The Approximate Solutions of Fractional Volterra-Fredholm Integro-Differential Equations by using Analytical Techniques

Ahmed Hamoud,Kirtiwant Ghadle

doi:10.15393/j3.art.2018.4350

Abstract

The Approximate Solutions of Fractional Volterra-Fredholm Integro-Differential Equations by using Analytical Techniques

Highlights

IntroductionWe consider the Caputo fractional Volterra – Fredholm integro-differential equations of the form: x cDαy(x) = g(x) + a(x)y(x) + K1(x, t)F1(y(t))dt + K2(x, t)F2(y(t))dt, with the initial conditions (1)
In this paper, we consider the Caputo fractional Volterra – Fredholm integro-differential equations of the form: x cDαy(x) = g(x) + a(x)y(x) + K1(x, t)F1(y(t))dt + K2(x, t)F2(y(t))dt, with the initial conditions (1)y(i)(0) = δi, i = 0, 1, · · ·, n − 1, (2)c Petrozavodsk State University, 2018where n − 1 < α ≤ n, n ∈ N, y : [0, 1] −→ R is the continuous function to be determined, g, a : [0, 1] −→ R and Ki : [0, 1] × [0, 1] −→ R, i = 1, 2, are continuous functions
The obtained results reveal that this method is very effective, Momani [19] and Qaralleh [20] applied Adomian polynomials to solve fractional integro-differential equations and systems of fractional integro-differential equations, Kadem and Kilicman [15] utilized the HPM and VIM methods for integrodifferential equation of fractional order with initial-boundary conditions, Yang [21] used the hybrid of block pulse function and Chebyshev polynomials to solve nonlinear Fredholm fractional integro-differential equations, Yang and Hou [22] applied the Laplace decomposition method to solve the fractional integro-differential equations, Mittal and Nigam [18] utilized the Adomian decomposition method to approximate solutions of fractional integro-differential equations, and Ma and Huang [17] applied hybrid collocation method to study integro-differential equations of fractional order

Summary

Introduction

We consider the Caputo fractional Volterra – Fredholm integro-differential equations of the form: x cDαy(x) = g(x) + a(x)y(x) + K1(x, t)F1(y(t))dt + K2(x, t)F2(y(t))dt, with the initial conditions (1). The obtained results reveal that this method is very effective, Momani [19] and Qaralleh [20] applied Adomian polynomials to solve fractional integro-differential equations and systems of fractional integro-differential equations, Kadem and Kilicman [15] utilized the HPM and VIM methods for integrodifferential equation of fractional order with initial-boundary conditions, Yang [21] used the hybrid of block pulse function and Chebyshev polynomials to solve nonlinear Fredholm fractional integro-differential equations, Yang and Hou [22] applied the Laplace decomposition method to solve the fractional integro-differential equations, Mittal and Nigam [18] utilized the Adomian decomposition method to approximate solutions of fractional integro-differential equations, and Ma and Huang [17] applied hybrid collocation method to study integro-differential equations of fractional order. Properties of the fractional integro-differential equations have been studied by several authors [4, 12, 16]

Objectives

Conclusion