Abstract
In this paper we developed a space discrete version of the homotopy analysis method (DHAM) to find the solutions of linear and nonlinear fractional partial differential equations with time derivative . The DHAM contains the auxiliary parameter , which provides a simple way to guarantee the convergence region of solution series. The efficiency and accuracy of the proposed method is demonstrated by test problems with initial conditions. The results obtained are compared with the exact solutions when . It is shown they are in good agreement with each other.
Highlights
Fractional calculus has been recognized as a powerful instrument to discover the secret directions of various material and physical processes that deal with derivatives and integrals of arbitrary orders [7,8,9,10,11,12,13,14,15,16]
homotopy analysis method (HAM) gives rapidly convergent successive approximations of the exact solutions, if such a solution exists, otherwise approximations can be used for numerical purposes
We develop the discrete homotopy analysis method (DHAM) for the fractional discrete diffusion equation, nonlinear fractional discrete Schrödinger equation and nonlinear fractional discrete Burgers’ equation with time derivative α(0 < α ≤ 1)
Summary
Fractional calculus has been of increasing interest to scientists and engineers, arising in mathematical physics, chemistry, modeling mechanical and electrical properties of real phenomena [1,2,3,4,5,6]. After the discrete ADM method [36], the discrete homotopy analysis method (DHAM) was introduced in 2010 by Zhu et al [37] This method can be applied to complex problems containing discontinuity in fluid characteristics and geometry of the problem. It needs little computational cost as numerical method in comparison to HAM; as an analytical approach DHAM has similar advantages to continuous HAM. By means of introducing an auxiliary parameter one can adjust and control the convergence region of the solution series This method should be employed for solving various differential equations to highlight its high capabilities in comparison with other numerical methods.
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