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We construct the conservation laws for a variable coefficient variant Boussinesq system, which is a third-order system of two partial differential equations. This system does not have a Lagrangian and so we transform it to a system of fourth-order, which admits a Lagrangian. Noether’s approach is then utilized to obtain the conservation laws. Lastly, the conservation laws are presented in terms of the original variables. Infinite numbers of both local and nonlocal conserved quantities are derived for the underlying system. http://www.hindawi.com/journals/aaa/2014/169694/ http://dx.doi.org/10.1155/2014/169694

MSc (Applied Mathematics), North-West University, Mafikeng Campus, 2006 In this dissertation we perform a Noether point symmetry classification of the generalized Lane-Emden equation which models several phenomena in mathematical physics and astrophysics. We then obtain first integrals of the various cases which admit Noether point symmetries. Moreover, we obtain reduction to quadratures for the cases that admit Noether symmetries. Three new cases are found and these correspond to cases 4.2, 5.1 and 5.2 of chapter 3. We also obtain a two-parameter family of solutions for case 5.1 (when r = 5 and n = 2), whereas in the literature only one-parameter family of solutions is usually given. Masters

MSc (Applied Mathematics), North-West University, Mafikeng Campus In this dissertation we study three nonlinear partial differential equations namely the Korteweg-de Vries (KdV) equation as an illustrative example, a generalized (2+1)- Dimesional Calogaro Bogoyavlenskii-Schiff equation and the (3+1)-Dimensional Boiti- Leon-Manna-Pempinelli equation. We invoke the multiplier approach to obtain conservation laws. In addition, we use the classical symmetry method to obtain group invariant solutions for the aforementioned equations. Masters

MSc (Applied Mathematics), North-West University, Mafikeng Campus In this dissertation we study two nonlinear partial differential equations namely, the longitudinal wave equation in a magnetoelectro-elastic circular rod and a new integrable fifth-order equation. We employ the multiplier method to find conservation laws and the Lie symmetry method to find symmetry reductions and exact solutions for the two underlying equations. Masters

PhD (Applied Mathematics ), North-West University, Mafikeng Campus In this thesis we study some nonlinear partial differential equations which appear in several physical phenomena of the real world. Exact solutions and conservation laws are obtained for such equations using various methods. The equations which are studied in this work are: a hyperbolic Lane-Emden system, a generalized hyperbolic Lane-Emden system, a coupled Jaulent-Miodek system and a (2+1)-dimensional Jaulent-Miodek equation power-law nonlinearity. We carry out a complete Noether and Lie group classification of the radial form of a coupled system of hyperbolic equations. From the Noether symmetries we establish the corresponding conserved vectors. We also determine constraints that the non-linearities should satisfy in order for the scaling symmetries to be Noetherian. This led us to a critical hyperbola for the systems under consideration. An explicit solution is also obtained for a particular choice of the parameters. We perform a complete Noether symmetry analysis of a generalized hyperbolic LaneEmden system. Several constraints for which Noether symmetries exist are derived. In addition, we construct conservation laws associated with the admitted Noether symmetries. Thereafter, we briefly discuss the physical meaning of the derived conserved vectors. We carry out a complete group classification of a generalized coupled hyperbolic Lane-Emden system. It is shown that the underling system admits six-dimensional equivalence Lie algebra. We further show that the principle Lie algebra which is one dimensional extends in several cases. We also carry out Lie reductions for some cases. Symmetry analysis is performed on a coupled Jaulent-Miodek system, which arises in many branches of physics such as particle physics and fluid dynamics. The similarity reductions and new exact solutions are constructed. Subsequently, con-servation laws are derived using the multiplier approach. We study complete Noether symmetry classification of a (2+1)-dimensional Jaulent-Miodek equation with power-law nonlinearity. Conservation laws for several cases which admit Noether point symmetries are established. Doctoral

MSc (Applied Mathematics), North-West University, Mafikeng Campus, 2017 In this dissertation we study two nonlinear partial differential equations namely; the Kudryashov-Sinclshchikov equation and the Benney-Luke equation. We employ the multiplier method to find conservation laws and Kudryashov method to obtain exact solutions for the generalized Kudryashov-Sinelshchikov equation. We derive the Noether symmetries of a generalized Benney-Luke equation. Thereafter, we construct the associated conserved vectors. In addition, we search for exact solutions for the generalized Benney-Luke equation via the extended tanh method. Masters

MSc (Applied Mathematics), North-West University, Mafikeng Campus, 2016 In this dissertation the generalized coupled Zakharov-Kuznetsov system and the (2+ 1 )-dimensional breaking soliton equation will be studied. Exact solutions for the coupled Zakharov-Kuznetsov equations are obtained using the Kudryashov method and the Jacobi elliptic function method while the exact solutions for the (2+1)dimensional breaking soliton equation are derived using the double reduction theory. Furthermore, N oether theorem is employed to construct conservation laws for the above mentioned partial differential equations. Since the coupled system is of thirdorder, it does not have a Lagrangian. Therefore, we use the transformations u = Ux and v = Vx to increase a third-order system to a fourth-order coupled system in U and V variables and let a = l. Thus, the new system of equations have a Lagrangian. However the (2+1)-dimensional breaking soliton equation has a Lagrangian in its natural form. Finally, the conservation laws are expressed in 'U and v variables for the generalized coupled Zakharov-Kuznetsov system. Some local and infinitely many nonlocal conserved quantities are found and the Kudryashov method and Jacobi elliptic function method are used to obtain the exact solutions for the coupled Zakharov-Kuznetsov system. The (2+1)-dimensional breaking soliton equation possesses only local conserved quantities and the double reduction theory is applied to obtain some exact solutions. Masters

PhD (Applied Mathematics), North-West University, Mafikeng Campus, 2014 In this work, we study the applications of Lie symmetry analysis to certain nonlinear wave equations. Exact solutions and conservation laws are obtained for such equations. The equations which are considered in this thesis are the generalized ( 2+ 1 )dimensional Klein-Gordon equation, the generalized double sinh-Gordon equation, the generalized double combined sinh-cosh-Gordon equation, the (2+ 1 )-dimensional nonlinear sinh-Gordon equation, the (3+ 1 )-dimensional nonlinear sinh-Gordon equat ion, the Boussinesq-double sine-Gordon equation, the Boussinesq-double sinh-Gordon equation, the Boussinesq-Liouville type I equation and the Boussinesq-Liouville type II equation. The generalized (2+ 1 )-dimensional Klein-Gordon equation is investigated from the point of view of Lie group classification. We show that this equation admits a nine-dimensional equivalence Lie algebra. It is also shown that the principal Lie algebra consists of six symmetries. Several possible extensions of the principal Lie algebra are computed and the group-invariant solutions-of the generalized (2+ 1 )-dimensional Klein-Gordon equation are presented for power law and exponential function cases. Thereafter, we illustrate that the generalized (2+ 1 )-dimensional Klein-Gordon equation is non-linearly self-adjoint. In addition, we derive conservation laws for the non-linearly self-adjoint sub-classes by using the new Ibragimov theorem. Lie symmetry method is performed on a generalized double sinh-Gordon equation. Exact solutions of a generalized double sinh-Gordon equation are obtained by using the Lie symmetry method in conjunction with the simplest equation method and the exponential function method. In addition to exact solutions we also present conservation laws which are derived using four different methods, namely the direct method, the Noether theorem, the new conservation theorem due to Ibragimov and the multiplier method. The generalized double combined sinh-cosh-Gordon equation is investigated using Lie group analysis. Exact solutions are obtained using the Lie group method together with the simplest equation method. Conservation laws are also obtained by using four different approaches, namely the direct method, the Noether theorem, the new conservation theorem due to Ibragimov and the multiplier method for the underlying equation. The (2+ 1 )-dimensional nonlinear sinh-Gordon equation and the (3+ 1 )-dimensional nonlinear sinh-Gordon equation are investigated by using Lie symmetry analysis. The similarity reductions and exact solutions with the aid of simplest equation method and ( G' / G)-expansion methods are computed. In addition to exact solutions, the conservation laws are derived as well for both the equations. Finally, the four Boussinesq-type equations, namely, the Boussinesq-double sineGordon equation, t he Boussinesq-double sinh-Gordon equation, the Boussinesq-Liouville type I equation and the Boussinesq-Liouville type II equation are analysed using Lie group analysis. Exact solutions for these equations are obtained using the Lie symmetry method in conjunction with the simplest equation. Conservation laws are also obtained for these equations by employing two methods, namely, the Noether theorem and the multiplier method. Doctoral