Approximate analytical solution of a class of highly nonlinear time–fractional-order partial differential equations - Richard Olu Awonusika (Partial Differential Equations in Applied Mathematics)
This article presents a power series technique for obtaining approximate solutions of the time–fractional-order version of a generalised Newell–Whitehead–Segel initial value problem, with the fractional-order derivative described in the Caputo sense. The method assumes a fractional power series in the time variable whose expansion coefficients are functions of the space variable. The proposed approach is based on the generalised Cauchy product of power series and does not require any kind of polynomial or digitisation in the simplification of the nonlinear terms. The application of the generalised Cauchy product enables us to construct explicit recursion formulae for the expansion coefficients of the series solution. The first expansion coefficients are nicely expressed in terms of appropriate integer sequences. Notable special cases of the proposed generalised problem, that include, Newell-Whitehead, Newell–Whitehead–Segel, and Cahn–Allen equations with suitable initial conditions are considered for the purpose of accuracy and reliability of the proposed method. Our numerical results are compared with the exact solutions and other existing results. Comparison of the absolute errors from our method and other published results indicates that the proposed technique is accurate and reliable. Two-dimensional and three-dimensional graphs of results are presented for different fractional-order values 0<μ≤1. It is observed that as the fractional-order μ gets closer to 1, the graphs of the approximate solutions gradually coincide with those of the exact solutions. The convergence rate of the proposed series solutions ranges between 10−15 and 10−18.