Wavelet methods for solving a class of nonlinear differential equations

Abstract

This thesis in six chapters develops an efficient wavelet operational matrix approach for solving various types of nonlinear partial differential equations and partial integro differential equations. While operational matrix wavelet methods have been studied earlier by Celik, Hariharan, Lepik, Mittal, Ray, Razzaghi, Rehman, Siraj, Yin, Yousefi and many others, our work majorly concentrates on discretizations exclusively based on wavelet methods associated with quasilinearization including differential and integral equations with more than two variables. This thesis also contributes theoretically an interesting unification of quasilineariztaion and independent existence and uniqueness theorems for q-initial and q-boundary value problems. To make the thesis self-contained, Chapter 1 gives a brief introduction to the basic concepts of wavelets and its development as a powerful tool in the area of numerical analysis. A short literature survey is also done to demonstrate its demand and effectiveness. Chapter 2 deals with numerical methods based on quasilinearization Haar wavelets and Legendre wavelets to solve a class of semi-linear parabolic initial boundary value problem. Through an appropriate illustration of the numerical scheme, it is shown that the proposed scheme is robust and easy to apply. This chapter also provides an interesting unification of quasilinearization in the abstract space setting. Two different approaches based on Haar and Legendre wavelets are studied in Chapter 3 to solve a class of two dimensional parabolic integro-differential equations that arises in nuclear reactor models and population models. A comparative numerical study is done to show the efficiency of the proposed schemes.In the first part of Chapter 4, new numerical techniques are proposed for solving nonlinear Klien/ Sine Gordon equation with initial and boundary conditions. The quasilinearization technique is carefully combined with Chebyshev and Legendre wavelet based collocation methods and numerical results obtained suggests that the proposed scheme is better than the methods available in the recent literature. The last and second part of this chapter, extends the previous section to a coupled sine-Gordon equation with initial and boundary conditions. Chapter 5 is a new attempt to solve a fourth order elliptic equations with nonlocal boundary conditions by coupling with two iterative procedures including quasilinearization and Legendre wavelets. The efficiency of the proposed scheme is illustrated through a comparative numerical study with the literature. The existence and uniqueness theorems for a q−initial and q−boundary value problem is obtained in Chapter 6 using classical Newton’s method. A Legendre wavelet technique is proposed to solve the equations numerically that produces higher accuracy and is straightforward to apply.