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three-variable Shifted Jacobi Polynomials Approach for Numerically Solving three-Dimensional Multi-Term Fractional-Order PDEs with variable Coefficients

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Computer Modeling in Engineering & Sciences 2018年第4期115卷 67-84页

作者： Jiaquan Xie Fuqiang Zhao Zhibin Yao Jun Zhang College of Mechanical Engineering Taiyuan University of Science and TechnologyTaiyuan 030024China Taiyuan Institute of China Coal Technology Engineering Group Taiyuan 030006China Collaborative Innovation Center of Taiyuan Heavy Machinery Equipment Taiyuan 030024China.

In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs with variable coefficients.The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem.The approximate solutions of nonlinear fractional PDEs with variable coefficients thus obtained by threevariable shifted Jacobi polynomials are compared with the exact solutions.Furthermore some theorems and lemmas are introduced to verify the convergence results of our algorithm.Lastly,several numerical examples are presented to test the superiority and efficiency of the proposed method.

关键词： three-variable shifted Jacobi polynomials multi-term fractional-order PDEs variable coefficients numerical solution convergence analysis

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A new class of three-variable orthogonal polynomials and their recurrences relations

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Science China Mathematics 2008年第6期51卷 1071-1092页

作者： SUN JiaChang State Key Laboratory of Computer Science,R&D Center for Parallel Computing,Institute of Software,Chinese Academy of Sciences,Beijing 100080,China State Key Laboratory of Computer Science R & D Center for Parallel Computing Institute of Software Chinese Academy of Sciences Beijing China

A new class of three-variable orthogonai polynomials,defined as eigenfunctions of a second order PDE operator,is studied.These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron,and can be taken as an extension of the 2-D Steiner domain.The polynomials can be viewed as Jacobi polynomials on such a domain.three- term relations are derived explicitly.The number of the individual terms,involved in the recurrences relations,are shown to be independent on the total degree of the polynomials.The numbers now are determined to be five and seven,with respect to two conjugate variables z,(?) and a real variable r, respectively.three examples are discussed in details,which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds,and Legendre polynomials.

关键词： 3-D PDE eigen-problem three-variable Chebyshev polynomials Legendre polynomial Jacobi polynomials recurrence relations

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