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

Authors:
E. H. Doha;A. H. Bhrawy;R. M. Hafez
Affiliations:
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt;Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt;Department of Basic Science, Institute of Information Technology, Modern Academy, Cairo, Egypt
Venue:
Mathematical and Computer Modelling: An International Journal
Year:
2011

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Abstract

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^(^@a^,^@b^) with @a,@b@?(-1,~) 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.