Numerical solutions of boundary value problems for variable coefficient generalized KdV equations using Lie symmetries

Abstract

The exhaustive group classification of a class of variable coefficient generalized KdV equations is presented, which completes and enhances results existing in the literature. Lie symmetries are used for solving an initial and boundary value problem for certain subclasses of the above class. Namely, the found Lie symmetries are applied in order to reduce the initial and boundary value problem for the generalized KdV equations (which are PDEs) to an initial value problem for nonlinear third-order ODEs. The latter problem is solved numerically using the finite difference method. Numerical solutions are computed and the vast parameter space is studied.

New solitary solutions and a conservative numerical method for the Rosenau–Kawahara equation with power law nonlinearity

“…(8)(9)(10). As we know, to keep the original physical feature is greatly important in constructing numerical schemes for different physical problems.…”

“…Biswas [5] studied the solitary wave solution for KdV equation with power law nonlinearity and time-dependent coefficients, [6] investigated the solitons, shock waves for the potential KdV equation, while [7] studied the solitary wave solutions for the generalized KdV equation. In addition to the theoretical studies, readers can refer to [8,9] for the numerical simulations of the KdV equation and the generalized KdV equation. The regularized long-wave (RLW) equation (also known as Benjamin-Bona-Mahony equation)…”

Group Classification of Variable Coefficient K(m,n) Equations

“…See [20], where a similar problem for generalized KdV equations was solved successfully using finite difference method, for details.…”

“…It is worthy to note that the group classification problem for the class of equations u t + umu x +f (t)u xxx = 0 withmf = 0, that are similar to equations of the form (1) with n = 1, was carried out in [12,20]. Table 2: Classification of the class (1) with n = 1.…”

Symmetry Reduction and Numerical Solution of Von K a ´ rm a ´ n Swirling Viscous Flow

“…Based on the symmetries of a PDE, many important properties of the equation such as Lie algebras [11,12], conservation laws [13][14][15][16][17][18], and exact solutions [16][17][18][19][20][21][22] can be considered successively. Recently, some researchers focus on the applications of the symmetry method for solving boundary value problems (BVP) of a PDE [2,[23][24][25].…”