17.1 Introduction

Laplace transform method, Adomian decomposition method (ADM), and homotopy perturbation method (HPM) have been implemented for solving various types of differential equations in Chapters 2, 11, and 12, respectively. This chapter deals with the methods that combine more than one method. These have been termed as hybrid methods. Two main such methods, which are getting more attention of research, are homotopy perturbation transform method (HPTM) and Laplace Adomian decomposition method (LADM). Although the Laplace transform method is an effective method for solving ordinary and partial differential equations (PDEs), a notable difficulty with this method is about handling nonlinear terms if appearing in the differential equations. This difficulty may be overcome by combining Laplace transform (LT) with the HPM and ADM.

First, we briefly describe the HPTM in the following section.

17.2 Homotopy Perturbation Transform Method

Here, the Laplace transform method and HPM are combined to have a method called the HPTM for solving nonlinear differential equations. It is also called as Laplace homotopy perturbation method (LHPM).

Let us consider a general nonlinear PDE with source term g(x, t) to illustrate the basic idea of HPTM as below [1–3]

(17.1)

subject to initial conditions

(17.2)

where D is the linear differential operator (or ), R is the linear differential operator whose order is less than that of D, and N is the nonlinear differential operator.

The HPTM methodology consists of mainly two steps. The first step is applying LT on both sides of Eq. (17.1) and the second step is applying the HPM where decomposition of the nonlinear term is done using He's polynomials.

First, by operating LT on both sides of Eq. ( 17.1 ), we obtain

Assuming that D is a second‐order differential operator and using differentiation property of LT we get

17.3

By applying inverse LT on both sides of Eq. (17.3), we have

(17.4)

where G(x, t) is the term arising from first three terms on right‐hand side of Eq. ( 17.3 ).

Next, to apply HPM (Chapter 12), first we need to assume solution as series that contains embedding parameter p ∈ [0, 1] as

(17.5)

and the nonlinear term may be decomposed using He's polynomials as

(17.6)

where H_n(u) represents the He's polynomials [ 1 ,2] which are defined as below

(17.7)

Interested readers are suggested to see Ref. [4] for the detailed derivation of He' polynomials (Eq. (17.7)) using the concept of HPM. By plugging Eqs. (17.5) and (17.6) in Eq. (17.4) and combining LT with the HPM, one may obtain the following expression:

17.8

By comparing the coefficients of like powers of “p” on both sides of Eq. (17.8) we may obtain the following successive approximations:

Finally, the solution of the differential equation 17.1) may be obtained as below

(17.9)

Next, we solve two test problems to demonstrate the present method.

17.3 Laplace Adomian Decomposition Method

Now, we briefly illustrate the LADM [7,8]. This method combines LT with the ADM. In this method nonlinear terms of the differential equations are handled by Adomian polynomials.

Let us consider nonlinear nonhomogeneous PDE ( 17.1 ) with same conditions as in PDE (17.2) for the explanation of the LADM.

Just like HPTM we apply LT on both sides of PDE ( 17.1 ) and using differentiation property of LT we obtain

(17.21)

Inverse LT on both sides of PDE (17.21) results in

(17.22)

where G(x, t) is the term arising from first three terms on right‐hand side of PDE ( 17.21 ).

In the LADM we assume a series solution for the PDE as below

(17.23)

Here, the nonlinear term is written in terms of Adomian polynomials as

(17.24)

where A_n represents the Adomian polynomials [9] and these are defined as

(17.25)

By plugging Eqs. (17.23) and (17.24) in Eq. (17.22), we obtain

(17.26)

An iterative algorithm may be obtained by matching both sides of Eq. (17.26) as below

(17.27)

17.28

Solution of the differential equation ( 17.1 ) using the LADM is

(17.29)

Next, we apply LADM to a nonlinear PDE to make the readers familiar with the present method.

References

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