Abstract
In this paper I present Numerical solutions of a one dimensional heat Equation together with initial condition and Dirichlet boundary conditions. Two methods are used to compute the numerical solutions, viz. Finite difference methods and Finite element methods. The finite element methods are implemented by Crank - Nicolson method. The numerical solutions of a one dimensional heat Equation together with initial condition and Dirichlet boundary conditions using finite difference methods do not always converge to the exact solutions. It indicates the occurrence of numerical instability in finite difference methods. Finally the numerical solutions obtained by these two methods are compared with the analytic solutions graphically into two and three dimensions.
Highlights
Many physical problems in the real world such as heat equation, wave equation, Poisson equation and Laplace equation are modeled by partial differential equations
Finite difference and finite element methods are the two numerical methods that are applied to compute the solutions of partial differential equations by discretizing the domain in to finite number of regions
The finite element methods approximate the exact solution of heat equation (1) together with (2) and (3)
Summary
Many physical problems in the real world such as heat equation, wave equation, Poisson equation and Laplace equation are modeled by partial differential equations. Some of these partial differential equations have exact solution in regular shape domain. In general if the domain has irregular shape, computing exact solution of such equations is difficult Due to this we use numerical methods to compute the solution of the modeled partial differential equations. Finite difference and finite element methods are the two numerical methods that are applied to compute the solutions of partial differential equations by discretizing the domain in to finite number of regions.
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