Nonlinear Fractional Integro-differential Equations Research Articles

On nonlinear mixed fractional integrodifferential equations with nonlocal condition in Banach spaces

Abstract In the present paper we investigate the existence and uniqueness of solutions of nonlinear mixed fractional integrodifferential equations with nonlocal condition in Banach spaces. The technique used in our analysis is based on fixed point theorems and Pachpatte's integral inequality.

Legendre wavelets method for solving fractional integro-differential equations

In this paper, based on the constructed Legendre wavelets operational matrix of integration of fractional order, a numerical method for solving linear and nonlinear fractional integro-differential equations is proposed. By using the operational matrix, the linear and nonlinear fractional integro-differential equations are reduced to a system of algebraic equations which are solved through known numerical algorithms. The upper bound of the error of the Legendre wavelets expansion is investigated in Theorem 5.1. Finally, four numerical examples are shown to illustrate the efficiency and accuracy of the approach.

Numerical Approximation of Fractional Integro-differential Equations by an Iterative Decomposition Method

In this paper, an attempt is made to approximate the solution of linear and nonlinear fractional integro-differential equations, by applying an iterative decomposition method. The approximate solution of each problem is presented as a rapidly convergent series of easily computable terms. Results obtained are compared favourably with known results to illustrate the accuracy and efficiency of the method.

Existence and Uniqueness of Mild Solutions for Fractional Integrodifferential Equations

The aim of this paper is to prove the existence and uniqueness of mild solution of a class of l nonlinear fractional integrodifferential equations {█((d^q u(t))/(dt^q )+Au(t)=∫_0^t▒f(t,s,x(s) )ds+∫_0^t▒〖a(t-s)g(s,y(s) )ds, t∈[0,T],〗@u(0)=u_(o.) )┤ in a Banach space X, where 0<q<1. Results are obtained by fixed point theorem. The results are established by using Krasnoselskii’s fixed point theorem and the contraction mapping principle.

Numerical Solution of Nonlinear Fractional Volterra Integro-Differential Equations via Bernoulli Polynomials

This paper presents a computational approach for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on the Bernoulli polynomials approximation. Our method consists of reducing the main problems to the solution of algebraic equations systems by expanding the required approximate solutions as the linear combination of the Bernoulli polynomials. Several examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method.

On a Couple System of Integro-Differential Equations of Arbitrary Orders

This paper studies the existence and the uniqueness of solutions for a coupled system of nonlinear fractional integro-differential equations. New existence and uniqueness results are established using Banach theorem. Other results are obtained using Schaefer fixed point theorem. Some concrete examples are also presented.

Homotopy perturbation and Variational iteration methods for nonlinear fractional integro-differential equations

Homotopy perturbation and Variational iteration methods for nonlinear fractional integro-differential equations

On exponential stability of nonlinear fractional multidelay integro-differential equations defined by pairwise permutable matrices

In this paper, systems of nonlinear differential equations with Caputo fractional derivative and multiple delays are considered. Using representation of a solution of differential equation with multiple delays in the form of matrix polynomial and stability results such as Gronwall’s and Pinto’s inequality, sufficient conditions for the exponential stability of a trivial solution of nonlinear multidelay fractional differential equations are proved.

Existence and uniqueness of solutions for multi-term nonlinear fractional integro-differential equations

In this manuscript, by using the fixed point theorems, the existence and the uniqueness of solutions for multi-term nonlinear fractional integro-differential equations are reported. Two examples are presented to illustrate our results.

Neutral fractional integro-differential equation with nonlinear term depending on lower order derivative

By applying an iterative technique, sufficient conditions are obtained for the existence of the unique solution of the nonlinear neutral fractional integro-differential equation involving two Riemann–Liouville derivatives of different fractional orders. Finally, an example is also given to illustrate the availability of our main results.

A nonlocal integral boundary value problem of nonlinear integro-differential equations of fractional order

This study discussed the existence of solutions for a nonlinear fractional integro-differential equation of order with four-point nonlocal integral boundary conditions. The given problem is transformed to an equivalent fixed point problem in terms of an operator equation. Then, by means of Banach contraction principle and a fixed point theorem due to Krasnoselskii, the existence results are obtained. The last existence result is based on nonlinear alternative of Leray-Schauder type. An illustrative example is also presented. Key words: Integro-differential equations, fractional order, four-point integral boundary conditions, fixed point theorems, Leray-Schauder degree.

Application of the collocation method for solving nonlinear fractional integro-differential equations

In this paper, using the collocation method we solve the nonlinear fractional integro-differential equations (NFIDE) of the form: f(t,y(t),aCDtα0y(t),…,aCDtαry(t))=λG(t,y(t),∫atk(t,s)F(s,y(s))ds),y(k)(a)=dk,k=0,1,…,m0−1. We study the convergence and the stability analysis of this method for f(t,y(t),aCDtα0y(t),…,aCDtαry(t))=y(t)+∑j=0rbjaCDtαjy(t)+g(t). Some numerical examples are given to show the efficiency of the presented method.

On the approximate solutions for system of fractional integro-differential equations using Chebyshev pseudo-spectral method

In this paper, we implement Chebyshev pseudo-spectral method for solving numerically system of linear and non-linear fractional integro-differential equations of Volterra type. The proposed technique is based on the new derived formula of the Caputo fractional derivative. The suggested method reduces this type of systems to the solution of system of linear or non-linear algebraic equations. We give the convergence analysis and derive an upper bound of the error for the derived formula. To demonstrate the validity and applicability of the suggested method, some test examples are given. Also, we present a comparison with the previous work using the homotopy perturbation method.

Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations

This paper presents a computational method for solving a class of system of nonlinear singular fractional Volterra integro-differential equations. First, existences of a unique solution for under studying problem is proved. Then, shifted Chebyshev polynomials and their properties are employed to derive a general procedure for forming the operational matrix of fractional derivative for Chebyshev wavelets. The application of this operational matrix for solving mentioned problem is explained. In the next step, the error analysis of the proposed method is investigated. Finally, some examples are included for demonstrating the efficiency of the proposed method.

Nonlinear fractional integro-differential Langevin equation involving two fractional orders with three-point multi-term fractional integral boundary conditions

In this article, we study a nonlinear fractional integro-differential Langevin equation involving two fractional orders with three-point multi-term fractional integral boundary conditions. By using fixed point theorems and Leray-Schauder degree theory, some new existence results are obtained. Two examples illustrate our results.

Nonlinear fractional integro-differential equations on unbounded domains in a Banach space

In this paper, by employing the fixed point theory and the monotone iterative technique, we investigate the existence of a unique solution for a class of nonlinear fractional integro-differential equations on semi-infinite domains in a Banach space. An explicit iterative sequence for approximating the solution of the boundary value problem is derived. An error estimate is also given.

On the Solution of Linear and Nonlinear Fractional Integro-Differential Equations in Nano-Transistors

On the Solution of Linear and Nonlinear Fractional Integro-Differential Equations in Nano-Transistors

Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations

This paper is devoted to considering the existence and uniqueness results for fuzzy fractional integral equations employing the method of upper and lower solutions. Moreover, the approach is followed to prove the existence of solutions for the fuzzy initial value problem of fractional integrodifferential equations involving Riemann–Liouville differential operators. The method is illustrated by two examples.

Approximate Controllability of Fractional Integrodifferential Evolution Equations

This paper addresses the issue of approximate controllability for a class of control system which is represented by nonlinear fractional integrodifferential equations with nonlocal conditions. By using semigroup theory,p-mean continuity and fractional calculations, a set of sufficient conditions, are formulated and proved for the nonlinear fractional control systems. More precisely, the results are established under the assumption that the corresponding linear system is approximately controllable and functions satisfy non-Lipschitz conditions. The results generalize and improve some known results.

On nonlinear fractional integrodifferential equations with non local condition in Banach spaces

On nonlinear fractional integrodifferential equations with non local condition in Banach spaces