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

Abstract

This paper demonstrates a study on some significant latest innovations in the approximation techniques to find the approximate solutions of Caputo fractional Volterra-Fredholm integro-differential equations. To apply this, the study uses Adomian decomposition method and modified Laplace Adomian decomposition method. A wider applicability of these techniques is based on their reliability and reduction in the size of the computational work. This study provides analytical approximate to determine the behavior of the solution. It proves the existence and uniqueness results and convergence of the solution. In addition, it brings an example to examine the validity and applicability of the proposed techniques.

Usage of the Modified Variational Iteration Technique for Solving Fredholm Integro-Differential Equations

“…In this section, we shall give an existence and uniqueness results of Equation (1.1), with the initial condition (1.2) and prove it [6][7][8]. We can write the equation (1.1) in the form of: Before starting and proving the main results, we introduce the following hypotheses:…”

“…In recent years, there has been a growing interest in the integro-differential equations, which are a combination of differential and integral equations. The nonlinear Fredholm integro-differential equations play an important role in many branches of nonlinear functional analysis and their applications in the theory of engineering, mechanics, physics, electrostatics, biology, chemistry and economics [1][2][3][4][5][6][22][23][24][25][26][27][28][29][30]. In this paper, we consider the Fredholm integro-differential equations of the type: where Z (j) (x) is the j th derivative of the unknown function Z(x) that will be determined, K(x,t) is the kernel of the equation, f(x) and ξ j (x) are an analytic function, G is nonlinear function of Z and a, b, γ, and b are real finite constants.…”

“…Recently, many authors focus on the development of numerical and analytical techniques for integrodifferential equations. For instance, we can remember the following works: Abbasbandy and Elyas [2] studied some applications on variational iteration method for solving system of nonlinear Volterra integro-differential equations, Hamoud and Ghadle [6] applied the hybrid methods for solving nonlinear Volterra-Fredholm integrodifferential equations, Alao et al [4] used Adomian decomposition and variational iteration methods for solving integro-differential equations, Yang and Hou [21] applied the Laplace decomposition method to solve the…”

Solving Fractional Fredholm Integro-Differential Equations by Laguerre Polynomials

“…The fractional calculus represents a powerful tool in applied mathematics to study a myriad of problems from different fields of science and engineering, with many break-through results found in mathematical physics, finance, hydrology, biophysics, thermodynamics, control theory, statistical mechanics, astrophysics, cosmology and bioengineering (Abbas et al 2015). Since the fractional calculus has attracted much more interest among mathematicians and other scientists, the solutions of the fractional integrodifferential equations have been studied frequently in recent years (Alkan & Hatipoglu 2017;Hamoud & Ghadle 2018a, 2018bIbrahim et al 2015;Kumar et al 2017;Ma & Huang 2014;Nemati et al 2016;Ordokhani & Dehestani 2016;Parand & Nikarya 2014;Pedas et al 2016;Shahooth et al 2016;Turmetov & Abdullaev 2017;Wang & Zhu 2016;Yi et al 2016). The methods that are used to find the solutions of the linear fractional Fredholm integro-differential equations are given as fractional pseudospectral integration matrices (Tang & Xu 2016), least squares with shifted Chebyshev polynomials Mohammed 2014), least squares method using Bernstein polynomials (Oyedepo et al 2016), fractional residual power series method (Syam 2017), Taylor matrix method (Gülsu et al 2013), reproducing kernel Hilbert space method (Bushnaq et al 2016), second kind Chebyshev wavelet method (Setia et al 2014), open Newton method (Al-Jamal & Rawashdeh 2009), modified Homotopy perturbation method (Elbeleze et al 2016), Sinc collocation method (Emiroglu 2015).…”

Local radial basis function scheme for solving a class of fractional integro‐differential equations based on the use of mixed integral equations

“…The fractional integro-differential equations have attracted much more interest of mathematicians and physicists because they provide an efficiency for the description of many practical dynamical problems arising in engineering and scientific disciplines such as, physics, biology, electrochemistry, chemistry, economy, electromagnetic, control theory and viscoelasticity. [2,33,55,56] Several methods have been proposed to approximate the solution of fractional integro-differential equations with various types of conditions. Taylor expansion approach has been utilized for numerically solving a class of linear fractional integro-differential equations in [36].…”