New Spectral Solutions for High Odd-Order Boundary Value Problems via Generalized Jacobi Polynomials
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In this article, some new efficient and accurate algorithms are developed for solving linear and nonlinear odd-order two-point boundary value problems. The algorithm in linear case is based on the application of Petrov–Galerkin method. For implementing this algorithm, two certain families of generalized Jacobi polynomials are introduced and employed as trial and test functions. The trial functions satisfy the underlying boundary conditions of the differential equations, and the test functions satisfy the dual boundary conditions. The developed algorithm leads to linear systems with band matrices which can be efficiently inverted. These special systems are carefully investigated, especially their complexities. Another algorithm based on the application of the typical collocation method is presented for handling nonlinear odd-order two-point boundary value problems. The use of generalized Jacobi polynomials leads to simplified analysis and very efficient numerical algorithms. Numerical results are presented for the sake of testing the efficiency and applicability of the two proposed algorithms.
KeywordsDual-Petrov–Galerkin method Collocation method Generalized Jacobi polynomials Legendre polynomials High odd-order boundary value problems
Mathematics Subject Classification65M70 65N35 35C10 42C10
The author is very grateful to the editor and to the referee for their constructive and useful comments which have improved the manuscript in its present form.
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