In the research, nonlinear volterra partial integro-differential equation is considered. This paper compares the Homotopy Perturbation Method (HPM) with Variational Iteration Method (VIM) for solving this equation. Compared with the Adomian Decomposition Method (ADM), the methods used for this equation need less work. The results of applying these methods show the simplicity and efficiency of t...

in this paper, we prove the existence and uniqueness results to the random fractional functional differential equations under assumptions more general than the lipschitz type condition. moreover, the distance between exact solution and appropriate solution, and the existence extremal solution of the problem is also considered.

Abstract: In this paper, a numerical method to solve nonlinear Fredholm integral equations of second kind is proposed and some numerical notes about this method are addressed. The method utilizes Chebyshev wavelets constructed on the unit interval as a basis in the Galerkin method. This approach reduces this type of integral equation to solve a nonlinear system of algebraic equation. The method...

The practical stability of a nonlinear nonautonomous Caputo fractional differential equation is studied using Lyapunov like functions. The novelty of this paper is based on the new definition of the derivative of a Lyapunov like function along the given fractional differential equation. Comparison results using this definition for scalar fractional differential equations are presented. Several ...

In this paper, a new numerical method for solving the fractional Riccati differential  equation is presented. The fractional derivatives are described in the Caputo sense. The method is based upon  fractional-order Bernoulli functions approximations. First, the  fractional-order Bernoulli functions and  their properties are  presented. Then, an operational matrix of fractional order integration...

In the recent paper [2], the authors obtained new proofs on the existence and uniqueness of the solution of the Volterra linear equation. Applying their results, in this paper we express the exact and approximate solution of the equation in the field of Mikusi´nski operators, F, which corresponds to an integro–differential equation.

This paper studies a fractional differential equation combined with Liouville–Caputo operator, namely, LCDηβ,γQ(t)=λϑ(t,Q(t)),t∈[c,d],β,γ∈(0,1],η∈[0,1], where Q(c)=qc is bounded and non-negative initial value. The function ϑ:[c,d]×R→R Lipschitz continuous in the second variable, λ>0 constant operator LCDηβ,γ convex combination of left right derivatives. We study well-posedness using fixed-po...

In this paper, a numerical method is proposed to solve FredholmVolterra fractional integro-differential equation with nonlocal boundary conditions. For this purpose, the Chebyshev wavelets of second kind are used in collocation method. It reduces the given fractional integro-differential equation (FIDE) with nonlocal boundary conditions in a linear system of equations which one can solve easily...

A collocation procedure is developed for the linear and nonlinear Fredholm and Volterra integro-differential equations, using the globally defined B-spline and auxiliary basis functions.The solution is collocated by cubic B-spline and the integrand is approximated by the Newton-Cotes formula. The error analysis of proposed numerical method is studied theoretically. Numerical results are given t...

The Tau method, produces approximate polynomial solution of differential, integral and integro-differential equations (see [E.l,Ortiz, The Tau method, SIAM J. Numer. Anal. 6 (3) (1969) 480–492; E.l. Ortiz, H. Samara, An operational approach to the Tau method for the numerical solution of non-linear differential equations, Computing 27 (1981) 15–25; S.M. Hosseini, S. Shahmorad, A matrix formulat...