Lie symmetry analysis, soliton and numerical solutions of boundary value problem for variable coefficients coupled KdV–Burgers equation

Abstract

In this article, an initial and boundary value problem for variable coefficients coupled KdV–Burgers equation is considered. With the help of Lie group approach, initial and boundary value problem for variable coefficients coupled KdV–Burgers equation reduced to an initial value problem for nonlinear third-order ordinary differential equations (ODEs). Moreover, the systems of ODEs are solved to obtain soliton solutions. Further, classical fourth-order Runge–Kutta method is applied to systems of ODEs for constructing numerical solutions of coupled KdV–Burgers equation. Numerical solutions are computed, and accuracy of numerical scheme is assessed by applying the scheme half mesh principal to calculate maximum errors.

Appendix

At the present appendix, we find the connection between the methods used in this paper to find the exact and numerical solutions. We compared the exact solutions obtained in Sect. 3 under the subalgebra 3.1, with the numerical solution by using the numerical technique RK4 method used in Sect. 4. The acquiring of the whole computational work is done with the help of the absolute errors $E_a$, relative error $E_{ r}$ and percentage error $E_{ p}$ given by the following formulas

$$\begin{aligned}&\hbox {Absolute error} E_a =\left| { u - \bar{{u}} } \right| \nonumber \\&\hbox {Relative error} E_r =\left| { \frac{u - \bar{{u}} }{u} } \right| \nonumber \\&\hbox {Percentage error} E_r =\left| { \frac{u - \bar{{u}} }{u} } \right| \times 100, \end{aligned}$$

where u and $\bar{{u}}$ denote the exact and numerical solution of the problem.

Table 9 Absolute error, Relative error and Percentage error

For numerical solution, we start with initial values at $\xi = 1,$ with the restriction on the arbitrary constants of ODEs (3.1) as $K_1 = 0, K_2 = \frac{1}{8 }, K_3 = \frac{3 K_5 }{2 K_2 }, K_4 = 0, \lambda _1 = -\frac{1}{2}, \lambda _2 = -2$ and initial approximation by solutions (3.3) with the restrictions on the arbitrary constants as $a_1 = b_1 = -8 K_2 .$ Then, the reduced ODEs system (3.1) for KdV–Burgers equation transforms into the following initial value problem (IVP)

$$\begin{aligned}&-\frac{1}{2} F - {F}' \xi - \frac{3}{2} F {F}' + \frac{1}{8} {F}''' = 0 \nonumber \\&-2 G - {G}' \xi + \frac{3}{16} G {G}' + {F}''' = 0. \end{aligned}$$

(5.1)

with initial conditions

$$\begin{aligned} F(1)= & {} 0, {F}'(1) = 3, {F}''(1) = -6\nonumber \\ G\left( 1 \right)= & {} -2, \quad {G}'\left( 1 \right) = 2. \end{aligned}$$

(5.2)

In that case, the comparison between exact and numerical solution with respect to IVP (5.1–5.2) is depicted below with the help of Fig. 6. Also, acquiring of the computational work is presented here with the help of Table 9. Form these results, we conclude that the numerical solutions are in good agreement with the exact solution. The results reported here provide further evidence of the usefulness of the fourth-order Runge–Kutta method for solving nonlinear equations.