Finite Element Approximation for Linear and Nonlinear Differential Equations

Abstract

The dissertation entitled ``FINITE ELEMENT APPROXIMATION FOR LINEAR AND NONLINEAR DIFFERENTIAL EQUATIONS" focuses on finding the numerical solutions of linear and nonlinear differential equations. Differential equations arise in almost all areas of science and engineering. But differential equations governing physical phenomenon are in general complex and nonlinear in nature and have complicated coefficients. Generally, analytical methods are not applicable on these type of problems. Therefore, we need numerical techniques to approximate the solutions of such type of problems. For this purpose, we have used finite element technique to approximate the solutions of the differential equations. \\ To brief, the first chapter of the dissertation focuses on the basic concepts related to differential equations. One example of each analytical and numerical techniques have been explained with the help of numerical example. Second chapter discusses the finite element formulation for linear differential equations. Proposed method has been explained in detail. Numerical examples have been solved using proposed method and MATLAB code has been generated for solving linear differential equations using finite element technique. In the third chapter, nonlinear differential equations have been considered. Quasilinearization technique has been employed to handle the nonlinearity. Convergence analysis of quasilinearization technique has also been discussed. Numerical results presented depicts that finite element method is capable of approximating the solution very nicely both for linear and nonlinear differential equations.