Generalised Kolmogorov-Petrovskii-Piskunov equation of fractional order: Power series and shifted Legendre collocation methods

The classical Kolmogorov-Petrovskii-Piskunov (KPP) equation describes physical phenomena such as combustion, chemical reaction, evolution of dominant genes, and propagation of nerve pulses. In this paper, we present solutions of a time-fractional order generalised KPP equation using a power series method and Legendre collocation method. The fractional order derivative is described in the Caputo sense. The proposed power series method assumes that the solution of the governing problem can be represented by a fractional power series in time variable, with space-variable expansion coefficients. The nonlinear term is assumed to be analytic in the unknown, and thus, admits a power series representation. The generalised Cauchy product is applied to transform the power series in the unknown to one in the time-dependent variable. An explicit recursion formula for the variable expansion coefficient is then constructed. On the other hand, the shifted Legendre collocation method assumes that the solution of the proposed problem can be expressed as a shifted Legendre polynomial series, with constant expansion coefficients to be determined. Collocating at the shifted Legendre-Gauss nodes, we obtain a set of nonlinear algebraic equations. These algebraic equations are then solved for the unknown Legendre expansion coefficients using Newton’s iteration method. Convergence analyses of both methods are presented. Examples of the proposed problem involving quadratic, cubic, and exponential nonlinearities are considered to demonstrate the efficiency, accuracy, and reliability of the proposed techniques. The results obtained from both methods are in excellent agreement with the exact solutions.