In this paper, the forward time centered space is applied to a simple problem involving one dimensional heat equation. Numerical solution to the heat equation using forward time centered space difference equation is obtained. The method of solving the problem is implemented by using Python Programming. We have discussed the local truncation error and stability analysis of explicit forward time centered space difference method of heat equation.
The Heat diffusion equation is a parabolic partial differential equation which describes the heat distribution in a given region and provides the basic tool for heat conduction analysis. Analytical and Numerical methods have gained the interest of researchers for finding approximate solution to partial differential equations. Numerical Methods have applied to calculate the approximate solutions using Finite Difference Method.
Xiao-Jun Yang and Feng Gao have proposed a new technology for combining the variation of iterative method and an integral transforms for solution of diffusion and heat equation for the first time. The authors have found that the method is accurate and efficient in development of approximate solutions for the partial differential equations . Alice Gorguis and Wai Kit Benny Chan have investigated a comparative study between the separation of variables and the Adomian method for heat equation. The study shows that Adomian has significant advantages and the method provides fast convergent series that gives exact solution for heat equation over the existing techniques . Gerald W. Recktenwald provided a practical overview of numerical solutions to the heat equation using the finite difference method. The forward time centered space and the backward time centered space applied to a simple problem involving one dimensional differential heat equation. MATLAB codes were used to implement the differences between forward time centered space and backward time centered space .
Clint N.Dawson, Qiang DU and Todd F. Dupont have presented a domain decomposition algorithm for numerical solution of heat equation . Abdulla-Al-Mamun et al. have obtained the analytical solution using MATLAB program by considering second order heat equation . Jesus Martin-Vaqueno and Svajunas Sajavicius have demonstrated that the explicit forward time centered space scheme can be stable while some implicit methods such as Crank-Nicolson are unstable . Hooshmandasl M.R., et al. have applied operational matrices of integration to get numerical solution of the one dimensional heat equation with Dirichlet boundary conditions. They have concluded that the use of Chebyshev wavelets is found to be accurate, simple and fast . Tahrich N.A.Shahid et al. have investigated that modified difference equation in specific problems are more convenient for discussing the solution behavior including physical interpretation of accuracy, stability and consistency . Jeffry Kusuma et al. have described a numerical solution for mathematical model of the transport equation in a simple rectangular box domain. Their mathematical model with various advection and diffusion parameters and boundary conditions is also solved numerically using finite difference forward time centered space method .
Wahida Zaman Loksar and Rama Sarkar have considered one dimensional heat equation as an initial boundary value problem for different materials. The forward time centered space is used with the stability conditions. The results were obtained by using MATLAB codes .
Hamzeh Zureigat et al. have discussed about an explicit finite difference scheme such as the forward time centered space was implemented to solve the complex fuzzy heat equations. The stability of the proposed approach indicated that the forward time centered space scheme was conditionally stable . This paper proposes the numerical solution for the heat equation using forward time centered space by Python programming. The paper is organized as follows: Section II presents the heat equation, Section III discusses the discrete grid, Section IV focuses on implementation and results, Section V discusses local truncation error, Section VI focuses on stability analysis and finally the conclusion is presented in Section VII.
We first introduced the one-dimensional heat equation to obtain the numerical approximation using forward time centered space difference method. We have proposed the initial value problem with boundary conditions to find the numerical solution. The region is discretized into a uniform mesh.
The solution is obtained by implementing Python programming by using initial and boundary conditions. The numerical solution for the heat equation is shown in Figure 4. We have investigated the local truncation error and stability analysis of the forward time centered space difference method of the heat equation.
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