In this thesis, solutions of variations of the following system of semi-linear coupled equations by classical (Lie [52]), nonclassical (Bluman and Yan [24]) and potential symmetry methods (Anco and Bluman [5]) are considered: ?u/?t + a ?u/?x = ?1uv, ?v/?t + b ?v/?x = ?2uv, (1) where a, b, ?1 and ?2 are constants, and where u and v are functions of x and t and may represent prey and predator densities respectively. The general solution of this system was first presented by Hasimoto [39] by means of a transformation analogous to that used by Hopf [41] and Cole [29] in their derivation of the solution of the Burgers' equation. Later, Chow [28] obtained the general solution by the classical symmetry technique. Chow's solution procedure is reviewed before solutions are found by use of the classical approach for the following modification of Hasimoto's system: ?u/?t + a ?u/?x = m(x, t, u, v), ?v/?t + b ?v/?x = n(x, t, u, v), (2) where m and n are functions of (x, t, u, v) and may model the interaction between prey and predator species along the x-axis at time t with respect to the densities of the species u and v. While general and exact solutions are not found for system (2) with general forms of m and n, exact solutions are obtained for special cases of m and n by use of the classical method. The first special case is ?u/?t + a ?u/?x = m(t)uv, ?v/?t + b ?v/?x = n(t)uv, (3) where the functions m and n may model time varying strengths of inter-species interaction. The second special case of (2) is ?u/?t + a ?u/?x = ?1F(u)G(v), ?v/?t + b ?v/?x = ?2F(u)G(v), (4) which may model more general time independent interactions between prey and predator. The solution process gave rise to special cases of system (4), one such system involving soliton solutions and another involving u and v to powers of constants on the right hand sides. Exact solutions are obtained by the classical approach and presented graphically. Some limitations of the classical approach are also considered. Two further modifications of Hasimoto's system are then explored: ?u/?t + u ?u/?x = ?1uv, ?v/?t + v ?v/?x = ?2uv, (5) and ?u/?t + u ?u/?x = ?1 ?/?x (uv), ?v/?t + v ?v/?x = ?2 ?/?x (uv). (6) Exact solutions are found by the classical method, while nonclassical and potential symmetry methods gave no new solutions. Finally, system (6) is modified to become a system of coupled Burgers' equations: ?u/?t - ?2u/?x2 + u ?u/?x = ?1 ?/?x (uv), ?v/?t - ?2v/?x2 + v ?v/?x = ?2 ?/?x (uv). (7) Classical, nonclassical and potential symmetry approaches are applied and new exact solutions obtained and presented graphically.
Date of Award | 2013 |
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Original language | English |
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- differential equations
- partial
- predation (biology)
- mathematical models
- symmetry methods
- predator-prey dynamical systems
Application of symmetry methods to partial differential equations
Hijazi, O. M. (Author). 2013
Western Sydney University thesis: Doctoral thesis