NUMERICAL SOLUTION OF DIFFERENTIAL EQUATION USING WAVELET

Abstract

In wavelet analysis a function or a signal is represented in different scales which is more amenable to analysis and processing. The concept of wavelet analysis is similar to that of Fourier analysis in that both techniques decompose the original signal into a linear combination of elementary functions. However, unlike the sine and cosine harmonics used in the Fourier analysis, wavelet analysis uses a more flexible function called a wavelet that is localized both in time and frequency. In this thesis, an attempt has been made to find numerical solution of some differential equations using wavelet bases. Various calculus operators are required in wavelet analysis of mathematical models. For wavelet analysis these operators have to be expressed in terms of the wavelet coefficients. Numerical solution of differential equation using wavelet bases requires efficient computation of the derivatives of the scaling function and integrals of the product of scaling functions and their derivatives called as con-nection coefficients. Due to the derivatives of compactly supported wavelets being highly oscillatory, it is difficult and unstable to compute the connection coefficients by the numerical evaluation of integrals. Also with the increase in nonlinearity, it becomes more difficult to solve the equation by wavelet Galerkin method. To deal 11 with this kind of difficulty, a new method named "wavelet Galerkin with Adomian linearisation" has been proposed in this thesis for numerical solution of nonlinear differential equations. The chapterwise summary of the thesis is as follows: Chapter 1 is introductory in nature. Besides stating the relevant definitions and numerical algorithms, it states the existing literature review. In Chapter 2, numerical solution of the non-linear singular boundary value problem of the form m, By yil y + k' 0 < x < 1, 0 > 0, k > 0 V(0) = 0 and ay(1) + by'(1) -y by two new approaches of wavelet-collocation method namely wavelet collocation with quasilinearisation and wavelet collocation with Adomian linearisation has been discussed. The boundary value problem describes the steady-state oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics. This problem arises in the study of the distribution of heat sources in the human head. Numerical results obtained by both the approaches agrees with the results reported in the literature. It has been established that wavelet collocation with Adomian linearisation converges faster than the quasilinearisation method. Chapter 3 is the numerical study of Burgers' equation with Dirichlet boundary con-dition by traditional wavelet Galerkin method and by the proposed method wavelet -Galerkin with Adomian linearisation. This is a special form of the momentum equa-tion for irrotational, incompressible flows in which pressure gradients are neglected. 111 It appears as a model equation in fluid mechanics to describe diffusive waves sub-jected to dissipation [132]. The Burgers' equation has also been used in discussions of shock structure in a Navier-Stokes fluid and it can be considered as a simplified form of Navier-Stokes equation due to the form of nonlinear convection term and the occurrence of the viscosity term. In this chapter computed results from both of the methods have been compared with the results reported in the literature and found to be good agreement with them. Moreover, both of the methods gave good solution for large Reynolds nos (Re = 2000). It is observed that while traditional wavelet Galerkin method gives more accurate results for large Reynolds nos it requires more computational cost in comparison to wavelet Galerkin with Adomian linearisation method. Chapter 4 presents the long time behavior of solution of Burgers' equation with Neumann boundary condition by wavelet Galerkin method to resolve the anomaly pointed out by Burns et al [34]. According to them, numerical solutions approaches non-constant shock type solution for moderately small viscosity and larger antisym-metric initial condition. However, theoretically stationary solution of Burgers' equa-tion with Neumann boundary condition is a constant. Numerical experiments show that wavelet Galerkin method give theoretically correct solution for the cases men-tioned by Burns et al. Wavelet-Galerkin method here gives quite good solutions because wavelets are able to approximate boundary condition more accurately in comparison to methods used by Burns. A part of this chapter has been accepted for iv publication in Inter. J. Comp. Math(2008). In Chapter 5, Fisher's equation, which describes the logistic growth-diffusion process and occurs in many biological and chemical processes, has been studied numerically by applying wavelet Galerkin method. This equation has the form ut — vuxx = icu(1 — u/k) where v, ic and k are diffusion constant, linear growth rate and carrying capacity of the environment respectively. The results obtained by wavelet Galerkin method is highly encouraging and have been computed for large value of diffusion constant ,c, even for n = 104, for which the results were reported poor in [137]. A part of this chapter has been published in Inter. J. Comp. Math(2006). In Chapter 6, the extended Fisher-Kolomogrov (EFK) equation has been studied. The equation ut 'Yttxxxx 'Uxx + U3 - U = 0, (x, t) E x (0, T] arises in a variety of applications such as pattern formation in bi-stable systems, prop-agation of domain walls in liquid crystals, travelling waves in reaction diffusion system and mezoscopic model of a phase transition in a binary system near Lipschitz point. In particular, in phase transitions near critical points (Lipschitz points), the fourth-order derivative becomes important. Solving this equation by traditional wavelet Galerkin method requires the computation of four term connection coefficients which is computationally expansive to evaluate. Therefore, the numerical solution has been obtained by the wavelet Galerkin with Adomian linearisation method. It is found that the implementation of the wavelet Galerkin algorithm with Adomian linearisa-tion requires significantly less computation time than wavelet Galerkin method. Chapter 7 presents the solution of the regularized long wave equation (RLW) equa-tion. This equation is introduced as an alternative model to the KdV equation, which describes the unidirectional propagation of shallow water waves of small amplitude and long waves. ut + uux — Suet = 0 a < x < b, 6 > 0 This nonlinear dispersive wave equation exhibit fascinating solutions, such as solitary waves, solitons and recurrence. In the present work the solution is obtained by wavelet Galerkin with Adomian linearisation method. Results are compared with those which are existing in the literature and found to be encouraging. Also the solution of two soliton collison problem is presented graphically. Finally, in Chapter 8, based on the present study, conclusions are drawn and future research work in this direction is suggested. The utility of the proposed wavelet Galerkin with Adomian linearisation method has been shown in the thesis