A study of certain multi-dimensional partial differential equations using Lie symmetry analysis

PhD (Applied Mathematics ), North-West University, Mafikeng Campus

In this thesis we study certain nonlinear multi-dimensional partial differential equations which are mathematical models of various physical phenomena of the real world. Closed-form solutions and conservation laws are obtained for such equa-tions using various methods. The multi-dimensional partial differential equations that are investigated in this thesis are (2+1) and (3+1)-dimensional Boussinesq equations, a generalized (3+1)-dimensional Kawahara equation, a (3 + 1)-dimensional KP-Boussinesq equation, a (3 + 1)-dimensional BKP-Boussinesq equation, two extended (3 + 1)-dimensional Jimbo-Miwa equations, the combined KdV−negative-order KdV equation and the Calogero-Bogoyavlenskii-Schiff equation. Exact solutions of the (2 + 1)-dimensional and (3 + 1)-dimensional Boussinesq equations are obtained using the Lie symmetry method along with the simplest equation method. The solutions obtained are solitary waves and non-topological soliton. Conservation laws for both equations are constructed using the new con-servation theorem due to Ibragimov. Lie symmetry analysis together with Kudryashov's method is used to obtained travelling wave solutions for the generalized (3+1)-dimensional Kawahara equation. Conservation laws are derived using the multiplier approach. Lie symmetry method is employed to perform symmetry reductions on the (3 + 1)-dimensional generalized KP-Boussinesq equation and thereafter Kudryashov's method is used to obtain exact solutions. Conservation laws are constructed using Ibragimov's theorem. Exact solutions of the (3 + 1)-dimensional BKP-Boussinesq equation are constructed using symmetry reductions and (G'/G)−expansion method. The new conservation theorem is employed to obtain conservation laws. Lie symmetry method together with the (G'/G)?expansion method and the simplest equation method are used to derive exact solutions of two generalized ex-tended (3 + 1)-dimensional Jimbo-Miwa equations. Conservation laws are constructed using Ibragimov's method. he (G'/G)−expansion method is used to obtain travelling wave solutions of a combined KdV−negative-order KdV equation. Multiplier approach is employed to derive the conservation laws. Noether's theorem is employed to construct conservation laws for the Calogero-Bogoyavlenskii-Schiff equation.

Doctoral