Solution of the advection equation using the iteration variation method

Abstract

Partial differential equations, particularly one-dimensional advection equations, frequently arise in various physics and engineering problems, including heat transfer and fluid flow. In this study, the Iteration Variation method was used to solve the one-dimensional advection equation. The Iteration Variation method was chosen due to its advantages in providing a solution that converges quickly without requiring complex linearisation. This study aims to analyse the effectiveness of the Iteration Variation method in producing analytical solutions and compare them with exact solutions for specific cases. The results show that the solutions obtained by the Iteration Variation method are close to the exact solution, with a relatively small error rate. Therefore, this method can be an efficient alternative for solving one-dimensional advection problems. The results of this study demonstrate that the Iteration Variation method can simplify the completion process and provide accurate results in the initial iteration, requiring only four iterations. The form of the solution is visualised using the MAPLE computer program. Although the complexity of the algebra increases with the number of iterations, it can be achieved with the help of the MaPLE program, which allows for the visualisation of the solution both in its algebraic formula and in its graphical representation.