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Volume 6, Issue 4, August 2018, Page: 142-148
The Adomian Decomposition Method of Volterra Integral Equation of Second Kind
Ali Elhrary Abaoub, Mathematics Department, Science Faculty, Sabratha University, Sabratha, Libya
Abejela Salem Shkheam, Mathematics Department, Science Faculty, Sabratha University, Sabratha, Libya
Suad Mawloud Zali, Mathematics Department, Science Faculty, Sabratha University, Sabratha, Libya
Received: Aug. 23, 2018;       Accepted: Sep. 20, 2018;       Published: Nov. 1, 2018
Abstract
In this work, we consider linear and nonlinear Volterra integral equations of the second kind. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The Adomian decomposition method or shortly (ADM) is used to find a solution to these equations. The Adomian decomposition method converts the Volterra integral equations into determination of computable components. The existence and uniqueness of solutions of linear (or nonlinear) Volterra integral equations of the second kind are expressed by theorems. If an exact solution exists for the problem, then the obtained series convergence very rabidly to that solution. A nonlinear term F(u) in nonlinear volterra integral equations is Lipschitz continuous and has polynomial representation. Finally, the sufficient condition that guarantees a unique solution of Volterra (linear and nonlinear) integral equations with the choice of the initial data is obtained, and the solution is found in series form. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method. A few examples including linear and nonlinear are provided to show validity and applicability of this approach. The results are taken from the works mentioned in the reference.
Keywords
Linear Integral Equations, Fredholm Integral Equations, Regularization Method, Direct Computation Method, Two - Dimensional Integral Equations
Ali Elhrary Abaoub, Abejela Salem Shkheam, Suad Mawloud Zali, The Adomian Decomposition Method of Volterra Integral Equation of Second Kind, American Journal of Applied Mathematics. Vol. 6, No. 4, 2018, pp. 142-148. doi: 10.11648/j.ajam.20180604.12
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