Modified Kudryashov Method to Solve Generalized Kuramoto-Sivashinsky Equation
Abstract
:1. Introduction
2. Analysis of the Modified Kudryashov Method
- Step 1.
- Consider the given NLPDE of the following form .
- Step 2.
- Apply the wave transformation in Equation (2), where:Here, is the wave variable and is the velocity; both are non-zero constants. Hence, Equation (2) transforms to the following ODE:
- Step 3.
- Let the initial solution guess of Equation (4) be,
- Step 4.
- Step 5.
3. MKM Application to Solve the Generalized Kuramoto–Sivashinsky Equation
- Case 1.
- For and in Equation (1), the unknown coefficients are given by,Further, for the same and value, the second set of unknown coefficients are given by,
- Case 2.
- For and in Equation (1), the unknown coefficients are given by,Further, for the same and value, the second set of unknown coefficients are given by,
- Case 3.
- For and in Equation (1), the unknown coefficients are given by,Further, for the same and value, the second set of unknown coefficients are given by,
- Case 4.
- For and in Equation (1), the unknown coefficients are given by,Further, for the same and , the second set of unknown coefficients are given by,
- Case 5.
- For and in Equation (1), the unknown coefficients are given by,Further, for the same and value, the second set of unknown coefficients are given by,
- Case 6.
- For and in Equation (1), the unknown coefficients are given by,Further, for the same and value, the second set of unknown coefficients are given by,
- Case 7.
- For and in Equation (1), the unknown coefficients are given by,Further, for the same and value, the second set of unknown coefficients are given by,
- Case 8.
- For and in Equation (1), the unknown coefficients are given by,Further, for the same and value, the second set of unknown coefficients are given by,
- Case 9.
- For and in Equation (1), the unknown coefficients are given by,Therefore, the exact complex solution of Equation (1) is given by,The 2D graph of real and imaginary parts of are drawn in Figure 9.Further, for the same and value, the second set of unknown coefficients are given by,
- Case 10.
- For and in Equation (1), the unknown coefficients are given by,Therefore, the exact complex solution of Equation (1) is given by,The 2D graphs of real and imaginary parts of are drawn in Figure 11.Further, for the same and value, the second set of unknown coefficients are given by,
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A. GKSE in the Previous Studies
Appendix B. Studying GKSE by GKM and SGEEM
- For solving Equation (1) by the generalized Kudryashov method [22,23,24], the homogeneous balancing of Equation (8) gives , which has infinite solutions. For the value , this gives . Therefore,
- Next, for solving Equation (1) by the sine-Gordon equation expansion method [26], the homogeneous balancing is the same as the MKM given by . Thus,Substituting the above equation in Equation (8) and following the steps in [26] lead to the polynomials in , , their products and powers. Collecting the coefficients, equating them to zero and solving in Maple result in the continuous execution. Thus, we conclude that Equation (1) cannot be solved by the sine-Gordon equation expansion method either.
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Kilicman, A.; Silambarasan, R. Modified Kudryashov Method to Solve Generalized Kuramoto-Sivashinsky Equation. Symmetry 2018, 10, 527. https://doi.org/10.3390/sym10100527
Kilicman A, Silambarasan R. Modified Kudryashov Method to Solve Generalized Kuramoto-Sivashinsky Equation. Symmetry. 2018; 10(10):527. https://doi.org/10.3390/sym10100527
Chicago/Turabian StyleKilicman, Adem, and Rathinavel Silambarasan. 2018. "Modified Kudryashov Method to Solve Generalized Kuramoto-Sivashinsky Equation" Symmetry 10, no. 10: 527. https://doi.org/10.3390/sym10100527