Numerical solutions and stability of some partial differential equations using finite difference methods

Abstract

Chapter 1 is introductory in nature. Besides stating some numerical techniques like Finite Difference methods, Finite Element method, Finite Volume method and methods of weighted residuals it gives an introduction to Finite Difference method and existing literature review. In chapter 2, we consider the one dimensional heat equation , 0 0 2 2 ≤ ≤ > ∂ ∂ = ∂ ∂ x L t x c t c This problem is one of the well-known second order linear partial differential equation [33, 34]. It shows that heat equation describes irreversible process and makes a distance between the previous and next steps. Such equations arise very often in various applications of science and engineering describing the variation of temperature (or heat distribution) in a given region over some time [34]. It can be expressed as the heat flow in the rod with diffusion xx c along the rod where the coefficient is the thermal diffusivity of the rod and L is the length of the rod. In this model, the flow of the heat in one-dimension that is insulated everywhere except at the two end points. In this chapter, finite difference method is proposed for the numerical solutions of one dimensional heat equation. Two test examples are considered to test the accuracy and efficiency of the method. In chapter 3, we consider the one dimensional advection-diffusion equation x L t T x c D x c u t c < < < ≤ ∂ ∂ = ∂ ∂ + ∂ ∂ , 0 , 0 2 2 The mathematical model describing the transport and diffusion processes is the one-dimensional advection-diffusion equation. Mathematical modeling of heat transport, pollutants and suspended matter in groundwater involves the solution of a convection–diffusion equation. In this chapter, finite difference method is proposed for the numerical solutions of one dimensional advection-diffusion equation. Two test examples are considered to test the accuracy and efficiency of the method. The absolute errors are calculated for the both examples.