Solution Of Fractional Integro-differential Equations Research Articles

Solution of fractional integro-differential equations using least squares and shifted Legendre methods

Solution of fractional integro-differential equations using least squares and shifted Legendre methods

Existence of solutions for fractional integro-differential equations with integral boundary conditions

"The aim of this study is to prove the existence of solutions for Caputo boundary value problems of nonlinear fractional integro-differential equations with integral boundary conditions, by using the measure of non compactness combined with Mönch's fixed point theorem. Two examples are offered to demonstrate our outcomes."

Theoretical and Numerical Study for Volterra−Fredholm Fractional Integro-Differential Equations Based on Chebyshev Polynomials of the Third Kind

In this paper, we develop an efficient numerical method to approximate the solution of fractional integro-differential equations (FI-DEs) of mixed Volterra−Fredholm type using spectral collocation method with shifted Chebyshev polynomials of the third kind (S-Cheb-3). The fractional derivative is described in the Caputo sense. A Chebyshev−Gauss quadrature is involved to evaluate integrals for more precision. Two types of equations are studied to obtain algebraic systems solvable using the Gauss elimination method for linear equations and the Newton algorithm for nonlinear ones. In addition, an error analysis is carried out. Six numerical examples are evaluated using different error values (maximum absolute error, root mean square error, and relative error) to compare the approximate and the exact solutions of each example. The experimental rate of convergence is calculated as well. The results validate the numerical approach’s efficiency, applicability, and performance.

Numerical Solution of Fractional Integro-Differential Equations Via Fourth-Degree Hat Functions

The goal of this paper is to construct new fourth-degree hat functions (FDHFs) and study their properties in order to develop a new numerical method for solving fractional integro-differential equations. The equation under consideration is transformed into a set of algebraic equations by using FDHFs, which makes it simple to solve the system using one of the iterative methods. In fact, this method’s advantage was that it was easy to use and had fifth-order convergence, as we showed in the section on error analysis. The numerical results demonstrate that the new technique works well through the presented examples.

A study on the existence of numerical and analytical solutions for fractional integrodifferential equations in Hilfer type with simulation

<abstract><p>Previous studies have shown that fractional derivative operators have become an integral part of modeling natural and physical phenomena. During the progress and evolution of these operators, it has become clear to researchers that each of these operators has special capacities for investigating phenomena in engineering sciences, physics, biological mathematics, etc. Fixed point theory and its famous contractions have always served as useful tools in these studies. In this regard, in this work, we considered the Hilfer-type fractional operator to study the proposed integrodifferential equation. We have used the capabilities of measure theory and fixed point techniques to provide the required space to guarantee the existence of the solution. The Schauder and Arzela-Ascoli theorems play a fundamental role in the existence of solutions. Finally, we provided two examples with some graphical and numerical simulation to make our results more objective.</p></abstract>

Spectral Collocation Methods for Fractional Integro-Differential Equations with Weakly Singular Kernels

In this paper, we propose and analyze a spectral approximation for the numerical solutions of fractional integro-differential equations with weakly kernels. First, the original equations are transformed into an equivalent weakly singular Volterra integral equation, which possesses nonsmooth solutions. To eliminate the singularity of the solution, we introduce some suitable smoothing transformations, and then use Jacobi spectral collocation method to approximate the resulting equation. Later, the spectral accuracy of the proposed method is investigated in the infinity norm and weighted L 2 norm. Finally, some numerical examples are considered to verify the obtained theoretical results.

Gegenbauer wavelet solutions of fractional integro-differential equations

The aim of this study is to use Gegenbauer wavelets in the solution of fractional integro-differential equations. The method is applied to several problems with different values of resolution parameter and the degree of the truncated polynomial. The results are compared with those obtained from other numerical methods. We observe that the current method is very effective and gives accurate results. One of the reasons for that is it enables us to improve accuracy by increasing resolution parameter, while keeping the degree of polynomial fixed. Another reason is nonlinear terms do not require linearization. Hence the method can be directly implemented and results in the system of algebraic equations which solved by Wolfram Mathematica. It can be asserted that this is the first application of the Gegenbauer wavelet method to the aforementioned types of problems.

Existence and Uniqueness of Solutions for Fractional Integro-Differential Equations Involving the Hadamard Derivatives

In this paper, we study the existence and uniqueness of solutions for the following fractional boundary value problem, consisting of the Hadamard fractional derivative: HDαx(t)=Af(t,x(t))+∑i=1kCiHIβigi(t,x(t)),t∈(1,e), supplemented with fractional Hadamard boundary conditions: HDξx(1)=0,HDξx(e)=aHDα−ξ−12(HDξx(t))|t=δ,δ∈(1,e), where 1<α≤2, 0<ξ≤12, a∈(0,∞), 1<α−ξ<2, 0<βi<1, A,Ci, 1≤i≤k, are real constants, HDα is the Hadamard fractional derivative of order α and HIβi is the Hadamard fractional integral of order βi. By using some fixed point theorems, existence and uniqueness results are obtained. Finally, an example is given for demonstration.

Analyticity for solution of fractional integro-differential equations

We prove that for a certain class of kernels K(y) viscosity solutions of the integro-differential equation ∫Rn(u(x+y)−2u(x)+u(x−y))K(y)dy=f(x,u(x)) are locally analytic if f is an analytic function. This extends results in Albanese et al. (2015) in which it was shown that such solutions belong to certain Gevrey classes.

Existence and Uniqueness of Solutions for Nonlinear Fractional Integro-Differential Equations with Nonlocal Boundary Conditions

In this paper, the existence and uniqueness of solutions of fractional integro-differential equations with nonlocal boundary conditions is investigated. We establish the existence of solution via Krasnoselskii fixed point theorem; however, the uniqueness results are obtained by applying the contraction mapping principle.

On the solutions of fractional integro-differential equations involving Ulam-Hyers-Rassias stability results via $\psi$-fractional derivative with boundary value conditions

In this paper, we study boundary value problems for the impulsive integro-differential equations via $\psi$-fractional derivative. The contraction mapping concept and Schaefer's fixed point theorem are used to produce the main results. The results reported here are more general than those found in the literature, and some special cases are presented. Furthermore, we discuss the Ulam-Hyers-Rassias stability of the solution to the proposed system.

A new approach for the approximate solution of fractional integro-differential equations

A new approach for the approximate solution of fractional integro-differential equations

New Existence of Solutions for Fractional Integro-Differential Equations with Nonseparated Boundary Conditions

Results reported in this article prove the existence and uniqueness of solutions for a class of nonlinear fractional integro-differential equations supplemented by nonseparated boundary value conditions. We consider a new norm to establish the existence of solution via Krasnoselskii fixed point theorem; however, the uniqueness results are obtained by applying the contraction mapping principle. Some examples are provided to illustrate the results.

Numerical solution of fractional integro-differential equations using Hermite polynomials

Numerical solution of fractional integro-differential equations using Hermite polynomials

Numerical Solution of Fractional Integro-differential Equations with Weakly Singular Kernels via Bernstein Polynomial

This paper is concerned with obtaining approximate numerical solutions of a class of fractional Volterra integro-differential equations with weakly singular kernels which using Bernstein Polynomials as basis in collocation methods(BPCM). The approximate solution is obtained by expanding the functions in terms of Bernstein Polynomial, whose unknown coefficients are determined by solving final system of linear equations. The convergence analysis of the proposed method are taken into consideration and Numerical examples are included to confirm the efficiency and accuracy of the method.

Existence and Uniqueness of Solutions for Fractional Integro-Differential Equations and Their Numerical Solutions

In this paper, we investigate the existence and uniqueness of solution for a multi-term fractional integro-differential problem with nonlocal four-point fractional boundary conditions via the Caputo differentiation. We obtain operational matrix of Riemann–Liouville fractional integral operator of Bernstein polynomials and investigate the numerical solutions of the problem by using the collocation method. By appling these matrices fractional integro-differential equations convert to a linear system of equations. In this way, we give some examples to illustrate our results. The numerical method is computer oriented and produces very accurate and stable numerical results.

Formula omitted]-spline for solution of second order fractional integro-differential equations

We construct the two unique piecewise C3-splines of degree six and eight to approximate the solution of fractional integro-differential equations. Existence and uniqueness of the solution of second order fractional integro-differential equation has been proved. Convergence analysis of the presented methods have been discussed. Illustrative examples are included to demonstrate the validity and applicability of the presented approaches.

Vieta–Fibonacci operational matrices for spectral solutions of variable-order fractional integro-differential equations

In this paper, we formulate a numerical method to find out the approximate solution for fractional integro-differential equations of variable order (FIDE-VO). The methodology that adopted here is converting the FIDE-VO problem into a system of ordinary differential equations and that has been transformed into a system of algebraic equations in the unknown coefficients. For this purpose, the shifted Vieta–Fibonacci polynomials will be used for constructing both new fractional variable-order operational matrices of differentiation and integration. The variable order operators of differentiation and integration will be used in the Caputo and Riemann–Liouville senses, respectively. The Tau method and the constructed operational matrices will be used at the collocation points for transforming the FIDE-VO into an algebraic system of equations that will be solved numerically. At the end, the applicability and accuracy of the recommended method will be demonstrated through some numerical applications.

An efficient algorithm for numerical solution of fractional integro-differential equations via Haar wavelet

In this paper, Haar wavelet collocation technique is developed for the solution of Volterra and Volterra–Fredholm fractional integro-differential equations. The Haar technique reduces the given equations to a system of linear algebraic equations. The derived system is then solved by Gauss elimination method. Some numerical examples are taken from literature for checking the validation and convergence of the proposed method. The maximum absolute errors are compared with the exact solution. The maximum absolute and mean square root errors for different number of collocation points are calculated. The results show that Haar method is efficient for solving these equations. The experimental rates of convergence for different number of collocation point is calculated which is approximately equal to 2. Fractional derivative is described in the Caputo sense. All algorithms for the developed method are implemented in MATLAB (R2009b) software.

A Spectral Method Based on Hahn Polynomials for Numerical Solution of Fractional Integro-Differential Equations with Weakly Singular Kernel

A Spectral Method Based on Hahn Polynomials for Numerical Solution of Fractional Integro-Differential Equations with Weakly Singular Kernel