The Homotopy Perturbation Method to Solve Initial Value Problems of First Order with Discontinuities

Abstract


INTRODUCTION
Ji-Huan He (1998He ( , 1999 published the so-called HPM [1][2][3] six years after Shijun Liao (1992) suggested the earliest homotopy analysis method (HAM) in his Ph.D. thesis. The HPM, such as the earliest HAM, is constructed Based on a homotopy equation [ ] ( ) where ( ) is an initial guess and is an auxiliary linear operator. The concept of homotopy (Hilton, 1953) in topology (Sen, 1983) theoretically gives us a lot of leeway in selecting the auxiliary linear operator and the initial guess ( ). The zeroth-order deformation equation is the same of Eq. (1).
The HPM is a semi-analytical method for solving both linear and non-linear differential and integral problems. A system of linear and non-linear differential equations may also be solved using this approach. Artificial parameters [4] were used to build this approach [5][6][7][8]. Almost every classic perturbation approach is predicated on the assumption of a small number of parameters. However, the vast majority of non-linear problems contain no tiny parameters at all, therefore determining small parameters appears to be a unique skill needing unique methodologies. Tiny changes in small factors might have a big impact on the outcomes. Unsuitable tiny parameter selection, on the other hand, has negative consequences, which can be severe.
Consider the general IVPs of first order [9,10] is a linear / non-linear function of and is a function with some discontinuity, whereas µ analpha arere real constants.
The major purpose of this study is to test the HPM for solving first-order IVPs with a derivative discontinuous, unit

Al-Rafidain Journal of Computer Sciences and Mathematics (RJCM)
www.csmj.mosuljournals.com step function and unit impulse function to achieve approximate-exact solutions in a variety of circumstances. Section 2 gives the fundamental idea of HPM, section 3 shows HPM Applied to an IVPs for linear and non-linear cases, and the conclusions in section 4.
is an initial approximation of Eq. [ ] is a tiny parameter, we consider Eq. (7) solution as a power series in , as shown below.
( ) The approximation solution of Eq. (3) can then be acquired as ( ) In [1] has given the convergence of the series solution (10).

HPM Applied to an IVPs
Applying the standard HPM as in [1][2][3], Eq. (2) can be written as [ ( ) ( )] ( ) We can utilize the embedding parameter as a tiny parameter and suppose that the solution of Eq. (2) can be represented as a power series in , according to the HPM.
and the non-linear term can be decomposed as for some He's polynomials ( ) [18] that are given by ( ) Substituting (12) and (13) into Eq. (11) we get and equating the terms in the same power of , we have a system IVPs of first order ( ) The closed form of this series is as follows: is the exact solution of the case 3.1.1.
In Table 1 we compare numerical results produced using the HPM ( ), the integral equation of the nth-Eq. for the system (16) (IEI), the numerical solution of the nth-Eq. for the system (16) using the Simpson rule (SIMPR) and trapezoidal rule (TRAPR) with the exact solution (17). We used twenty points in the Simpson and trapezoidal rules. Table 2 shows the maximum absolute error (MAE), the maximum relative error (MRE) and the maximum residual error (MRR) obtained by the HPM with the exact solution (17) on [ ] Table 3 shows the estimated orders of convergence (EOC) for various values of the constant . Fig. 1 gives the exact solution ( ) with our approximation HPM ( ) on . The application of the HPM for , necessitates order approximants if we want to get over the (at ) discontinuity.

Case 3.1.2. Taking
, and is the unit step function at . Following that, from the system (16)  The closed form of this series is as follows: is the exact solution of the case 3.1.2. Table 4 compares numerical results obtained by the HPM ( ), the integral equation of the nth-Eq. for the system (16) (IEI), the numerical solution of the nth-Eq. for the system (16) using the Simpson rule (SIMPR) and trapezoidal rule (TRAPR) with the exact solution (18). We used twenty points in the Simpson and trapezoidal rules.
and etc, obtaining the rest of the iterations in this manner. As a result, the series form of the approximate answer is The closed form of this series is as follows: is the exact solution of the case 3.1.4. The EOC for both sides of the discontinuity are given in Table 7. Figs. 4 and 5 show the exact solution ( ) as well as our HPM ( ) approximation. The approximation HPM ( ) is only valid until the second discontinuity, as can be seen in Fig. 5.

Non-Linear Case: Let ( )
and . The non-linear term is calculated using He's polynomials [18] as follows:

Conclusions
In this paper, the HPM was effectively used to solve first-order initial value problems with discontinuities. The size of the jump (given by µ), which performs equally well on both sides of the discontinuity, has no effect on the method's convergence. The HPM for does not converge in these initial value problems even for small values of the parameter, such as In the non-linear cases with large values of µ, sometimes a computation with more digits is required in order to avoid unstable oscillations. The approximate solution obtained by HPM is compatible with analytical approximation approaches in the literature, such as Adomian decomposition method. The HPM has been confirmed by applying it to a linear situation to yield approximation exact results. The method's dependability is demonstrated by the outcomes obtained in all scenarios.