Discrete homotopy analysis method on heat conduction with radiation problem via fredholm integral equation

Abstract

Science and engineering problems can be modelled in Fredholm integral equations (FIE) whether it is a linear or nonlinear problem. There is method called homotopy analysis method (HAM) can be used to solve even highly nonlinear equations ensuring convergence by introducing convergence control parameter, ?. HAM has been applied to solve linear and nonlinear Fredholm integral equations with high accuracy. There is also discretized version of HAM which evaluate definite integrals in FIE using numerical integration method which is called discrete Homotopy analysis method (DHAM). We applied DHAM on highly nonlinear Fredholm integral equation and compared the results with Nystrom method. It shows that convergence control parameter, ? introduced in DHAM provides a convenient way of ensuring convergent series solutions. DHAM convergence is also faster than Nystrom method in solving nonlinear equations. We proved that accuracy of both DHAM and Nystrom method are dependent on numerical integration. Therefore, the DHAM is superior than Nystrom method in solving nonlinear Fredholm integral equations.