B and C be two closed linear operators on X satisfying the condition D(B)?D(C),and let d∈L^(1)(R_(+))and 0≤β<α≤2.We characterize the well-posedness of the fractional integro-differential equations D^(α)u(t)+CD^(β)u(t)=Bu(t)+∫_(-∞)td(t-s)Bu(s)ds+f(t),(0≤t≤2π)on periodic Lebesgue-Bochner spaces L^(p)(T;X)and periodic Besov spaces B_(p,q)^(s)(T;X).

DISCRETE GALERKIN METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

In this article, we develop a fully Discrete Galerkin（DG） method for solving ini- tial value fractional integro-differential equations（FIDEs）. We consider Generalized Jacobi polynomials（CJPs） with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the approximate solution. The fractional derivatives are used in the Caputo sense. The numerical solvability of algebraic system obtained from implementation of proposed method for a special case of FIDEs is investigated. We also provide a suitable convergence analysis to approximate solutions under a more general regularity assumption on the exact solution. Numerical results are presented to demonstrate the effectiveness of the proposed method.

A Jacobi Spectral Collocation Method for Solving Fractional Integro-Differential Equations

The aim of this paper is to obtain the numerical solutions of fractional Volterra integrodifferential equations by the Jacobi spectral collocation method using the Jacobi-Gauss collocation points.We convert the fractional order integro-differential equation into integral equation by fractional order integral,and transfer the integro equations into a system of linear equations by the Gausssian quadrature.We furthermore perform the convergence analysis and prove the spectral accuracy of the proposed method in L∞norm.Two numerical examples demonstrate the high accuracy and fast convergence of the method at last.

On Fractional Integro-Differential Equation with Nonlinear Time Varying Delay

In this manuscript,we analyze the solution for class of linear and nonlinear Caputo fractional Volterra Fredholm integro-differential equations with nonlinear time varying delay.Also,we demonstrate the stability analysis for these equations.Our paper provides a convergence of semi-analytical approximate method for these equations.It would be desirable to point out approximate results.

CONVERGENCE ANALYSIS OF THE JACOBI SPECTRAL-COLLOCATION METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L^2-norm. The numerical examples are given to illustrate the theoretical results.

EXISTENCE AND UNIQUENESS RESULTS FOR BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS INVOLVING GRONWALL'S INEQUALITY IN BANACH SPACES

We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder＇s inequality, a suitable singular Cronwall＇s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.

Solving Fractional Integro-Differential Equations by Using Sumudu Transform Method and Hermite Spectral Collocation Method

In this paper we are looking forward to finding the approximate analytical solutions for fractional integro-differential equations by using Sumudu transform method and Hermite spectral collocation method.The fractional derivatives are described in the Caputo sense.The applications related to Sumudu transform method and Hermite spectral collocation method have been developed for differential equations to the extent of access to approximate analytical solutions of fractional integro-differential equations.

Existence of Solution for a Coupled System of Fractional Integro-Differential Equations on an Unbounded Domain

We present the existence of solution for a coupled system of fractional integro-differential equations. The differential operator is taken in the Caputo fractional sense. We combine the diagonalization method with Arzela-Ascoli theorem to show a fixed point theorem of Schauder.

Analytical solutions fractional order partial differential equations arising in fluid dynamics

This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectional propagation of long-wave in dispersive media and GSEs are used to model the interaction between one-dimensional high,and low-frequency waves.Classes of trigonometric and hyperbolic function solutions in fractional calculus are discussed.Graphical simulations of the numerical solutions are flaunted by MATLAB.

Memory effect in time fractional Schrödinger equation

A significant obstacle impeding the advancement of the time fractional Schrodinger equation lies in the challenge of determining its precise mathematical formulation.In order to address this,we undertake an exploration of the time fractional Schrodinger equation within the context of a non-Markovian environment.By leveraging a two-level atom as an illustrative case,we find that the choice to raise i to the order of the time derivative is inappropriate.In contrast to the conventional approach used to depict the dynamic evolution of quantum states in a non-Markovian environment,the time fractional Schrodinger equation,when devoid of fractional-order operations on the imaginary unit i,emerges as a more intuitively comprehensible framework in physics and offers greater simplicity in computational aspects.Meanwhile,we also prove that it is meaningless to study the memory of time fractional Schrodinger equation with time derivative 1<α≤2.It should be noted that we have not yet constructed an open system that can be fully described by the time fractional Schrodinger equation.This will be the focus of future research.Our study might provide a new perspective on the role of time fractional Schrodinger equation.

A Novel Accurate Method forMulti-Term Time-Fractional Nonlinear Diffusion Equations in Arbitrary Domains

Anovel accuratemethod is proposed to solve a broad variety of linear and nonlinear(1+1)-dimensional and(2+1)-dimensional multi-term time-fractional partial differential equations with spatial operators of anisotropic diffusivity.For(1+1)-dimensional problems,analytical solutions that satisfy the boundary requirements are derived.Such solutions are numerically calculated using the trigonometric basis approximation for(2+1)-dimensional problems.With the aid of these analytical or numerical approximations,the original problems can be converted into the fractional ordinary differential equations,and solutions to the fractional ordinary differential equations are approximated by modified radial basis functions with time-dependent coefficients.An efficient backward substitution strategy that was previously provided for a single fractional ordinary differential equation is then used to solve the corresponding systems.The straightforward quasilinearization technique is applied to handle nonlinear issues.Numerical experiments demonstrate the suggested algorithm’s superior accuracy and efficiency.

THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS

This study introduces a pre-orthogonal adaptive Fourier decomposition(POAFD)to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre(generalized Poisson equation).As a first step,the method expands the initial data function into a sparse series of the fundamental solutions with fast convergence,and,as a second step,makes use of the semigroup or the reproducing kernel property of each of the expanding entries.Experiments show the effectiveness and efficiency of the proposed series solutions.

A Collocation Technique via Pell-Lucas Polynomials to Solve Fractional Differential EquationModel for HIV/AIDS with Treatment Compartment

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 200 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(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].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.

Pseudo S-Asymptotically(ω,c)-Periodic Solutions to Fractional Differential Equations of Sobolev Type

In this paper,we firstly recall some basic results on pseudo S-asymptotically(ω,c)-periodic functions and Sobolev type fractional differential equation.We secondly investigate some existence of pseudo S-asymptotically(ω,c)-periodic solutions for a semilinear fractional differential equations of Sobolev type.We finally present a simple example.

Legendre-Weighted Residual Methods for System of Fractional Order Differential Equations

The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.

Crank-Nicolson Quasi-Compact Scheme for the Nonlinear Two-Sided Spatial Fractional Advection-Diffusion Equations

The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L2-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.

Numerical studies for solving fractional integro-differential equations

In this paper,we give a new numerical method for solving a linear system of fractional integro-differential equations.The fractional derivative is considered in the Caputo sense.The proposed method is least squares method aid of Hermite polynomials.The suggested method reduces this type of systems to the solution of systems of linear algebraic equations.To demonstrate the accuracy and applicability of the presented method some test examples are provided.Numerical results show that this approach is easy to implement and accurate when applied to integro-differential equations.We show that the solutions approach to classical solutions as the order of the fractional derivatives approach.

A Compact Difference Scheme on Graded Meshes for the Nonlinear Fractional Integro-differential Equation with Non-smooth Solutions

In this paper,a compact finite difference scheme for the nonlinear fractional integro-differential equation with weak singularity at the initial time is developed,with O(N^(-(2-α))+M^(-4))accuracy order,where N;M denote the numbers of grids in temporal and spatial direction,α ∈(0,1)is the fractional order.To recover the full accuracy based on the regularity requirement of the solution,we adopt the L1 method and the trapezoidal product integration(PI)rule with graded meshes to discretize the Caputo derivative and the Riemann-Liouville integral,respectively,further handle the nonlinear term carefully by the Newton linearized method.Based on the discrete fractional Gr¨onwall inequality and preserved discrete coefficients of Riemann-Liouville fractional integral,the stability and convergence of the proposed scheme are analyzed by the energy method.Theoretical results are also confirmed by a numerical example.

An analytical solution with existence and uniqueness conditions for fractional integro-differential equations

This study aims to apply the two-step Adomian decomposition method(TSADM)to find an analytical solution of integro-differential equations for fractional order without discretization/approximation with less number of iterations and reduce the computational efforts.Moreover,we have established the results for the existence and uniqueness of a solution with the help of some fixed point theorems and the Banach contraction principle.Furthermore,the method is demonstrated on different test examples arising in real life situations.It is concluded that the TSADM provides exact solution of the fractional integro-differential equations in one iteration.At the same time,the other existing methods furnish an approximate solution and require lots of computation to solve the problem applying discretization/approximation on fractional operators.

Existence and Uniqueness of Solution for a Fractional Order Integro-Differential Equation with Non-Local and Global Boundary Conditions

In this paper, we prove an important existence and uniqueness theorem for a fractional order Fredholm – Volterra integro-differential equation with non-local and global boundary conditions by converting it to the corresponding well known Fredholm integral equation of second kind. The considered in this paper has been solved already numerically in [1].