Semi-Analytical Method with Laplace Transform for Certain Types of Nonlinear Problems

In this paper, the approximate solution is found for the Fornberg-Whitham equation (F-W) by using two analytical methods which are the Laplace decomposition method (LDM) and modified Laplace decomposition method (MLDM) with comparison between these methods for which gave the best approximate solution near to the exact solution, The analytical results of these methods have been received in terms of convergent series with easily calculable components. The results show that the modified method was found to be efficient, accurate and fast compared to the second method used in this research.


1.
Introduction Many important phenomena can be represented by nonlinear equations, both ordinary and partial, such as population models, chemical kinetics and fluid dynamics. Many efforts have been made to implement either approximate or analytical methods to solve the nonlinear equations such as [1] and [2]. The F-W gave as [3,4] v − v + v = vv − vv + 3v v (1.1) It consists of a type of travelling wave solution called a kink-like wave solution and anti-kink-like wave solutions. No such sorts of travel wave solutions have been found for F-W. These days, numerous distinct methods have been presented to solve the F-W such as homotopy analysis method (HAM) [5], variational iteration method (VIM) [6], Daftardar-Jafari iterative method (DJM) and homotopy perturbation transform method (HPTM) [7]. Temimi and Ansari method (TAM) and Banach contraction method (BCM) [8].
In this paper, we implemented the LDM introduced by wazwaz [9] and MLDM introduced by Khuri [10,11] to solve F-W, and the solution will be compared in both methods, those iterative methods have been successfully used to solve several kinds of problems. For example the linear and nonlinear fractional diffusion-wave equation was solved by applying the LDM [12], MLDM used to solve lane-Emden type differential equations [13]. In the following sections, the LDM and MLDM application are presented to solve the F-W and the validity of these methods to find the appropriate approximate solution.

5)
Then the solution can be represented as an infinite series mentioned below: The nonlinear operator is disintegrating as Where Ai are Adomian polynomials [14] of v1, v2, …, vi and it can be evaluated by the following formula  Wherein h (b, t) depict the term originating from origin term and define initial conditions. Now, first of all, we stratifying LT of the terms on the right-hand facet of Eq. (2.16) then stratifying inverse LT we get the values of v1, v2... vi each in order. To applied MLDM, we imposed that ℎ( , ) = ℎ ( , ) + ℎ ( , ).
(2.17) According to this assumption, a small change should be made on the components , . The difference we suggest is that only part ℎ ( , ) is set to , at the same time as the ultimate part ℎ ( , ) is combined with other terms in eq. (2.16) to fined . Based totally on those suggestions, we formulate the modified iterative algorithm is as follows The solution using the modified Adomian analysis method in large part relies upon on the choice of h (b, t) and h (b, t).

3.
The application of methods We will discuss the use of LDM and MLDM for the solution of the F-W in this section.   ∑ v (b, t) = e + e − C Cv + C Cv

4.
Numerical analysis's In Table 1, absolute errors are calculated for the differences between the exact solution (3.2) and the approximate solutions (3.26) and (3.49) obtained by LDM and MLDM. Besides, Figure 1, Figure 2 and Figure 3 show the approximate and the exact solutions for the Fornberg-Whitham problem respectively, Figure 4 and Figure 5 show the behaviour of exact and approximate solutions obtained by the LDM and MLDM.

Conclusion
In this paper, we dealt with analytical solutions include the LDM and the MLDM, which we discussed convergence and compared to the exact solution where we found that the convergence achieved by the modification method is more efficient and accurate than the Laplace decomposition method.