A coupled scheme based on uniform algebraic trigonometric tension B-spline and a hybrid block method for Camassa-Holm and Degasperis-Procesi equations.

[EN]In this article, high temporal and spatial resolution schemes are combined to solve the Camassa-Holm and Degasperis-Procesi equations. The differential quadrature method is strengthened by using modified uniform algebraic trigonometric tension B-splines of order four to transform the partial differential equation (PDE) into a system of ordinary differential equations. Later, this system is solved considering an optimized hybrid block method. The good performance of the proposed strategy is shown through some numerical examples. The stability analysis of the presented method is discussed. This strategy produces a saving of CPU-time as it involves a reduced number of grid points

Simulation Modeling

The book presents some recent specialized works of a theoretical and practical nature in the field of simulation modeling, which is being addressed to a large number of specialists, mathematicians, doctors, engineers, economists, professors, and students. The book comprises 11 chapters that promote modern mathematical algorithms and simulation modeling techniques, in practical applications, in the following thematic areas: mathematics, biomedicine, systems of systems, materials science and engineering, energy systems, and economics. This project presents scientific papers and applications that emphasize the capabilities of simulation modeling methods, helping readers to understand the phenomena that take place in the real world, the conditions of their development, and their effects, at a high scientific and technical level. The authors have published work examples and case studies that resulted from their researches in the field. The readers get new solutions and answers to questions related to the emerging applications of simulation modeling and their advantages

On new and improved semi-numerical techniques for solving nonlinear fluid flow problems.

Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.Most real world phenomena is modeled by ordinary and/or partial differential equations. Most of these equations are highly nonlinear and exact solutions are not always possible. Exact solutions always give a good account of the physical nature of the phenomena modeled. However, existing analytical methods can only handle a limited range of these equations. Semi-numerical and numerical methods give approximate solutions where exact solutions are impossible to find. However, some common numerical methods give low accuracy and may lack stability. In general, the character and qualitative behaviour of the solutions may not always be fully revealed by numerical approximations, hence the need for improved semi-numerical methods that are accurate, computational efficient and robust. In this study we introduce innovative techniques for finding solutions of highly nonlinear coupled boundary value problems. These techniques aim to combine the strengths of both analytical and numerical methods to produce efficient hybrid algorithms. In this work, the homotopy analysis method is blended with spectral methods to improve its accuracy. Spectral methods are well known for their high levels of accuracy. The new spectral homotopy analysis method is further improved by using a more accurate initial approximation to accelerate convergence. Furthermore, a quasi-linearisation technique is introduced in which spectral methods are used to solve the linearised equations. The new techniques were used to solve mathematical models in fluid dynamics. The thesis comprises of an introductory Chapter that gives an overview of common numerical methods currently in use. In Chapter 2 we give an overview of the methods used in this work. The methods are used in Chapter 3 to solve the nonlinear equation governing two-dimensional squeezing flow of a viscous fluid between two approaching parallel plates and the steady laminar flow of a third grade fluid with heat transfer through a flat channel. In Chapter 4 the methods were used to find solutions of the laminar heat transfer problem in a rotating disk, the steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation and the classical von Kάrmάn equations for boundary layer flow induced by a rotating disk. In Chapter 5 solutions of steady two-dimensional flow of a viscous incompressible fluid in a rectangular domain bounded by two permeable surfaces and the MHD viscous flow problem due to a shrinking sheet with a chemical reaction, were solved using the new methods

Symmetry reductions and group-invariant solutions for models arising in water and contaminant transport

A thesis submitted to the Faculty of Science in fulfilment of the requirement for the degree Doctor of Philosophy (PhD), University of the Witwatersrand, School of Computer Science and Applied Mathematics, Johannesburg, 2018In this work we consider convection-diffusion equation (CDE) arising in the theory of contamination of water by oil spill. Furthermore, these equations arise in so lute transport and groundwater. Group classification of the one dimensional CDE which depends on time t and space x is performed. Lie point symmetries of the one-dimensional CDE are obtained. Group invariant solutions are constructed using admitted Lie point symmetries and these solutions are used to reduce the CDE to the ordinary differential equations (ODEs), which in most cases are solvable. In cases where a number of symmetries are obtained, we will construct the one-dimensional optimal systems of sub-algebras. The two-dimensional and three dimensional CDE in solute transport with constant dispersion coefficient is considered. In some of these cases, double reduction meth ods will be used. Exact solutions are obtained using the Lie symmetry method in conjunction with the (G0/G)-expansion method and the substitution w(z) = (z0)−1 . To further our studies, we apply the method of potential symmetries to determine group invariant solutions that cannot be obtained using point symmetries. Finally, the non-classical symmetries are obtained and comparison study is done between the results obtained through nonlocal and nonclassical symmetry methods.GR201