In this thesis, we make use of numerical schemes in order to solve Fisher’s and FitzHugh-Nagumo equations with specified initial conditions. The thesis is made up of six chapters. Chapter 1 gives some literatures on partial differential equations and chapter 2 provides some concepts on finite difference methods, nonstandard finite difference methods and their proper-ties, reaction-diffusion equations and singularly perturbed equations. In chapter 3, we obtain the numerical solution of Fisher’s equation when the coefficient of diffu-sion term is much smaller than the coefficient of reaction (Li et al., 1998). Li et al. (1998) used the Moving Mesh Partial Differential Equation (MMPDE) method to solve a scaled Fisher’s equation with coefficient of reaction being 104 and coefficient of diffusion equal to one and the initial condition consisted of an exponential function. The problem considered is quite challeng-ing and the results obtained by Li et al. (1998) are not accurate due to the fact that MMPDE is based on familiar arc-length or curvature monitor function. Qiu and Sloan (1998) constructed a suitable monitor function called modified monitor function and used it with the Moving Mesh Differential Algebraic Equation (MMDAE) method in order to solve the same problem as Li et al. (1998) and better result were obtained. However, each problem has its own choice of monitor function which makes the choice of the monitor function an open question. We use the Forward in Time Central Space (FTCS) scheme and the Nonstandard Finite Difference (NSFD) to solve the scaled Fisher’s equation and we find that the temporal step size must be very small in order to obtain accurate results and comparable to Qiu and Sloan (1998). This causes the computational time to be long if the domain is large. We use two techniques to modify these two schemes either by introducing artificial viscosity or using the approach of Ruxun et al. (1999). These techniques are efficient and give accurate results with a larger temporal step size. We prove that these four methods are consistent with the partial differential equation and we also obtain the region of stability. Chapter 4 is an improvement and extension of the work from Namjoo and Zibaei (2018) whereby the standard FitzHugh-Nagumo equation with specified initial and boundary conditions is solved. Namjoo and Zibaei (2018) constructed two versions of nonstandard finite difference (NSFD1, NSFD2) and also derived two schemes (one explicit and the other implicit) constructed from the exact solution. However, they presented results using the nonstandard finite difference schemes only. We showed that one of the nonstandard finite difference schemes (NSFD1) has convergence issues and we obtain an improvement for NSFD1 which we call NSFD3. We per-form a stability analysis of the schemes constructed from the exact solution and found that the explicit scheme is not stable for this problem. We study some properties of the five methods (NSFD1, NSFD2, NSFD3, two schemes obtained using the exact solution) such as stability, positivity and boundedness. The performance of the five methods is compared by computing L1, L∞ errors and the rate of convergence for two values of the threshold of Affect effect, γ namely; 0.001 and 0.5 for small and large spatial domains at time, T = 1.0. Tests on rate of convergence are important here as we are dealing with nonlinear partial differential equations and therefore the Lax-Equivalence theorem cannot be used. In chapter 5, we consider FitzHugh-Nagumo equation with the parameter β referred to as in-trinsic growth rate. We chose a numerical experiment which is quite challenging for simulation due to shock-like profiles. We construct four versions of nonstandard finite difference schemes and compared the performance by computing L1, L∞ errors, rate of convergence with respect to time and CPU time at given time, T = 0.5 using three values of the intrinsic growth rate, β namely; β = 0.5, 1.0, 2.0. Chapter 6 highlights the salient features of this work. / Thesis (PhD)--University of Pretoria, 2020. / South African DST/NRF SARChI / Mathematics and Applied Mathematics / PhD / Unrestricted

Two solution strategies for large, sparse, and structured algebraic systems of equations are considered. The first strategy is to construct efficient preconditioners for iterative solvers. The second is to reduce the sparse algebraic system to a smaller, dense system of equations, which are called the boundary summation equations. The proposed preconditioners perform well when applied to equations that are discretizations of linear first order partial differential equations. Analysis shows that also very simple iterative methods converge in a number of iterations that is independent of the number of unknowns, if our preconditioners are applied to certain scalar model problems. Numerical experiments indicate that this property holds also for more complicated cases, and a flow problem modeled by the nonlinear Euler equations is treated successfully. The reduction process is applicable to a large class of difference equations. There is no approximation involved in the reduction, so the solution of the original algebraic equations is determined exactly if the reduced system is solved exactly. The reduced system is well suited for iterative solution, especially if the original system of equations is a discretization of a first order differential equation. The technique is used for several problems, ranging from scalar model problems to a semi-implicit discretization of the compressible Navier-Stokes equations. Both strategies use the concept of fundamental solutions, either of differential or difference operators. An algorithm for computing fundamental solutions of difference operators is also presented.