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10
Exp‐Function Method

10.1 Introduction

In Chapter 9, we have studied Akbari–Ganji's Method (AGM), which provides the algebraic solution of linear and nonlinear differential equations. In the present chapter we study another straightforward and simple method called the exp‐function method by which one can find the solution (analytical/semi‐analytical) of differential equations. This method was first proposed by He and Wu [1] and was successfully applied to obtain the solitary and periodic solutions of nonlinear partial differential equations. Further, this method was used by many researchers for handling various other equations like stochastic equations , system of partial differential equations , nonlinear evaluation equation of high dimension [4], difference‐differential equation , and nonlinear dispersive long‐wave equation .

10.2 Basics of Exp‐Function Method

This method will be illustrated for partial differential equations as ordinary differential equations are straightforward while solving partial differential equations. Let us consider a nonlinear partial differential equation

10.1

to understand briefly the exp‐function method [1 7–10].

Here, subscripts indicate partial differentiation with respect to indicated variables in the subscript.

Next, using traveling wave transformation u = u(η), η = kx + ωt (where k and ω are constants) in Eq. (10.1), the partial differential equation may be transformed to a nonlinear ordinary differential equation

10.2

where the prime indicates derivative with respect to η.

The exp‐function method is based on the assumption that the solution of ordinary differential Eq. (10.2) can be expressed as

10.3

where c, d, p, and q are positive integers yet to be determined, and a_n and b_n are unknown constants.

The values of c and p are determined by balancing exp‐functions of the linear term of lowest order in Eq. (10.2) with the lowest‐order nonlinear term. Similarly, balancing the exp‐functions of highest‐order linear term in Eq. (10.2) with the highest‐order nonlinear term, values of d and q can be obtained.

Then, we may obtain an equation in terms of exp(nη) by substituting Eq. (10.3) along with the above‐determined values of c, d, p, and q into Eq. (10.2). Setting all the coefficients of the various powers of exp(nη) to zero lead to a set of algebraic equations for a_n, b_n, k, and ω.

By solving these algebraic equations, the values of a_n, b_n, k, and ω may be found. Further, by incorporating these values into Eq. (10.3), the solution of the nonlinear PDE (10.1) can be obtained.

10.3 Numerical Examples

We solve now two example problems to make the readers understand the exp‐function method.

Example 10.1

Let us solve nonlinear parabolic PDE from Ref. [11] as below

10.4

Solution

Using the transformation u = u(η), η = kx + ωt in Eq. (10.4), it reduces to

10.5

where the prime denotes the differentiation with respect to η.

We assume the solution of the ordinary differential equation 10.5) as Eq. (10.3). So, we may assume the solution as

10.6

The highest‐order linear and nonlinear terms in Eq. (10.5) are u″ and u³.

Differentiating and simplifying Eq. (10.6) we get

10.7

and

10.8

It may be noted that after differentiation and simplification we will get various terms containing different combinations of a_−c, a_d, b_−p, …, etc. which for simplicity we assumed as C_AN, …, C_LN, C_AD, …, C_LD, D_AN, …, D_LN, and D_AD, …, D_LD.

The lowest‐order exp‐function term of Eq. (10.8) is

Multiplying and dividing by exp(−pη), we get

10.9

Similarly, the highest‐order exp‐function term of Eq. (10.8) is

Multiplying and dividing by exp(qη), we get

10.10

Balancing the lowest‐order exp‐functions in Eqs. (10.7) and (10.9), we get

Balancing the highest‐order of exp‐functions in Eqs. (10.7) and (10.10), we have

Here, for simplicity, we consider p = c = 1 and d = q = 1 [1 7 – 10 ]. Then, the trial solution (10.3) takes the form as

10.11

For the case b₁ ≠ 0, Eq. (10.11) can be simplified further as below

10.12

Here, , , , , and are coefficients to be determined next.

Plugging Eq. (10.12) in ODE (10.5), we have

10.13

where A = [exp(η) + d + exp(−η)]³,

Equating the coefficients of exp(nη) in Eq. (10.13) to zero, we have the following system of equations:

10.14

By solving the above system of equations 10.14), we obtain

10.15

Case (i): For λ = − μ where μ > 0 is an arbitrary constant, Eq. (10.4) becomes the Newell–Whitehead equation [11]. Then, we have

The solution may be written from Eq. (10.12) as
10.16
where d is the free parameter.
Case (ii): For λ = − 1, Eq. (10.4) is referred to as Cahn–Allen equation [11,12]. Then the constants in Eq. (10.15) are a = ± 1, b = 0, c = 0, d = d, f = 0, , and .
Accordingly, the solution of Eq. (10.4) for free parameter d and λ = − 1 is
10.17

The solution (10.16) is the same as the solution given in Refs. [11] and [12] for d = ± 1.

Example 10.2

Let us now consider classical Burger's equation

10.18

Solution

Equation (10.18) may be reduced to ordinary differential equation in η by using the above discussed transformation u = u(η), η = kx + ωt as

10.19

where the prime denotes the differentiation with respect to η.

Next, we assume the solution of the ordinary differential equation 10.19) as Eq. (10.3). This may be rewritten as

10.20

The highest‐order linear and nonlinear terms in Eq. (10.19) are u″ and uu′.

By differentiating and simplifying Eq. (10.20), we get

10.21

and

10.22

Again, as mentioned in Example 10.1, after differentiation and simplification we will get various terms containing different combinations of a_−c, a_d, b_−p, …, etc. which for simplicity, we assumed as C_AN, …, C_LN, C_AD, …, C_LD, D_AN, …, D_LN, and D_AD, …, D_LD.

The lowest‐order exp‐function term of Eq. (10.8) is

Multiplying and dividing by exp(−pη), we get

10.23

Similarly, the highest order exp‐function term of Eq. (10.8) is

Multiplying and dividing by exp(qη), we get

10.24

Next, we balance the lowest‐order exp‐functions in Eqs. (10.21) and (10.23) to have relation between p and c as below

Similarly, we balance the highest‐order exp‐functions in Eqs. (10.21) and (10.24) and we have

Here, for simplicity, we consider p = c = 1 and d = q = 1. Then the trial solution (10.3) takes the form

10.25

For the case b₁ ≠ 0, Eq. (10.25) can be simplified as below

10.26

where a, b, c, d, and f are coefficients as explained in Example 10.1, which are to be determined next.

Plugging Eq. (10.26) in the ordinary differential equation 10.19) and simplifying, we get

10.27

where A = [exp(η) + d + exp(−η)]³,

Equating the coefficients of exp(nη) in Eq. (10.27) to zero, we have

10.28

By solving the above system of equations 10.28), we obtain a = a, b = 0, , d = 0, f = f, k = k, and ω = λka − 2k².

By substituting the above obtained values of b, c, d, and ω in Eq. (10.26), we get the final solution as

where a, f, and k are free parameters which may be found from initial and boundary conditions.

Exercise

1 Solve the KdV equation using the exp‐function method.
2 Solve the modified KdV equation using the exp‐function method.

References

1 He, J.H. and Wu, X.H. (2006). Exp‐function method for nonlinear wave equations. Chaos, Solitons and Fractals 30 (3): 700–708.
2 Dai, C.Q. and Zhang, J.F. (2009). Application of He's exp‐function method to the stochastic mKdV equation. International Journal of Nonlinear Sciences and Numerical Simulation 10 (5): 675–680.
3 Misirli, E. and Gurefe, Y. (2010). Exact solutions of the Drinfel'd–Sokolov–Wilson equation using the exp‐function method. Applied Mathematics and Computation 216 (9): 2623–2627.
4 Boz, A. and Bekir, A. (2008). Application of exp‐function method for (3+1)‐dimensional nonlinear evolution equations. Computers and Mathematics with Applications 56 (5): 1451–1456.
5 Bekir, A. (2010). Application of the exp‐function method for nonlinear differential‐difference equations. Applied Mathematics and Computation 215 (11): 4049–4053.
6 Zhang, S., Tong, J.L., and Wang, W. (2009). Exp‐function method for a nonlinear ordinary differential equation and new exact solutions of the dispersive long wave equations. Computers and Mathematics with Applications 58 (11–12): 2294–2299.
7 He, J.H. and Abdou, M.A. (2007). New periodic solutions for nonlinear evolution equations using Exp‐function method. Chaos, Solitons and Fractals 34 (5): 1421–1429.
8 Wu, X.H.B. and He, J.H. (2008). Exp‐function method and its application to nonlinear equations. Chaos, Solitons and Fractals 38 (3): 903–910.
9 Bekir, A. and Boz, A. (2009). Application of He's exp‐function method for nonlinear evolution equations. Computers and Mathematics with Applications 58 (11): 2286–2293.
10 Biazar, J. and Ayati, Z. (2009). Extension of the Exp‐function method for systems of two‐dimensional Burgers equations. Computers and Mathematics with Applications 58 (11): 2103–2106.
11 Wazwaz, A.M. (2007). The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations. Applied Mathematics and Computation 188 (2): 1467–1475.
12 Taşcan, F. and Bekir, A. (2009). Travelling wave solutions of the Cahn–Allen equation by using first integral method. Applied Mathematics and Computation 207 (1): 279–282.
13 Yıldırım, A. and Pınar, Z. (2010). Application of the exp‐function method for solving nonlinear reaction–diffusion equations arising in mathematical biology. Computers and Mathematics with Applications 60 (7): 1873–1880.

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Table of Contents for 10 Exp‐Function Method

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10.1 Introduction

10.2 Basics of Exp‐Function Method

10.3 Numerical Examples

Exercise

References

Table of Contents for
10 Exp‐Function Method