Numerical Solution of Heat Equation by Variational Iteration Method

In this article, we study the one-dimensional heat equation, which models the diffusion of heat through a medium over time. To solve this equation numerically, we employ He’s Variational Iteration Method (VIM), a semi-analytical technique particularly effective for problems where exact solutions are intractable. The VIM relies on a correction functional that iteratively minimizes the equation’s residuals using a Lagrange multiplier. By repeating this process until convergence is achieved, we obtain an approximate solution to the heat equation.

Heat equation, numerical solution, variational iteration method, Lagrange multiplier

1. He, J. (1997). Variational iteration method for delay differential equations. Communications in Nonlinear Science and Numerical Simulation, 2(4), 235-236. [Google Scholar] [Crossref]

2. He, J. H. (2000). Variational iteration method for autonomous ordinary differential systems. Applied mathematics and computation, 114(2-3), 115-123. [Google Scholar] [Crossref]

3. He, J. H., & Wu, X. H. (2007). Variational iteration method: new development and applications. Computers & Mathematics with Applications, 54(7-8), 881-894. [Google Scholar] [Crossref]

4. Lu, J. (2007). Variational iteration method for solving a nonlinear system of second-order boundary value problems. Computers & Mathematics with Applications, 54(7-8), 1133-1138. [Google Scholar] [Crossref]

5. Sontakke, B. R., Shelke, A. S., & Shaikh, A. S. (2019). Solution of non-linear fractional differential equations by variational iteration method and applications. Far East J. Math. Sci, 110(1), 113-129. [Google Scholar] [Crossref]

6. Tatari, M., & Dehghan, M. (2007). On the convergence of He's variational iteration method. Journal of Computational and Applied Mathematics, 207(1), 121-128. [Google Scholar] [Crossref]

7. Wazwaz, A. M. (2009). The variational iteration method for analytic treatment for linear and nonlinear ODEs. Applied Mathematics and Computation, 212(1), 120-134. [Google Scholar] [Crossref]

8. Wazwaz, A. M. (2014). The variational iteration method for solving linear and nonlinear ODEs and scientific models with variable coefficients. Central European Journal of Engineering, 4, 64-71. [Google Scholar] [Crossref]

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