Homotopy analysis method for space-time fractional differential equations

Purpose - In this article, the authors aim to present the homotopy analysis method (HAM) for obtaining the approximate solutions of space-time fractional differential equations with initial conditions. Design/methodology/approach - The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be determined by imposing the initial conditions. Findings - The comparison of the HAM results with the exact solutions is made; the results reveal that the HAM is very effective and simple. The HAM contains the auxiliary parameter h, which provides a simple way to adjust and control the convergence region of series solution. Numerical examples demonstrate the effect of changing homotopy auxiliary parameter h on the convergence of the approximate solution. Also, they illustrate the effect of the fractional derivative orders a and b on the solution behavior. Originality/value - The idea can be used to find the numerical solutions of other fractional differential equations.