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A number of experimental implementations of the multigrid algorithm for the solution of systems of partial differential equations have been produced. One program is applicable to simple nonlinear scalar equations, the others to linear equations, scalar and systems, which may be mildly stiff. All use nested grids and residual extrapolation techniques to compute solution and error estimates very economically. One version implements list based adaptive grids to further decrease both computation and storage needed for comparable problems. Each experiment was demonstrated using a set of problems with known solutions and the program performance or nonperformance discussed. Several techniques were examined to ensure that the system of difference equations representing a given problem would be convergent. The use of artificial viscosity was found to be practical in the general case, though for linear problems the use of one-sided differencing may be superior. / Science, Faculty of / Computer Science, Department of / Graduate

M.Sc. (Mathematics) / Please refer to full text to view abstract

This paper will be devoted to an examination of several general and specific equations and systems of equations of the diophantine type. Only algebraic equations with integral coefficients, not all zero, will considered. The elementary properties of the integers will be assumed.

A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy to the Faculty of Science, University of the Witwatersrand, Johannesburg, 2019 / We will study the symmetry, invariance properties and conservation laws of partial dif ferential equations (pdes) that arise in a number of situations in mathematical physics. These will be range from Image Processing and noise removal algorithms to Timoshenko beam systems. Furthermore, we will study the invariance properties and approximate conservation laws of some nonlinear Schro¨dinger equation with PT-symmetric potentials with inhomogeneous nonlinearity and some nonlinear Schro¨dinger equation involving a spatially extended system consisting of two coupled elements. / TL (2019)

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This thesis gives an account of most known methods which can be utilized to solve various integral equations of the first kind. / Thesis / Master of Science (MS)

It is our purpose here to investigate the method of solving equations for real roots by Newton's Method and to indicate a generalization arising from this method.

This research studies two nonlinear problems arising in mathematical physics. Firstly the Korteweg-de Vrics-Burgers equation is considered. Lie symmetry method is used to obtain t he exact solutions of Korteweg-de Vries-Burgers equation. Also conservation laws are obtained for this equation using the new conservation theorem. Secondly, we consider the generalized (2+ 1)-dimensional Zakharov-Kuznctsov (ZK) equation of time dependent variable coefficients from the Lie group-theoretic point of view. We classify the Lie point symmetry generators to obtain the optimal system of one-dimensional subalgebras of t he Lie symmetry algebras. These subalgebras arc then used to construct a number of symmetry reductions and exact group-invariant solutions of the ZK equation. We utilize the new conservation theorem to construct the conservation laws of t he ZK equation. / Thesis (M. Sci in Applied Mathematics) North-West University, Mafikeng Campus, 2011

This research studies two nonlinear differential equations arising in fluid mechanics. Firstly, the Zakharov-Kuznetsov's equation in (3+1) dimensions with an arbitrary power law nonlinearity is considered. The method of Lie symmetry analysis is used to carry out the integration of Zakharov-Kuznetsov's equation. Also, the extended tanh-function method and t he G'/G method are used to integrate the Zakharov-Kuznetsov's equation. The non-topological soliton solution is obtained by the aid of solitary wave ansatz method. Numerical simulation is given to support the analytical development. Secondly. the nonlinear flow problem of an incompressible viscous fluid is considered. The fluid is taken in a channel having two weakly permeable moving porous walls. An incompressible fluid fills the porous space inside the channel. The fluid is magnetohydrodynamic in the presence of a time-dependent magnetic field. Lie group method is applied along with perturbation method in the derivation of analytic solution. The effects of the magnetic field, porous medium, permeation Reynolds number and wall dilation rate on the axial velocity arc shown and discussed. / Thesis (M.Sc.(Applied Mathematics) North-West University, Mafikeng Campus, 2010