A study of certain multi-dimensional partial differential equations using Lie symmetry analysis
Moleleki, Letlhogonolo Daddy
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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 equations 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 solution. Conservation laws for both equations are constructed using the new conservation 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 extended (3 + 1)-dimensional Jimbo-Miwa equations. Conservation laws are constructed using Ibragimov's method. The ( 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.