An efficient pseudo‐spectral Legendre–Galerkin method for solving a nonlinear partial integro‐differential equation arising in population dynamics

The pseudo-spectral Legendre–Galerkin method (PS-LGM) is applied to solve a nonlinear partial integro-differential equation arising in population dynamics. This equation is a competition model in which similar individuals are competing for the same resources. It is a kind of reaction–diffusion equation with integral term corresponding to nonlocal consumption of resources. The proposed method is based on the Legendre–Galerkin formulation for the linear terms and interpolation operator at the Chebyshev–Gauss–Lobatto (CGL) points for the nonlinear terms. Also, the integral term, which is a kind of convolution, is directly computed by a fast and accurate method based on CGL interpolation operator, and thus, the use of any quadrature formula in its computation is avoided. The main difference of the PS-LGM presented in the current paper with the classic LGM is in treating the nonlinear terms and imposing boundary conditions. Indeed, in the PS-LGM, the nonlinear terms are efficiently handled using the CGL points, and also the boundary conditions are imposed strongly as collocation methods. Combination of the PS-LGM with a semi-implicit time integration method such as second-order backward differentiation formula and Adams-Bashforth method leads to reducing the complexity of computations and obtaining a linear algebraic system of equations with banded coefficient matrix. The desired equation is considered on one and two-dimensional spatial domains. Efficiency, accuracy, and convergence of the proposed method are demonstrated numerically in both cases. Copyright © 2012 John Wiley & Sons, Ltd.