Lazhar Bougoffa

A convenient technique for solving linear and nonlinear Abel integral equations by the Adomian decomposition method

Abstract In this paper, linear and nonlinear Abel integral equations are transformed in such a manner that the Adomian decomposition method can be applied. Some examples with closed-form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate this approach is indeed practical and efficient.

An approximate method for solving a class of weakly-singular Volterra integro-differential equations

Abstract In this paper, we present a new approach to resolve linear and nonlinear weakly-singular Volterra integro-differential equations of first- or second-order by first removing the singularity using Taylor’s approximation and then transforming the given first- or second-order integro-differential equations into an ordinary differential equation such as the well-known Legendre, degenerate hypergeometric, Euler or Abel equations in such a manner that Adomian’s asymptotic decomposition method can be applied, which permits convenient resolution of these equations. Some examples with closed-form solutions are studied in detail to further illustrate the proposed technique, and the results obtained demonstrate this approach is indeed practical and efficient.

Adomian method for solving some coupled systems of two equations

Coupled systems of two linear and nonlinear differential equations for second- and first-orders, respectively, can be solved with some techniques and Adomian decomposition method. A few simple examples are also studied to show with analytical results how the (ADM) works efficiently.

Adomian method for a class of hyperbolic equations with an integral condition

Abstract In this paper Adomian Decomposition Method (ADM) is applied to the solvability of a boundary problem with a nonlocal condition for a class of hyperbolic equations and illustrated with a few simple examples.

A convenient technique for solving integral equations of the first kind by the Adomian decomposition method

Purpose – In this paper, Fredholm integral equations of the first kind, the Schlomilch integral equation, and a class of related integral equations of the first kind with constant limits of integration are transformed in such a manner that the Adomian decomposition method (ADM) can be applied. Some examples with closed‐form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate this approach is indeed practical and efficient. The purpose of this paper is to develop a new iterative procedure where the integral equations of the first kind are recast into a canonical form suitable for the ADM. Hence it examines how this new procedure provides the exact solution.Design/methodology/approach – The new technique, as presented in this paper in extending the applicability of the ADM, has been shown to be very efficient for solving Fredholm integral equations of the first kind, the Schlomilch integral equation and a related class of nonlinear integral equatio...

A convergence analysis of the Adomian decomposition method for an abstract Cauchy problem of a system of first-order nonlinear differential equations

In this article, we study some fundamental results concerning the convergence of the Adomian decomposition method (ADM) for an abstract Cauchy problem of a system of first-order nonlinear differential equations. Under certain conditions, we obtain upper estimates for the norm of solutions of this system. We also obtain results about the error estimates for the approximate solutions by the ADM and discuss their applications.

Solvability of the predator and prey system with variable coefficients and comparison of the results with modified decomposition

The proof of a new result concerning the conditions for solvability of predator and prey system with variable coefficients is presented and illustrated with a few simple examples and comparison of the results obtained by the present method with that obtained by modified decomposition method.

On the exact solutions for initial value problems of second-order differential equations

In this paper, the solutions of initial value problems for a class of second-order linear differential equations are obtained in the exact form by writing the equations in the general operator form and finding an inverse differential operator for this general operator form.

Note on the explicit solutions for a boundary value problem of a second-order nonlinear equation

Abstract In this note, we consider a boundary value problem for a second-order nonlinear equation: d 2 y d x 2 = λ ( x ) e μ ( x ) y , y ( 0 ) = y ( 1 ) = 0 , where μ ( x ) > 0 and λ ( x ) are two functions satisfying certain conditions. The explicit solutions to this problem are obtained.

A Convergence Analysis of an Iterative Method for a Coupled Nonlinear System of Functional Equations

In this paper, we study some fundamental results concerning the convergence of an iterative method for the coupled nonlinear system of functional equations and discuss their applications.