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.
Highlights
Many problems from physics and other disciplines lead to linear and nonlinear integral equations
The most standard form of Volterra linear integral equations are of the form φ x u x = f x + λ k x, t u t dt
If the differential equation is linear, we are led in this way to a linear integral equation of the second kind
Summary
Many problems from physics and other disciplines lead to linear and nonlinear integral equations. The solution is found as infinite series which convergence rapidly to accurate solutions This method has proven successful in dealing with linear as well as nonlinear problems, as it yields analytical solutions and offers certain advantages over standered numerical solutions. Where k x, t is called the kernel of integral equation, λ 0 the parameter of integral equation, α x and β x are the limits of integration. Definition Volterra integral equations are written in a form (1) where the upper limit of integration β x = x (independent variable). The most standard form of Volterra linear integral equations are of the form φ x u x = f x + λ k x, t u t dt
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