Author(s)

Mariam Almahdi Mohammed Mu’lla¹, Amal Mohammed Ahmed Gaweash², Hayat Yousuf Ismail Bakur³

¹Department of Mathematics, University of Hafr Al-Batin (UoHB), Hafar Al-Batin, KSA.
²Department of Mathematics, University of Taif, Taif, KSA.
³Department of Mathematics, University of Kordofan, El-Obeid, Sudan.

In this paper, we shall be concerned with the numerical solution of parabolic equations in one space variable and the time variable t. We expand Taylor series to derive a higher-order approximation for U_t. We begin with the simplest model problem, for heat conduction in a uniform medium. For this model problem, an explicit difference method is very straightforward in use, and the analysis of its error is easily accomplished by the use of a maximum principle. As we shall show, however, the numerical solution becomes unstable unless the time step is severely restricted, so we shall go on to consider other, more elaborate, numerical methods which can avoid such a restriction. The additional complication in the numerical calculation is more than offset by the smaller number of time steps needed. We then extend the methods to problems with more general boundary conditions, then to more general linear parabolic equations. Finally, we shall discuss the more difficult problem of the solution of nonlinear equations.

Partial Differential Equations (PDEs), Homentropic, Spatial Derivatives with Finite Differences, Central Differences, Finite Differences

Mu’lla, M. , Gaweash, A. and Bakur, H. (2022) Numerical Solution of Parabolic in Partial Differential Equations (PDEs) in One and Two Space Variable. Journal of Applied Mathematics and Physics, 10, 311-321. doi: 10.4236/jamp.2022.102024.