Mathematics Letters
Volume 2, Issue 1, February 2016, Pages: 1-18

Solitary Wave Solutions for the Boussinesq and Fisher Equations by the Modified Simple Equation Method

Md. Ashrafuzzaman Khan1, M. Ali Akbar1, Fethi Bin Muhammad Belgacem2

1Department of Applied Mathematics, University of Rajshahi, Rajshahi, Bangladesh

2Department of Mathematics, Faculty of Basic Education, PAAET, Al-Ardhyia, Kuwait

(M. A. Khan)
(M. A. Akbar)
(F. B. M. Belgacem)

Md. Ashrafuzzaman Khan, M. Ali Akbar, Fethi Bin Muhammad Belgacem. Solitary Wave Solutions for the Boussinesq and Fisher Equations by the Modified Simple Equation Method. Mathematics Letters. Vol. 2, No. 1, 2016, pp. 1-18. doi: 10.11648/j.ml.20160201.11

Received: November 14, 2016; Accepted: March 30, 2016; Published: June 3, 2016

Abstract: Although the modified simple equation method effectively provides exact traveling wave solutions to nonlinear evolution equations in the field of engineering and mathematical physics, it has some drawbacks. Particularly, if the balance number is greater than 1, the method cannot be expected to yield any solution. In this article, we present a process to implement the modified simple equation method to solve nonlinear evolution equations for balance number greater than 1, namely with balance number equal to 2. To validate our theory through applications, two equations have been chosen to undergo the proposed process, the Boussinesq and the Fisher equations, to which traveling wave are found and analyzed.  For special parameters values, solitary wave solutions are originated from the exact solutions. We analyze the solitary wave properties by the graphs of the solutions. This shows the validity, usefulness, and necessity of the process.

Keywords: Boussinesq Equation, Fisher Equation, Modified Simple Equation Method, Nonlinear Evolution Equations, Solitary Wave Solutions

1. Introduction

The mathematical modeling of complex phenomena that change over time depends closely on the study of a variety of systems of ordinary and partial differential equations. Similar models are developed in diverse fields of study, ranging from the natural and physical sciences, population ecology to economics, infectious disease epidemiology, neural networks, biology, mechanics etc. In spite of the eclectic nature of the fields wherein these models are formulated, different groups of them contribute adequate common attributes that make it possible to examine them within a unified theoretical structure. Such studies make for a large area of functional analysis, usually called the theory of evolution equations (EEs) which may be linear or nonlinear. The latter are usually more challenging than their linear counterparts, and richer in terms of adequately modeling and describing complex phenomena. Therefore, the investigation of solutions to nonlinear evolution equations (NLEEs) plays a very important role to uncover the obscurity of many phenomena and processes throughout the natural sciences. However, one of the essential problems is to obtain their exact solutions. Therefore, in order to find out exact solutions to NLEEs different groups of mathematicians, physicist, and engineers have been working tirelessly. Accordingly, in the recent years, they establish several methods to search exact solutions, for instance, the inverse scattering method [1], the Hirota’s bilinear transformation method [2], the Backlund transformation method [3], [4], the Darboux transformation method [5], the Painleve expansion method [6], the Adomian decomposition method [7], [8], the He’s homotopy perturbation method [9], [10], the Jacobi’s elliptic function method [11], [12], the Miura transformation method [13], the sine-cosine method [14], [15], the homogeneous balance method [16], the tanh-function method [17], [18], the extended tanh-function method [19], [20], the first integration method [21], the F-expansion method [22], the auxiliary equation method [23], the Lie group symmetry method [24], the variational iteration method [25], the ansatz method [26], [27], the Exp-function method [28], [29], the -expansion method [30], [31], [32], [33], [34], [35], the -expansion method [41], [42], and the various versions and improvements of the -expansion method [45], [46], [47], [48], [49], and [50].

The modified simple equation method, [36], [37], [38], [39], [40], being recently developed, is rising in use. Its computation is straightforward, systematic, and needs not the symbolic computation software to manipulate the algebraic equations. However, the method has some shortcomings. The main problem is that when the balance number is greater than one, the method usually does not give any solution. To the best of our knowledge, till now only two articles are available in the literature concerning higher balance number (for balance number two). In [43], Salam used the MSE method to the modified Liouville equation (wherein the balance number is two) and written down a solution to this equation. However, unfortunately the obtained solution does not satisfy the equation. Also, in Ref. [44], Zayed and Arnous solved the KP-BBM equation by means of the MSE method and found some solutions of this equation. Unfortunately, there is no guideline in this article, how one can solve other NLEEs for the higher balance number. In the present article, we have considered two equations; the balance number for each of these equations is two. If the balance number is greater than one, usually there arise difficulties in solving the NLEEs by means of the MSE method. One cannot use the MSE method in straight away. In this case, we need to take in some strategy. Inserting the assumed solution to the corresponding ordinary differential equation and then equating the coefficients of  yields an over-determined set of algebraic and differential equations. During determination of the unknown function, there born a third order linear ordinary differential equation in and .A polynomial appearing in the solution of , will make it ineligible for a solitary wave solution, because in this case, we have as  [7]. Therefore, the coefficients of the polynomial must be zero. This constraint is essential to solve NLEEs for higher balance number.

The article is organized as follows: In section 2, we summarize the description of the method. In section 3, we employ the method to NLEEs with balance number is 2, and in section 4, we give conclusions.

2. The Modified Simple Equation (MSE) Method

To elaborate on the MSE method, let us consider the nonlinear evolution equation of the form,

(1)

where is an unidentified function,  is a polynomial in  and its partial derivatives, which include the highest order derivatives and nonlinear terms of the highest order, and the subscripts denote partial derivatives. In order to solve Equation (1) by means of the MES method [36], [37], [38], [39], and [40], we have to execute the following steps:

Step 1: The traveling wave variable,

,with, ,        (2)

Permits for the change of Eq. (1), into the following ordinary differential equation (ODE):

(3)

Where, is a polynomial in  and its derivatives, wherein.

Step 2: We suppose that Eq. (3) has the solution in the form,

(4)

where are unknown constants to be determined, such that , and  is an unknown function to be evaluated. In sine-cosine method, tanh-function method, -expansion method, Jacobi elliptic function method, Exp-function method etc., the solutions are proposed in terms of some functions established in advance, but in the MSE method, is not pre-defined or not a solution of any pre-defined differential equation. Therefore, it is not possible to conjecture from earlier what kind of solutions one may get through this method. This is the individuality and distinction of this method. Therefore, some fresh solutions might be found by this method.

Step 3: The positive integer appearing in Eq. (4) can be determined by taking into account the homogeneous balance between the highest order nonlinear terms and the derivatives of highest order occurring in Eq. (3). If the degree of is, therefore, the degree of the other expressions will be as follows:

,

and,

.

Step 4: We substitute Eq. (4) into Eq. (3) and then we account the function. As a result of this substitution, we get a polynomial of  and its derivatives. In the resultant polynomial, we equate all the coefficients of  to zero. This procedure yields a system of algebraic and differential equations which can be solved for getting the values of, and the value of the other needful parameters.

3. Applications of the MSE Method

In this section, we will execute the MSE method to extract solitary wave solutions to the Boussinesq equation and the Fisher equations which are very important in the fields of surface wave propagation in coastal regions, heat and mass transfer, biology, ecology, gene propagation, physiology, crystallization, plasma physics, and reaction-diffusion systems.

3.1. The Boussinesq Equation

In this sub-section, we will implement the MSE method to find the exact solutions and then the solitary wave solutions to the Boussinesq equation in the form:

(5)

where and  are real constants.

To construct solitary wave solutions of the Boussinesq equation by applying the MSE method, we use the wave variable

(6)

The traveling wave transformation (3.2) reduces Eq. (3.1) to the following ODE:

(7)

where prime denotes the ordinary derivatives with respect to . Now, integrating Eq. (7) twice with respect to  and setting the constant of integration to zero, we obtain a new ODE in the form:

(8)

Balancing the highest order derivative term  and the nonlinear term of the highest order , we obtain . Therefore, the solution (4) takes the form,

(9)

where and  are constants such that and  is an unknown function to be determined. Now, it is easy to make out,

(10)

(11)

Substituting the values of  and  from (9)-(11) into Eq. (8) and then equating the coefficients of  to zero, we respectively obtain

(12)

(13)

(14)

(15)

(16)

From Eq. (12) and Eq. (16), we obtain

andsince .

Therefore, we obtain the following two cases arises for the values of .

Case 1: When , then From Eq. (13)-(15), we get

And

where andare integrating constant.

Now, using the values of and into Eq. (9), we obtain the solution

(17)

Simplifying the required solution (17), we derive the following close-form solution of the Boussinesq equation:

(18)

Since  and  are arbitrary constants, one may arbitrarily pick their values. If we choose and then from solution (18), we obtain

(19)

Again if we chooseand then from solution (18), we have the following solitary wave solution:

(20)

On the other hand, if and , from solution (18), we derive the solitary wave solutions in the form:

(21)

Also whenand , then from solution (18) can be written as the following solitary wave solutions in the form:

(22)

The solutions (19)-(20) are plotted and shown in Figure 1 and solutions (21)-(22) are plotted and shown in Figure 2 and 3 respectively for.

Case 2: Whenthen from Eqs.(13)-(15), we obtain,

, while

Where, and are integrating constant.

Now, using the values of  and  in Eq. (9), we obtain the solution in the form:

(23)

Switching the exponential solution (23) into trigonometric function, we derive the solution of the Eq. (5):

(24)

Thus, we get the exact solution (24) to the Boussinesq equation (5). But, since  and  are arbitrary constants, one may randomly pick their values. So, if we take  and  then the solitary wave solution (24) becomes,

(25)

Again, if we choose  and  then from (24), we obtain the following solitary wave solutions in the form:

(26)

On the other hand, if we take and  then from (24), we get the solution in the form:

(27)

Also, if and  then we derive the solitary wave solution (24) can be written in the form:

(28)

The solutions (25)-(26) are plotted and shown in Figure 4 and solutions (27)-(28) are plotted and shown in Figure 5, for .

The major advantage of the MSE method is that the calculations are not sophisticated and easy to control. It is not required any computer algebra system to facilitate the calculations as it take to the Exp-function method, the -expansion, the tanh-function method, the homotopy analysis method etc. But the solutions obtained by the MSE method are equivalent to those solutions obtained by the above mentioned method. Since  and  are arbitrary constants for other choices of and , we might obtain much new and more general exact solutions of Eq. (5) by the MSE method without any aid of symbolic computation software.

Remark 1: Solutions (19)-(22) and (25)-(28) have been verified by putting them back into the original equation and found correct.

3.2. Physical Interpretations of the Boussinesq Equation Solutions

In this sub-section, we will depict the graph and signify the obtained solutions to the Boussinesq equation. The solutions (19) and (20) represent the singular periodic solutions. Periodic solutions are traveling wave solutions that are periodic, such that Fig. 1 shows the shape of the solutions (19) and (20) for within Solitons are solitary waves with resilient scattering property. Solutions (21) and (22) are complex solutions, therefore, the modulus and arguments of these solutions have been plotted. The graph of modulus of the solutions (21) and (22) have been shown in Fig. 2, and their arguments have been shown in Figs. 3 and 4 respectively for  within  On the other hand, Fig. 5 shows that the solution (25) is the bell shape soliton and solution (26) is singular bell shape soliton. Also, the solutions (27) and (28) are complex solutions, therefore, the modulus and arguments of these solutions have been plotted. The graph of modulus and arguments of that solutions have been shown in Fig. 6 and 7 respectively for , such that,

Fig. 1. Periodic solutions  and in (19) and (20) to the Boussinesq equation (5).

Fig. 2. Plot of the modulus of the solutions and in (21) and (22) to the Boussinesq equation (5).

Fig. 3. Plot of the arguments of the solutionin (21) to the Boussinesq equation (5).

Fig. 4. Plot of the arguments of the solutionin (22) to the Boussinesq equation (5).

Fig. 5. Bell shape solution in (25) and singular bell shape solutionin (26) to the Boussinesq equation.

Fig. 6. Plot of the modulus and arguments of the solutionsin (27) to the Boussinesq equation (5).

Fig. 7. Plot of the modulus and arguments of the solutionsin (28) to the Boussinesq equation (5).

3.3. The Fisher Equation

In this sub-section, we will make use of the MSE method to obtain new and more general exact solutions and then the solitary wave solutions to the Fisher equation in the form:

(29)

The traveling wave transformation in Eq. (6) helps reduce Eq. (29) to the following ODE,

(30)

where prime denotes the derivatives with respect to . Balancing the highest order derivative term  and highest order nonlinear term , we have . Therefore, the form of the solution of Eq. (30) is similar to the form of the solution in Eq. (9).

Substituting the values of  and  from (9)-(11) into Eq. (30) and then equating the coefficients of  to zero, we respectively obtain the following equations,

(31)

(32)

(33)

(34)

(35)

From Eq. (31), (34) and (35), we obtain

; , since , and,,

where andare constants of integration. Hence the following two cases arise for the values of.

Case 1: When , then from Eq. (32)-(33), we obtain

Therefore, three cases arises depending on the values for.

Case 1(a): When and, then using the values of and in Eq. (3.5), we get

(36)

Simplifying the solution (36) of the Fisher equation (29), we obtain

(37)

Hence we get the more general exact solution (37) of the Eq. (29). Since,  and  are arbitrary constants, we might randomly opt their values. Therefore, if we take and , we derive the solitary wave solution of the Fisher equation (29) from Eq. (37) as follows:

(38)

Again, if choose  and then from the solution Eq. (37),we obtain the following solitary wave solution:

(39)

Solution (38)is plotted and shown in Figure 8 and solution (39)is plotted and shown in Figure 9.

Case 1(b): When and then substituting the values of  and  in Eq. (9), we get

(40)

Now, simplifying the exponential solution (40), we obtain

(41)

Since, in this case , i.e. the wave speed is zero, the wave is no more traveling wave. Therefore, we are not interested to discuss this case. Hence it is rejected.

Case 1(c): When and then substituted the values of and in Eq. (9), we get

(42)

Now, simplifying the solution (42), we obtain the following trigonometric function solution:

(43)

Thus, we get the more general exact solution (43) of the Fisher equation. Since  and  are constants of integration, we may intuitively choose their values. Therefore, if we choose, and,, then the solution(43)is simplified to yield,

(44)

Again if  and then from the exact solution (43) we derived the following solitary wave solution,

(45)

Another way is that we choose  and  then from solution (43), we get,

(46)

Also, if we take  and  then the solution (43) becomes

(47)

These solutions (44)-(47) are plotted as shown in Figure 10 to 15, respectively.

Case 2: When   then from Eq. (32)-(33), we obtain

Again for values ofwe can discuss the following three cases.

Case 2(a): When,, and,, then using the values of using the values of and  in Eq. (9), we obtain exponential solution:

(48)

After simplification, from solution (48) we obtain the following trigonometric solution of the Fisher equation (29):

(49)

The solution (49)is more general exact solution of the Fisher equation(29). If we pickand , from solution(49), we obtain the following solitary wave solution:

(50)

Again, if we put,, and,  then the solution(49) of the Fisher equation becomes,

(51)

On the other hand, if and  then we derive the following exact solution of Eq. (49):

(52)

Also, if we choose  and  then from Eq. (49), we obtain the following solitary wave solution:

(53)

On the other hand, if we set and , from Eq. (3.45), we get

(54)

Finally if we take and  then the exact solution (49) can be written as:

(55)

The solutions (50)-(55) are drawn in Figure 16-24, respectively.

Case 2(b): When and then substituting the values of  and in Eq. (3.5), we get

(56)

Now, simplifying the solution (56) we obtain the following close-form of the Fisher equation:

(57)

Since  and  are arbitrary constants, so we may takeand , then the general solution(57) can be written as the following form:

(58)

Again, if we set and  then from Eq. (57), we obtain the following solitary wave solution,

(59)

Solutions (58) - (59) are plotted as shown in Figure 21-22, respectively.

Case 2(c): Whenand , then substituting the values of and  into Eq. (3.5), we derive the exponential form general solution:

(60)

Simplifying the exponential solution transformed to the trigonometric function in the following close-form solution of the Eq. (29):

(61)

This is the more general exact solution (61)to the Fisher equation(29). But, this is not the traveling wave solution, since the wave velocity is null. So, we are not interested to discuss the general solution (61) to the Fisher equation (29).

Imperative is it now to point out that solutions derived by the MSE method are equipotential to those solutions obtained by the previously mentioned method. Since, and  are arbitrary constants, we may obtain new and/or more general exact solutions to Eq. (29) by the MSE method without any aid of symbolic computation software.

Remark 2: Solutions (38)-(39), (41),(44)-(47), (50)-(55), (58)-(59) and (61) have been confirmed by setting them into the original equation.

3.4. Physical Interpretations of the Fisher Equation Solutions

In this sub-section, we discuss the physical interpretation of the solutions to the Fisher equation. The solution (38) represents the bell shape soliton and the solution (39) represents the kink. The bell shape soliton is a localized surface "wave envelope" that causes a temporary increase in wave amplitude and the kink waves are traveling waves which arise from one asymptotic state to another. The kink solutions are approach to a constant at infinity. Figs. 8 and 9 show the shape of the solutions (38) and (39) within .Solutions (44)-(47) and (50)-(55) are complex solutions, therefore, the modulus and arguments of these solutions have been plotted. The graph of modulus of the solutions (44)-(47) and (50)-(55) have been shown in Figs. 10, 13, 16, 19 and 22 respectively. On the other hand, the graph of arguments of the solutions (44)-(47) and (50)-(55) have been shown in Figs. 11, 12, 14, 15, 17, 18, 20,21, 23 and 24 respectively. These solutions are plotted for .within . Solutions (58) represent the bell shape solitons and solution (59) represents the kink. The shapes of these solutions are plotted in Figs. 25 and 26 within.

Fig. 8. Bell shape solitary wave solution given in (38) to the Fisher equation (29).

Fig. 9. Kink soliton obtained from (39) to the Fisher equation (29).

Fig. 10. Plot of the modulus of the solitary wave solutions and given in 44) and (45) to the Fisher equation (29).

Fig. 11. Plot of the arguments of the solitary wave solutionin (44) to the Fisher equation (29).

Fig. 12. Plot of the arguments of the soliton (45) to the Fisher equation (29).

Fig. 13. Graph of the modulus of the solitary wave solitons and to the Fisher equation.

Fig. 14. Plot of the arguments of the solitary wave solutionin (46) to the Fisher equation (29).

Fig. 15. Figure of the arguments of the solitonin (47) to the Fisher equation.

Fig. 16. Figure of the modulus of the solitons in (50) andin (51) to the Fisher equation (29).

Fig. 17. Sketch of the arguments of the solitary wave solutionin (50) to the Fisher equation.

Fig. 18. Sketch of the arguments of the solitonin (51) to the Fisher equation.

Fig. 19. Plot of the modulus of the solitons (52) and (53) to the Fisher equation.

Fig. 20. To the figure of the arguments of the solutionin (52) to the Fisher equation.

Fig. 21. Sketch of the arguments of the solitonin (53) to the Fisher equation.

Fig. 22. Plot of the modulus of the solutionin (54) andin (55) to the Fisher equation (29).

Fig. 23. Plot the arguments of the solitonin (54) to the Fisher equation.

Fig. 24. Plot of the arguments of the solitary wave solutionin (55) to the Fisher equation (29).

Fig. 25. Sketch of the Bell shape solitary wave soliton (58) to the Fisher equation (29).

Fig. 26. Kink solitons obtained from the solitary wave solutionin (59) to the Fisher equation (29).

4. Conclusions

In this article, we considered the Boussinesq equation and the Fisher equations for both of them the balance number is two. If the balance number is greater than one, in general the MSE method does not provide any solution. For this case, we have established the procedure in order to implement the MSE method to solve NLEEs for balance number two. If the solution of  consists of polynomial of the wave variable, it will not be the solitary wave solution, since it does not meet the condition  as  for solitary wave solution. In this case, each coefficient of the polynomial must be zero. This constraint is crucial to solve NLEEs for higher balance number. By using this achieved process, we solved the above mentioned NLEEs and found some new traveling wave solutions. When the parameters receive special values, solitary wave solutions are derived from the exact solutions. Although the method has been applied in two equations, it can clearly be applied to many other nonlinear evolution equations whose balance number is equal to 2.

Acknowledgments

Fethi Bin Muhammad Belgacem wishes to acknowledge the continued support of the Public Authority for Applied Education and Training Research Department, (PAAET RD), Kuwait.

The Authors wish to thank the ML Chief Editor, Professor Onur Ilhan, and the ML Referees, whose professional support and detailed comments helped improve the original manuscript.

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 Contents 1. 2. 3. 3.1. 3.2. 3.3. 3.4. 4.
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