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

  • J. Kafle Tribhuvan University, Kathmandu, Nepal
  • L. P. Bagale Tribhuvan University, Kathmandu, Nepal
  • D. J. K. C. Tribhuvan University, Kathmandu, Nepal
Keywords: Finite Difference Methods; Forward Time Central Space Scheme; Heat conduction in non-homogeneous material; Numerical solution; Partial Differential Equations; Thermal diffusivity

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 utα 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.

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Author Biographies

J. Kafle, Tribhuvan University, Kathmandu, Nepal

Central Department of Mathematics

L. P. Bagale, Tribhuvan University, Kathmandu, Nepal

Central Department of Mathematics

D. J. K. C., Tribhuvan University, Kathmandu, Nepal

Central Department of Mathematics

Published
2020-12-31
How to Cite
Kafle, J., Bagale, L., & K. C., D. (2020). Numerical Solution of Parabolic Partial Differential Equation by Using Finite Difference Method. Journal of Nepal Physical Society, 6(2), 57-65. https://doi.org/10.3126/jnphyssoc.v6i2.34858
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Articles