In this paper, we discuss the relation between the partial sums of Jacobi serier on an elliptic region and the corresponding partial sums of Fourier series. From this we derive a precise approximation formula by the p...In this paper, we discuss the relation between the partial sums of Jacobi serier on an elliptic region and the corresponding partial sums of Fourier series. From this we derive a precise approximation formula by the partial sums of Jacobi series on an elliptic region.展开更多
The paper deals with growth estimates and approximation（not necessarily entire） of solutions of certain elliptic partial differential equations. These solutions are called generalized bi-axially symmetric potentials...The paper deals with growth estimates and approximation（not necessarily entire） of solutions of certain elliptic partial differential equations. These solutions are called generalized bi-axially symmetric potentials（GBASP＇s）. To obtain more refined measure of growth, we have defined q-proximate order and obtained the characterization of generalized q-type and generalized lower q-type with respect to q-proximate order of a GBASP in terms of approximation errors and ratio of these errors in sup norm.展开更多
Presents a study which investigated some Jacobi approximations which are used for numerical solutions of differential equations on the half line. Application of the approximations to problems on unbounded domains; Alg...Presents a study which investigated some Jacobi approximations which are used for numerical solutions of differential equations on the half line. Application of the approximations to problems on unbounded domains; Algorithm to prove the stability and convergence of the approximations.展开更多
Jacobi polynomial approximations in multiple dimensions are investigated. They are applied to numerical solutions of singular differential equations. The convergence analysis and numerical results show their advantages.
A new set of generalized Jacobi rational functions of the first and second kinds,GJRFs-1 and GJRFs-2,which are mutually orthogonal in L^(2)(0,∞),are introduced and they are analytical eigensolutions to a new family o...A new set of generalized Jacobi rational functions of the first and second kinds,GJRFs-1 and GJRFs-2,which are mutually orthogonal in L^(2)(0,∞),are introduced and they are analytical eigensolutions to a new family of singular fractional Sturm-Liouville problems(SFSLPs)of the first and second kinds as non-polynomial functions.We establish some properties and optimal approximation results for these GJRFs-1 and GJRFs-2 in non-uniformly weighted Sobolev spaces involving fractional derivatives,which play important roles in the related spectral methods for a class of fractional differential equations.We develop Jacobi rational-Gauss quadrature type formulae and L^(2)-orthogonal projections based on GJRFs-1 and GJRFs-2.As examples of applications,the two quadrature rules are proposed for Fermi-Dirac and Bose-Einstein integrals.Using various orthogonal properties of GJRFs-1 and GJRFs-2,the Petrov-Galerkin methods are proposed for fractional initial value problems and fractional boundary value problems.Numerical results demonstrate its efficient algorithm,and spectral accuracy for treating the above-mentioned classes of problems.The suggested numerical scheme provides an applicable substitutional to other competitive methods in the recent method-related accuracy.展开更多
In this paper, we investigate spectral method for mixed inhomogeneous boundary value problems in three dimensions. Some results on the three-dimensional Legendre approxima- tion in Jacobi weighted Sobolev space are es...In this paper, we investigate spectral method for mixed inhomogeneous boundary value problems in three dimensions. Some results on the three-dimensional Legendre approxima- tion in Jacobi weighted Sobolev space are established, which improve and generalize the existing results, and play an important role in numerical solutions of partial differential equations. We also develop a lifting technique, with which we could handle mixed inho- mogeneous boundary conditions easily. As examples of applications, spectral schemes are provided for three model problems with mixed inhomogeneous boundary conditions. The spectral accuracy in space of proposed algorithms is proved. Efficient implementations are presented. Numerical results demonstrate their high accuracy, and confirm the theoretical analysis well.展开更多
文摘In this paper, we discuss the relation between the partial sums of Jacobi serier on an elliptic region and the corresponding partial sums of Fourier series. From this we derive a precise approximation formula by the partial sums of Jacobi series on an elliptic region.
文摘The paper deals with growth estimates and approximation（not necessarily entire） of solutions of certain elliptic partial differential equations. These solutions are called generalized bi-axially symmetric potentials（GBASP＇s）. To obtain more refined measure of growth, we have defined q-proximate order and obtained the characterization of generalized q-type and generalized lower q-type with respect to q-proximate order of a GBASP in terms of approximation errors and ratio of these errors in sup norm.
文摘Presents a study which investigated some Jacobi approximations which are used for numerical solutions of differential equations on the half line. Application of the approximations to problems on unbounded domains; Algorithm to prove the stability and convergence of the approximations.
基金This work is supported by The Special Funds for State Major Basic Research Projects of China N.199903 2804,The Shanghai Natural Science Fundation N.00JC14057,and The Special Funds for Major Specialites of Shanghai Education Committee N.00JC14057
文摘Jacobi polynomial approximations in multiple dimensions are investigated. They are applied to numerical solutions of singular differential equations. The convergence analysis and numerical results show their advantages.
文摘A new set of generalized Jacobi rational functions of the first and second kinds,GJRFs-1 and GJRFs-2,which are mutually orthogonal in L^(2)(0,∞),are introduced and they are analytical eigensolutions to a new family of singular fractional Sturm-Liouville problems(SFSLPs)of the first and second kinds as non-polynomial functions.We establish some properties and optimal approximation results for these GJRFs-1 and GJRFs-2 in non-uniformly weighted Sobolev spaces involving fractional derivatives,which play important roles in the related spectral methods for a class of fractional differential equations.We develop Jacobi rational-Gauss quadrature type formulae and L^(2)-orthogonal projections based on GJRFs-1 and GJRFs-2.As examples of applications,the two quadrature rules are proposed for Fermi-Dirac and Bose-Einstein integrals.Using various orthogonal properties of GJRFs-1 and GJRFs-2,the Petrov-Galerkin methods are proposed for fractional initial value problems and fractional boundary value problems.Numerical results demonstrate its efficient algorithm,and spectral accuracy for treating the above-mentioned classes of problems.The suggested numerical scheme provides an applicable substitutional to other competitive methods in the recent method-related accuracy.
文摘In this paper, we investigate spectral method for mixed inhomogeneous boundary value problems in three dimensions. Some results on the three-dimensional Legendre approxima- tion in Jacobi weighted Sobolev space are established, which improve and generalize the existing results, and play an important role in numerical solutions of partial differential equations. We also develop a lifting technique, with which we could handle mixed inho- mogeneous boundary conditions easily. As examples of applications, spectral schemes are provided for three model problems with mixed inhomogeneous boundary conditions. The spectral accuracy in space of proposed algorithms is proved. Efficient implementations are presented. Numerical results demonstrate their high accuracy, and confirm the theoretical analysis well.
