Numerical Solution of Parabolic Partial Differential Equation by Using Finite Difference Method

Abstract

In the real world, many physical problems like heat equation, wave equation, Laplace equation and Poisson equation are modeled by partial differential equations (PDEs). A PDE of the form ut = α uxx, (α > 0) where x and t are independent variables and u is a dependent variable; is a one-dimensional heat equation. This is an example of a prototypical parabolic equation. The heat equation has analytic solution in regular shape domain. If the domain has irregular shape, computing analytic solution of such equations is difficult. In this case, we can use numerical methods to compute the solution of such PDEs. Finite difference method is one of the numerical methods that is used to compute the solutions of PDEs by discretizing the given domain into finite number of regions. Here, we derived the Forward Time Central Space Scheme (FTCSS) for this heat equation. We also computed its numerical solution by using FTCSS. We compared the analytic solution and numerical solution for different homogeneous materials (for different values of diffusivity α). There is instantaneous heat transfer and heat loss for the materials with higher diffusivity (α) as compared to the materials of lower diffusivity. Finally, we compared simulation results of different non-homogeneous materials.

Numerical Modelling on the Influence of Source in the Heat Transformation: An Application in the Metal Heating for Blacksmithing

“…The FTCS scheme of the above heat equation is [6] where, This FTCSS is consistent with the order of accuracy (1, 2) and is stable iff [6,20]. We can maintain the condition of stability by resizing the lengths of space and time intervals.…”

“…We can maintain the condition of stability by resizing the lengths of space and time intervals. To find the more accurate approximation we have to increase the number of space and time partitions [6,20]. Let the length of space and time intervals be and respectively.…”

Application of Numerical Methods for the Analysis of Damped Parallel RLC Circuit

“…In this section, we introduce the Euler method (explicit), Third-order Ruge-Kutta method (RK3), and Butcher's fifth-order Runge-Kutta (BRK5) method to solve initial value problems (IVP) for ordinary differential equations (ODE) (Boyce & Di Prima, 2012;Butcher & Goodwin, 2008;Kafle et al, 2020).…”

“…They briefly reviewed the stability and accuracy of this RK method. Kafle et al (2020) compared the different iterative methods to analyze the damping conditions of series RLC circuits under the transient situation with DC source and they found the best iterative method as BRK5 to solve the second-order ODE of the series RLC circuit. They observed the three damping conditions by using the BRK5 method.…”

Formulative Visualization of Numerical Methods for Solving Non-Linear Ordinary Differential Equations

“…For such types of equations, we use the alternative approach to approximate the analytical solution. One such method is a numerical method [9,13].…”