Solution Of Fractional Integro-differential Equations Research Articles

Analytic solutions of fractional integro-differential equations of Volterra type with variable coefficients

Abstract In this paper we study explicit solutions of fractional integro-differential equations with variable coefficients involving Prabhakar-type operators. Analytic solutions to equations involving Prabhakar operators and Laguerre derivatives are obtained by means of operational methods. Cauchy type problems for fractional integro-differential equations of Volterra type with generalized Riemann-Liouville derivative operator, which contain generalized Mittag-Leffler function in the kernel are also considered. Using the Laplace transform method, explicit solutions of the fractional integro-differential equations of Volterra type with variable coefficients, proposed by Srivastava and Tomovski in [28] are established in terms of the multinomial Wright function.

Comparison of Adomian Decomposition and Taylor Expansion Methods for the Solutions of Fractional Integro-Differential Equations

In this paper will be compared between Adomian decomposition method (ADM) and Taylor expansion method (TEM) for solving (approximately) a class of fractional integro-differential equations. Numerical examples are presented to illustrate the efficiency and accuracy of the proposed methods. General Terms Numerical solutions, Fractional integro-differential equations.

Numerical solution of fractional integro-differential equations by a hybrid collocation method

Fractional integro-differential equations have been recently solved by many methods, such as Adomian decomposition method, differential transform method, collocation method and Taylor expansion approach. In this paper a hybrid collocation method is used which combines a non-polynomial collocation used on the first subinterval and graded piecewise polynomial collocation used on the rest of the interval. A theoretical analysis for the convergence order of the method is presented. Some numerical examples are given which confirm the theoretical results.

Approximate solution of fractional integro-differential equation by Taylor expansion and Legendre wavelets methods

This paper, deals with the approximate solution of fractional integro-differential equations of the type by Taylor expansion and Legendre wavelet methods. In addition, illustrative example are presented to demonstrate the efficiency and accuracy of this methods.

On convergence of homotopy analysis method and its application to fractional integro-differential equations

In this paper, we have used the homotopy analysis method (HAM) to obtain approximate solution of fractional integro-differential equations (FIDEs). Convergence of HAM is considered for this kind of equations. Also some examples are given to illustrate the high efficiency and precision of HAM.

Numerical Solutions of Fractional Integrodifferential Equations of Bratu Type by Using CAS Wavelets

A numerical method based on the CAS wavelets is presented for the fractional integrodifferential equations of Bratu type. The CAS wavelets operational matrix of fractional order integration is derived. A truncated CAS wavelets series together with this operational matrix is utilized to reduce the fractional integrodifferential equations to a system of algebraic equations. The solution of this system gives the approximation solution for the truncated limited2k(2M+1). The convergence and error estimation of CAS wavelets are also given. Two examples are included to demonstrate the validity and applicability of the approach.

Existence and uniqueness of mild solution for fractional integrodifferential equations of neutral type with nonlocal conditions

Abstract In this paper, we prove the existence and uniqueness of mild solution of a class of nonlinear fractional integrodifferential equations of neutral type with nonlocal conditions in a Banach space. New results are obtained by fixed point theorem.

Approximate Solution of Fractional Integro-Differential Equations by Using Bernstein Polynomials

Approximate Solution of Fractional Integro-Differential Equations by Using Bernstein Polynomials

A numerical scheme for the solution of fractional integro-differential equations using the Adomian decomposition method.

A numerical scheme for the solution of fractional integro-differential equations using the Adomian decomposition method.

Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions

This paper is concerned with the fractional integrodifferential equations of Sobolev type with nonlocal condition in a separable Banach space. With the help of the theory of propagation family as well as the theory of the measures of noncompactness and the condensing maps, we obtain the existence result of mild solutions for above equations. Two examples are given to illustrate the existence theorem.

Existence of the Mild Solutions for Delay Fractional Integrodifferential Equations with Almost Sectorial Operators

This paper is concerned with the existence of mild solutions for the fractional integrodifferential equations with finite delay and almost sectorial operators in a separable Banach space X. We obtain existence theorem for mild solutions to the above‐mentioned equations, by means of measure of noncompactness and the resolvent operators associated with almost sectorial operators. As an application, the existence of mild solutions for some integrodifferential equation is obtained.

On the existence of solutions of fractional integro-differential equations

Abstract Under some suitable conditions, we prove the solvability of a large class of nonlinear fractional integro-differential equations by establishing some fractional integral inequalities and using the nonlinear alternative Leray-Schauder type. The uniqueness of solutions is also proved in some situations.

Existence results for fractional integrodifferential equations with nonlocal condition via resolvent operators

In this paper, we prove the existence of solutions of fractional integrodifferential equations by using the resolvent operators and fixed point theorem. An example is given to illustrate the abstract results.

Approximate solution of fractional integro-differential equations by Taylor expansion method

In this paper, Taylor expansion approach is presented for solving (approximately) a class of linear fractional integro-differential equations including those of Fredholm and of Volterra types. By means of the m th-order Taylor expansion of the unknown function at an arbitrary point, the linear fractional integro-differential equation can be converted approximately to a system of equations for the unknown function itself and its m derivatives under initial conditions. This method gives a simple and closed form solution for a linear fractional integro-differential equation. In addition, illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method.

Optimal mild solutions and weighted pseudo-almost periodic classical solutions of fractional integro-differential equations

In this paper, we investigate a class of fractional integro-differential equations given by d α x ( t ) d t α = A x ( t ) + f ( t , x ( t ) , G x ( t ) ) . Our main results concern the existence, uniqueness of optimal mild solutions and weighted pseudo-almost periodic classical solutions. An example is given to illustrate our results.

Existence and Uniqueness of Mild Solution for Fractional Integrodifferential Equations

Existence and Uniqueness of Mild Solution for Fractional Integrodifferential Equations

Existence and Uniqueness of Mild Solution for Fractional Integrodifferential Equations

We study the existence and uniqueness of mild solution of a class of nonlinear fractional integrodifferential equations , , , in a Banach space , where . New results are obtained by fixed point theorem. An application of the abstract results is also given.

Solution of fractional integro-differential equations by using fractional differential transform method

In this study, fractional differential transform method (FDTM), which is a semi analytical numerical technique, is extended to solve fractional integro-differential equations of Volterra type. New theorems for the transformation of integral terms having degenerate kernels that never existed before are introduced with their proofs. This implemented new technique is validated by solving and comparing four different examples that exist in the literature. It is observed that, FDTM can be utilized as a powerful and reliable tool for the solution of fractional integro-differential equations.

Numerical solution of fractional integro-differential equations by collocation method

This paper deals with the numerical solution of fractional integro-differential equations of the type D q y ( t ) = p ( t ) y ( t ) + f ( t ) + ∫ 0 t K ( t , s ) y ( s ) d s , t ∈ I = [ 0 , 1 ] by polynomial spline functions. We derive a system of equations that characterizing the numerical solution. Some numerical examples are also provided to illustrated our results.

Lyapunov stability solutions of fractional integrodifferential equations

Lyapunov stability and asymptotic stability conditions for the solutions of the fractional integrodiffrential equations , 0 < α ≤ 1, with the initial condition x(α−1)(t0) = x0, have been investigated. Our methods are applications of Gronwall′s lemma and Schwartz inequality.