Quadrature solution of ordinary and partial differential equations

Abstract

An entirely new alternative orthogonal polynomial (or Bellman-type) quadrature method is presented for the solution of ordinary and partial differential equations. The quadrature method, like finite difference and finite element methods, obtains an approximate solution in terms of unknown nodal values of the dependent variable(s), but, like the Clenshaw, tau, and spectral methods, expresses the solution in terms of orthogonal polynomials. (For simple linear problems using Chebyshev polynomials, the method is effectively equivalent to Clenshaw's method or the tau method). The quadrature is readily applicable to a broad range of linear and nonlinear ordinary and partial differential equations for a variety of initial, boundary or mixed initial-boundary conditions including Dirichlet, Neumann or the Robin type boundary conditions, and non-linear cases. It does not involve recourse to intermediate computer codes, such as are used in spectral methods. The method may be readily generalized to the case of integral equations. Quadrature solutions of linear partial differential equations in two independent variables are obtained on a grid N × N in 0(N′) operations with a correspondingly little amount of computer storage indicating a considerable saving of time and effort.