Abstract
The main purpose of this paper is to use a method with a free parameter, named the optimal asymptotic homotopy method (OHAM), in order to obtain the solution of delay differential equations, delay partial differential equations, and a system of coupled delay equations featuring fractional derivative. This method is preferable to others since it has faster convergence toward homotopy perturbation method, and the convergence rate can be set as a controlled area. Various examples are given to better understand the use of this method. The approximate solutions are compared with exact solutions as well.
Highlights
1 Introduction Fractional arithmetic and fractional differential equations appeared in many disciplines, including medicine [1], economics [2], dynamical problems [3, 4], chemistry [5], mathematical physics [6], traffic model [7], fluid flow [8], and so on
To find the approximate solution for delay differential equations with fractional derivative that we explore in this paper is presented as follows: Dαx u(x) + A x, u(p0x), ux(p1x), uxx(p3x), . . . , ux · · · x(pnx) = g(x)
We look for an approximate solution for delay differential equations with fractional derivative of the following form: Dαt u(x, t) + A x, t, u(p0x, q0t), ux(p1x, q1t), uxx(p3x, q3t), . . . , ux · · · x(pnx, qnt)
Summary
Fractional arithmetic and fractional differential equations appeared in many disciplines, including medicine [1], economics [2], dynamical problems [3, 4], chemistry [5], mathematical physics [6], traffic model [7], fluid flow [8], and so on. To find the approximate solution for delay differential equations with fractional derivative that we explore in this paper is presented as follows: Dαx u(x) + A x, u(p0x), ux(p1x), uxx(p3x), . We look for an approximate solution for delay differential equations with fractional derivative of the following form: Dαt u(x, t) + A x, t, u(p0x, q0t), ux(p1x, q1t), uxx(p3x, q3t), . A relatively large number of approximate solutions expressed by the scholars are not possible if they find the accurate analytical solutions with the existing procedures For such differential equations, we have to consider some direct and iterative methods. According to least square method for the calculations of the constants c1 and c2, we get c1 = –2.01508, c2 = 7.91742, which are called convergent control parameters.
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