Writing the thesis was made easier with the help of Shaun Moodley and the school's Administrative Assistant Faith Nzimande. In general, the character and qualitative behavior of solutions may not always be fully revealed by numerical approximations, hence the need for improved semi-numerical methods that are accurate, computationally efficient, and robust. The methods are used in Chapter 3 to solve the nonlinear equation governing the two-dimensional squeeze flow of a viscous fluid between two approaching parallel plates and the steady laminar flow of a third-order fluid with heat transfer through a flat channel.
In Chapter 4, methods were used to find solutions to the laminar heat transfer problem on a rotating disk, the steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation, and the classical von K´arm´an equations for the flow of boundary layer. driven by a rotating disk. A spectral-homotopic analysis method for the heat transfer flow of a third class fluid between parallel plates. A note on the solution of von Karm´an equations using Chebyshev series and spectral methods.
Solutions of differential equations are important in predicting the future conditions of the phenomenon under study (Hale and Moore, 2008).
Numerical methods for fluid flow problems
- Discretization approaches
- Runge-Kutta schemes
- The Keller-box scheme
- WENO schemes
- The bvp4c algorithm
- The shooting method
- Spectral methods
In the early 1940s, Hrennikov and Courant laid the mathematical foundations of FEM (Elishakoff and Ren, 2003). Also, unlike the FDM case, the transformation of the equations in terms of body coordinate systems is not required (Tu et al., 2008). The mathematical basis of the boundary element method (BEM) is the method of integral equations (Sato, 1992; Grecu et al., 2009).
The DRBEM was successfully used to find solutions to the Laplace equation of Eldho and Young (2001). Qiu and Shu (2005) proposed a class of WENO schemes based on Hermite polynomials (HWENO), which further improved the compactness of the WENO schemes. The HWENO schemes were used to find solutions to the Hamilton–Jacobi equations by Qiu and Shu (2005).
These trial functions are used as basis functions of the truncated string expansion of the solution (Babolian et al., 2007).
Perturbation methods
The SEM combines the accuracy of spectral methods with the flexibility of the finite element method (van de Vosse and Minev, 2002). The disturbance amount can be part of the differential equation, the boundary conditions, or both (Nayfeh, 1973; Liao, 2003a). In general, the solution of the differential equation at û= 0 should be known (Bellman, 1966; Kevorkian and Cole, 1981).
The approximate solutions are then generated using asymptotic expansions of appropriate sequences of the perturbation parameter (Bellman, 1966). The accuracy of perturbation approximations does not depend on the value of the independent variable, but on the perturbation parameter (Liao, 2003b,a). Selection of appropriate sequences of the perturbation parameter requires prior knowledge of the general nature of the solution (Nayfeh, 1973).
Some perturbation methods may fail for expansions near an irregular point and thus make complete analysis of the solution impossible (Kevorkian and Cole, 1981).
Non-perturbation methods
- Adomian decomposition method
- The variational iteration method
- The homotopy analysis method
- The homotopy perturbation method
Consequently, ADM provides the true solution of the problem and is not affected by discretization errors. This version successfully extended the convergence regions of the DTM and also captured the periodic behavior of solutions. Another modification of DTM based on Laplace transforms and Pad´e approximants is DTM-Pad´e.
Solutions of the Camassa-Holm equation were obtained using the DTM-Paths by Zou et al. Moghimi and Hejazi (2007) used the VIM to find solutions of the generalized Burger-Fisher and Burger equations. They introduced improvements using Path'e approach (Abassy et al., 2007c) to extend the convergence region of the VIM.
Geng (2010) presented a modification of VIM whose purpose was to expand the convergence region of the solutions.
Objectives of this study
Thesis outline
Strengths and weaknesses of the HAM
These include the freedom to choose different basis functions, the ability to control the convergence rate of the series solution, and the ability to efficiently handle both weakly and strongly nonlinear problems with or without nested small or large quantities. The Lyapunov artificial small parameter method and the δ-expansion method are special cases of HAM (Liao, 2003b). Nevertheless, Liao (2003b) shows that even with such freedom to choose the basis functions, for a meaningful solution it is important for the user to have some prior knowledge of the physics of the problem.
Furthermore, the choice of the initial guess is limited to practical and useful features (van Gorder and Vajravelu, 2009). Such a constraint then degrades the choice of the initial guess and forces the user to use only an adequate initial guess instead of the best possible initial guess (van Gorder and Vajravelu, 2009). It has previously been pointed out that there is no foolproof guide to help with the selection of the optimum.
This is also a disadvantage of the method, as a user may end up not using the best possible value of ~.
The spectral-homotopy analysis method
Construction of the SHAM algorithm
The solution to equation (2.14) is generally a "better" choice compared to the one chosen to satisfy only the boundary conditions. In more recent studies, MSHAM has been used, but called SHAM instead of MSHAM. We note again that, unlike the case of HAM, the auxiliary function H(x) is not needed, since there is no need for the solution of the higher-order deformation equation to conform to some rule of solution expression.
The initial approximation0(x) is the solution to the equation. 2.22) Consequently, asq increases from 0 to 1, the unknown function U(x;q) changes from the initial approximation u0(x) to the solutionnu(x). In the collocation method, the unknown sum of the function (ξ) is approximated as a truncated series of Chebyshev polynomials of the form Applying the Chebyshev approximations to the higher order deformation equations (2.25) results in a matrix equation of the form.
Equation (2.34) gives a recursive formula used to find solutions of the higher order approximations um(x), (m≥1).
Convergence theorem for the SHAM
The recursive formula for HAM involves a series of ordinary differential equations, equation (2.34) gives a series of algebraic equations, and as Boyd (2000) points out, it is easier to evaluate a function than to integrate a differential equation.
Strengths and weaknesses of the SHAM
It is also not clear which combination of N and L would give the optimal performance of SHAM for a particular problem.
The successive linearisation method
The SLM for nonlinear ODEs in one variable
However, since the equation is non-linear, it may not be possible to find an exact solution. We are therefore looking for an approximate solution, which is obtained by solving the linear part of the equation and assuming that Y1(x) and its derivatives are small. If Y1(x) is the solution of equation (2.39), we let y1(x) denote the solution of the linear part of (2.39), which has the following composite form.
Since the left-hand side of equation (2.40) is linear and the right-hand side is known, a solution for y1(x) can be found. Equation (2.49) is replaced into the nonlinear equation (2.45) and the linear part of the equation is solved. Starting with an initial guess0(x), the solutions syi(x) are obtained by successively linearizing equation (2.36) and solving the resulting linear equation.
The general form of the linearized equation that is successively solved foryi(x) is given by.
The SLM for systems of nonlinear ODEs
An approximate solution is obtained by solving the linear part of the equation (2.70) assuming that Z1 and its derivatives are small. Since the right-hand side of equation (2.71) is known and the left-hand side is linear, the equation can be solved for Y1(x). We note that when i is large, Zi+1 is small, so for large i, we can approximate the ide-order solution of Y(x) with.
Starting from a known initial guess Y0(x), the solutions Yi(x) (i ≥ 2) can be obtained by successively solving the resulting linear part of the governing equation (2.62) for Yi(x). The general form of the linear part of the equation to be solved for Yi(x) is given by.
Strengths and weaknesses of the SLM
Improved spectral homotopy analysis method
Comparison between the results obtained by the method of successive linearization, with the literature and the numerical solution in terms of accuracy and efficiency of the method. The comparison revealed the efficiency and accuracy of the method compared to the homotopy analysis method. We demonstrated the computational efficiency and accuracy of the spectral-homotope analysis method by comparing the results with the results obtained by the homotopy analysis method.
In Section 3.1, the sequential linearization method was used to find solutions to the compressive flow problem between two parallel plates. The problem was successfully solved and the numerical results demonstrated the ability of the method to generate convergent results at low degrees of approximation. In Section 3.2, the spectral homotopy analysis method was used to solve the third-order fluid heat transfer flow problem between parallel plates, while in Section 3.3, sequential linearization methods and improved spectral homotopy analysis methods were used to solve the same problem.
The robustness and efficiency of the successive linearization method, the spectral homotopy analysis method and the improved spectral homotopy analysis method have been demonstrated. The improved spectral homotopy analysis method was used to study solutions of the steady flow problem of a Reiner-Rivlin fluid with Joule heating and viscous dissipation in section 4.1. Convergence to the numerical solutions was achieved at second order, while the SHAM for some of the flow parameters converged at eighth order.
The spectral homotopy analysis method together with the successive linearization method was used to find numerical solutions of the von K´arm´an non-linear equations for swirling flow with and without suction/injection over the disc walls and an applied magnetic field in Section 4.2. The study revealed that the SLM is accurate and converges at very low orders of the iteration scheme. A note on the solution of the von K' arm' and equations using series and Chebyshev spectral methods 2.
The following corrections and further clarifications have been added in this section;. i) Below is the geometry of the problem;. The ISHAM was shown to be an improvement over the SHAM, as it generated converging results at lower approximation orders than the SHAM. The ISHAM converged to second order for all simulations and the size of the parameter values used did not affect its performance.
However, the ISHAM is expensive in terms of the size of the code and computer time. In this study, it was indicated that the ISHAM outperforms the SHAM and SLM in terms of the accuracy of the results and speed of convergence. A parametric study of the effects of different parameters was conducted and results were found to be in good agreement with those in the literature.
