Abstract
In this paper, using the implicit trapezoidal rule in conjunction with Newton's method to solve nonlinear system.We have used a Maple 17 program to solve the System of two nonlinear Volterra integral equations. Finally, several illustrative examples are presented to show the effectiveness and accuracy of this method.
Highlights
In this paper, we consider the Volterra integral equation of the second kind ( ) ( ) ∫ ( ( )) (1)Where and are vector-valued functions with components
We present the computation of numerical solution of system of two nonlinear Volterra integral equation of the second kind
If ( ) is an by matrix-valued function that is invertible in a neighborhood of, is a fixed point of Assuming the components of ( ) have continuous first and second order partial derivatives and that the first order partial derivatives and that the first order partial derivatives at are equal to zero, it can be shown that if ( ) is set equal to the Jacobian matrix of the function, the iterates ( )
Summary
We consider the Volterra integral equation of the second kind ( ) ( ) ∫ ( ( )). If and are continuous and ( Lipcshitz condition with respect to , a unique solution ( ) of (1) exists[1,4,7]. We present the computation of numerical solution of system of two nonlinear Volterra integral equation of the second kind. Theorem 1.Consider the equation () () ∫ ( ) ( ) (). Are continuous functions of , 4) ( ) is absolutely integrable with respect to for all. (2) has a unique continuous solution in Theorem 2.Consider the equation ( ) ( ) ∫ ( ) ( ( )). (3) has a unique continuous solution in replaced by
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