Existence and Uniqueness Theorems for Fractional Volterra-Fredholm Integro-Differential Equations

Abstract

In this article, the homotopy perturbation method has been successfully applied to find the approximate solution of a Caputo fractional Volterra-Fredholm integro-differential equation. The reliability of the method and reduction in the size of the computational work give this method a wider applicability. Also, the behavior of the solution can be formally determined by the analytical approximate. Moreover, we proved the existence and uniqueness results of the solution. Finally, an example is included to demonstrate the validity and applicability of the proposed technique.

Laplace discrete decomposition method for solving nonlinear Volterra-Fredholm integro-differential equations

“…The integro-differential equations have attracted much more interest of mathematicians and physicists which provides an efficiency for the description of many practical dynamical arising in engineering and scientific disciplines such as, physics, biology, electrochemistry, chemistry, economy, electromagnetic, control theory and viscoelasticity [2,3,5,12,13,15,16,19,[21][22][23]. In recent years, many authors focus on the development of numerical and analytical techniques for fractional integro-differential equations.…”

“…In addition to these numerical methods, Khuri [29] used Laplace transform numerical scheme. Moreover, properties of the integro-differential equations have been studied by several authors [1,10,11,14,17,18,20,[24][25][26][27]33].…”

On the solution of time-fractional coupled system of partial differential equations

“…Later RDTM was introduced in order to overcome the complex calculation of DTM. One may see the references [25][26][27][28] for different methods and their application on fractional calculus. This paper deals with the implementation of a method which is the amalgamation of reduced differential transform method, Laplace transform method, and PA [29] to obtain the exact solutions of PDEs [30][31][32][33][34].…”

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

“…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.…”