A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations

作者：Doha E H; Bhrawy A H^*; Hafez R M

来源：Mathematical and Computer Modelling, 2011, 53(9-10): 1820-1832.

DOI：10.1016/j.mcm.2011.01.002

• Petrov-Galerkin method
• Jacobi collocation method
• Jacobi polynomials
• Jacobi-Gauss-Lobatto quadrature
• Fast Fourier transform
• Jacobi-Jacobi Galerkin method

This paper analyzes a method for solving the third-and fifth-order differential equations with constant coefficients using a Jacobi dual-Petrov-Galerkin method, which is more reasonable than the standard Galerkin one. The spatial approximation is based on Jacobi polynomials P(n)((alpha,beta)) with alpha,beta is an element of (-1,infinity) and n is the polynomial degree. By choosing appropriate base functions, the resulting system is sparse and the method can be implemented efficiently. A Jacobi-Jacobi dual-Petrov-Galerkin method for the differential equations with variable coefficients is developed. This method is based on the Petrov-Galerkin variational form of one Jacobi polynomial class, but the variable coefficients and the right-hand terms are treated by using the Gauss-Lobatto quadrature form of another Jacobi class. Numerical results illustrate the theory and constitute a convincing argument for the feasibility of the proposed numerical methods.

• 出版日期2011-5