Analysis of Fractional Advection Equation with Improved Homotopy Analysis Method

Abstract

The enhanced homotopy analysis method was employed to solve the nonlinear time-fractional advection equation, resulting in the derivation of a series of solutions. The objective of this study is to minimize the absolute error by identifying the ideal value for a certain parameter, indicated as h, by utilizing the residual error (RE) function associated with that parameter. The 3-dimensional graphs illustrating the absolute discrepancies between the solutions obtained from the exact approach and the modified homotopy analysis method have been made using the MAPLE software. Hence, the proposed approach demonstrates efficacy and appropriateness in tackling fractional partial differential equations. It has been discovered that all optimal homotopy-analysis methods significantly enhance the speed at which series solutions converge. It is highly recommended to use the most effective methods that involve four unknown convergence-control parameters. This efficient method possesses broad implications and can be employed to obtain rapidly converging series solutions for various types of equations exhibiting significant nonlinearity.