High Order Numerical Solution of a Volterra Integro - Differential Equation Arising in Oscillating Magnetic Fields using Variational Iteration Method

Abstract

In this paperwe have considered an integro-differential equation which describes the charged particle motion for certain configurations of oscillating magnetic fields. This equation contains variable coefficients with large expressions which complicate the application of any numerical method. We have used variational iteration method to find its numerical solution by developing MATHEMATICA modulae and solved a number of numerical examples. The results show high accuracy and efficiency ofour approach.

Numerical Solution of an Integro-Differential Equation Arising in Oscillating Magnetic Fields

“…In Figure 3(b) to show the convergence of the present method, we showed that by increasing the m the residual function decreases, where α = 0.50. Table 2 compares the obtained values of y(t) by the present method and the values given by Khan [18] (Legendre multi-wavelets) and Pathak [21] (Variational iteration method), it shows that the results obtained in the present method are more accurate. …”

“…Table 1 compares the error norm y − y m 2 by the present method and Dehghan [16] for examples 1-3. Table 3 compares the obtained values of y(t) by the present method and the values given by Khan [18] (Legendre multi-wavelets) and Pathak [21] (Variational iteration method), it shows that the results obtained in the present method are more accurate. The exact solution of this equation is y(t) = −t 3 + t 2 − 5t + 2.…”

“…The present method is applied to solve the integro-differential equation (2.1) and their outputs are compared with the corresponding analytical solution. These examples studied also by Dehghan [16], Khan [18] and Pathak [21], we will compare our results with their results to show the effectiveness of the present method. Example 1.…”

“…There are methods to solve this equation, such as, He's Homotopy perturbation method [16], Chebyshev wavelet [17], Legendre multi-wavelets [18], Local polynomial regression [19], Shannon wavelets [20], Variational iteration method [21] and Homotopy analysis method [22]. In this paper, we attempt to introduce a new method based on the generalized fractional order Chebyshev orthogonal functions (GFCFs) of the first kind for solving the equation (2.1) with the initial conditions (2.8).…”

“…Example 1. Consider the equation (2.1) with [16,18,21] w p = 2, a(t) = cos(t), b(t) = sin(t/2), β 0 = 1, β 1 = 0…”