摘要

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.

  • 出版日期2013-8