Exact Solutions of Kuramoto-Sivashinsky Equation

Abstract

Many nonlinear partial differential equations admit traveling wave solutions that move at a constant speed without changing their shapes. It is very important and difficult to search the exact travelling wave solutions. In this work, the auxiliary Riccati equation method and the computer symbolic system Maple are used to study exact solutions for the nonlinear Kuramoto-Sivashinsky equation. Maple can help us solve tedious algebraic calculation. Therefore many exact traveling wave solutions are successfully obtained which include some new kink (or anti-kink) wave solutions and periodic wave solutions.

Modified Kudryashov Method to Solve Generalized Kuramoto-Sivashinsky Equation

“…Khalique reduced Equation (1) by Lie symmetry and solved exactly by the simplest equation method with Riccati and Bernoulli equations separately. D. Feng in [10] by taking β = 0 and uu x = γuu x in Equation (1) solved using the Riccati equation as the auxiliary differential equation. M. Lakestani et al used the B-spline approximation function and solved Equation (1) numerically in [11], where they used tanh exact solutions for error estimations.…”

“…Ravi et al in [33], A. R. Seadawy et al in [34] and M. Nur Alam et al in [35]; the Jacobi elliptic function method by S. Liu et al in [36]; the F-expansion method by A. Ebaid et al in [37]; and the extended G G method by E. M. E. Zayed and S. Al-Joudi et al in [38]. The GKSE Equation (1) does not have the solution for general α and β; however, for the different values of α and β, the solution exists for (1), which can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. In this work, we apply the modified Kudryashov method (MKM) to solve the GKSE in which we compute the constants α and β by the MKM.…”

Modified Kudrayshov Method to Solve Generalized Kuramoto–Sivashinsky Equation

“…Khalique reduced (1) by Lie symmetry and solved exactly by simplest equation methd with Riccati and Bernoulli equations seperately. D. Feng in [10] solved GKSE using Riccati equation where they taken β = 0 and uu x = γuu x in (1).…”

Modified Kudrayshov Method to Solve Generalized Kuramoto–Sivashinsky Equation

“…Khalique reduced Eq (1) by Lie symmetry and solved exactly by simplest equation methd with Riccati and Bernoulli equations seperately. D. Feng in [10] solved GKSE using Riccati equation where they taken β = 0 and uu x = γuu x in Eq (1). M. Lakestani et al used B-spline approximations function and solved Eq (1) numerically in [11] where they used tanh exact solutions for error estimations.…”