Computational and Numerical Solutions for 2+1-Dimensional Integrable Schwarz–Korteweg–de Vries Equation with Miura Transform

Mostafa M A Khater,Raghda A M Attia,S H Alfalqi,Dianchen Lu,J F Alzaidi

doi:10.1155/2020/2394030

Abstract

This paper investigates the analytical, semianalytical, and numerical solutions of the 2+1–dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation. The extended simplest equation method, the sech-tanh method, the Adomian decomposition method, and cubic spline scheme are employed to obtain distinct formulas of solitary waves that are employed to calculate the initial and boundary conditions. Consequently, the numerical solutions of this model can be investigated. Moreover, their stability properties are also analyzed. The solutions obtained by means of these techniques are compared to unravel relations between them and their characteristics illustrated under the suitable choice of the parameter values.

Highlights

Introduction e Korteweg–de Vries (KdV) equation is a seminal model in fluid mechanics. is model was introduced by Boussinesq in 1877 and reintroduced by Diederik Korteweg and Gustav de Vries in 1895. e KdV has the following formula [1,2,3,4,5,6,7,8,9]: Qt + Uxxx − 6UUx 0, (1)
Semianalytical and Numerical Solutions is section applies semianalytical and numerical schemes for deriving the solutions of the (2 + 1)-dimensional integrable Schwarz–Korteweg–de Vries (SKdV) model. e Adomian decomposition method and cubic b-spline schemes are employed to the method to investigate the accuracy of the obtained analytical solutions
(vii) Tables 1 and 2 show calculated values of the exact, semianalytical, and numerical solutions with different values of Z. ese values show the accuracy of the obtained analytical solutions via the sechtanh expansion method over the obtained analytical solutions via the extended simplest equation method where the absolute values of error in the sech-tanh method is smaller that those obtained by the extended simplest equation method

Summary

If we adopt the wave transformation

Λ2ρ􏼑, we convert equation (7) into an ordinary differential a1 ⟶ 0, equation (NLODE). e integration of the obtained NLODE with zero constant of integration leads to a2 ⟶ 0,. Having these ideas in mind, this paper is organized as follows: Section 2 presents the two methods and derives the solutions of the SKdV equation. We apply two analytical techniques for deriving the solutions of the (2 + 1)-dimensional integrable SKdV model. Us, the general solution of equation (10) is given by a1 ⟶ − λμρ, a2 ⟶ − μ2ρ, (14) From these two families, the solitary wave solutions of equation (7) can be obtained. E general solution of equation (10) according to the sech-tanh method and calculated value of balance is given by n. Applying this scheme gives equation (10) in the following form:. E Adomian decomposition method and cubic b-spline schemes are employed to the method to investigate the accuracy of the obtained analytical solutions. (ii) Figure 2 shows the dark solitary for (31) in the three-dimensional plot (a) to illustrate the perspective view of the solution, the two-dimensional plot (b) to present the wave propagation pattern of the wave along the x-axis, and the contour plot (c) to explain the overhead view of the solution when

Value of ʒ

Exact Numerical

Conclusion