This series has the closed form as
which is exactly the exact solution for the case 3.1.1.
In Table 1 show a comparison of the numerical results applying the HAM (
), Iteration of the Integral Equation (IIE) (3.9), and the numerical solution of (3.9) with Simpson rule (SIMP) with the exact solution (3.13). Twenty points have been used in the Simpson rule. In Table 2 we list the Maximum Absolute Error (MAE),
, the Maximum Relative Error (MRE), the Maximum Residual Error (MRR), obtained by the HAM with the exact solution (3.13) on the interval
. The Estimated Order of Convergence (EOC) for different values of the constant k are given in Table 3.
Figure 1 represents both the exact solution
and our approximation by HAM (
) within the interval
, the application of the HAM requires approximants of order
if we want to arrive beyond the discontinuity (at
Case 3.1.2 Taking
The Heaviside step function at
. We now successively obtain
In this work, the HAM has been successfully applied to solve IVPs of second order with discontinuities. The size of the jump (given by
) does not affect the convergence of the method, which behaves equally well on both sides of the discontinuity. In this IVPs, the application by the HAM with k, does not converge even for small values of the parameter like
The proposed scheme of the HAM has been applied directly without any need for transformation formulae or restrictive assumptions. The solution process by the HAM is compatible with the method in the literature providing analytical approximation such as ADM. The approach of the HAM has been tested by employing the method to obtain approximate-exact solutions of the linear case. The results obtained in all cases demonstrate the reliability and the efficiency of this method.
Cite this paper
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