The application of discrete homotopy analysis method in one-dimensional thermal problem

Abstract

Many engineering phenomena can be performed in mathematical modelling. Many thermal problems can be presented in Fredholm integral equation. A discretized version of Homotopy analysis method which introduced by Behiry et al (2010) is applied for solving the Fredholm integral equation. Besides that, we used a numerical method which is trapezoidal rule to solve the problem and make comparison with DHAM results. Comparison of approximation results from both methods demonstrates that DHAM is more accurate than trapezoidal rule as the DHAM convergent series solutions are well coincide with the exact solution. The convergence control parameter ℏ in DHAM is able to find the convergence region and control the convergence region of series solution. Moreover, DHAM can help to quicken the convergence of series solution. Therefore, DHAM is a powerful tool to solve the non-linear problem and can be applied to many others linear problems as well. Last but not least, DHAM is a good technique in solving non-linear problems in the science and engineering field.