Generalized Jacobi polynomials/functions and their applications

We introduce a family of generalized Jacobi polynomials/functions with indexes @a,@[email protected]?R which are mutually orthogonal with respect to the corresponding Jacobi weights and which inherit selected important properties of the classical Jacobi polynomials. We establish their basic approximation properties in suitably weighted Sobolev spaces. As an example of their applications, we show that the generalized Jacobi polynomials/functions, with indexes corresponding to the number of homogeneous boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the use of generalized Jacobi polynomials/functions leads to much simplified analysis, more precise error estimates and well conditioned algorithms.

[1]  Stationary solitons of the fifth order KdV-type. Equations and their stabilization , 1996, hep-th/9604122.

[2]  P. Olver,et al.  Existence and Nonexistence of Solitary Wave Solutions to Higher-Order Model Evolution Equations , 1992 .

[3]  Leon M. Hall,et al.  Special Functions , 1998 .

[4]  L. Nikolova,et al.  On ψ- interpolation spaces , 2009 .

[5]  Ben-yu Guo,et al.  Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces , 2004, J. Approx. Theory.

[6]  A. Timan.  Theory of Approximation of Functions of a Real Variable , 1994 .

[7]  E. V. Krishnan,et al.  On shallow water waves , 1990 .

[8]  E. J. Parkes,et al.  SECH-POLYNOMIAL TRAVELLING SOLITARY-WAVE SOLUTIONS OF ODD-ORDER GENERALIZED KDV EQUATIONS , 1998 .

[9]  J. E. Chappelear,et al.  Shallow‐water waves , 1962 .

[10]  Ben-Yu Guo,et al.  Jacobi Approximations in Certain Hilbert Spaces and Their Applications to Singular Differential Equations , 2000 .

[11]  Ivo Babuska,et al.  Direct and Inverse Approximation Theorems for the p-Version of the Finite Element Method in the Framework of Weighted Besov Spaces. Part I: Approximability of Functions in the Weighted Besov Spaces , 2001, SIAM J. Numer. Anal..

[12]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[13]  R. Askey.  Orthogonal Polynomials and Special Functions , 1975 .

[14]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[15]  Ivo Babuska,et al.  Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two dimensions , 2000, Numerische Mathematik.

[16]  C. Bernardi,et al.  Approximations spectrales de problèmes aux limites elliptiques , 2003 .

[17]  Mohamed El-Gamel,et al.  Sinc-Galerkin method for solving linear sixth-order boundary-value problems , 2004, Math. Comput..

[18]  Monique Dauge,et al.  Spectral Methods for Axisymmetric Domains , 1999 .

[19]  Jöran Bergh,et al.  Interpolation Spaces: An Introduction , 2011 .

[20]  Jie Shen,et al.  A New Dual-Petrov-Galerkin Method for Third and Higher Odd-Order Differential Equations: Application to the KDV Equation , 2003, SIAM J. Numer. Anal..

[21]  N. J. Zabusky,et al.  Shallow-water waves, the Korteweg-deVries equation and solitons , 1971, Journal of Fluid Mechanics.

[22]  T. A. Zang,et al.  Spectral Methods: Fundamentals in Single Domains , 2010 .

[23]  Dr. M. G. Worster.  Methods of Mathematical Physics , 1947, Nature.

[24]  F. Smith,et al.  Conservative, high-order numerical schemes for the generalized Korteweg—de Vries equation , 1995, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[25]  Jie Shen,et al.  Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials , 1994, SIAM J. Sci. Comput..

[26]  A. Quarteroni,et al.  Approximation results for orthogonal polynomials in Sobolev spaces , 1982 .

[27]  D. Funaro.  Polynomial Approximation of Differential Equations , 1992 .

[28]  Huo-Yuan Duan,et al.  Nonconforming elements in least-squares mixed finite element methods , 2004, Math. Comput..

[29]  Guo Ben-Yu,et al.  Gegenbauer Approximation and Its Applications to Differential Equations on the Whole Line , 1998 .

[30]  Bengt Fornberg,et al.  A Pseudospectral Fictitious Point Method for High Order Initial-Boundary Value Problems , 2006, SIAM J. Sci. Comput..

[31]  Jie Shen,et al.  Legendre and Chebyshev dual-Petrov–Galerkin methods for Hyperbolic equations , 2007 .

[32]  ShenJie.  Efficient spectral-Galerkin method I , 1994 .

[33]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[34]  Weizhang Huang,et al.  The pseudospectral method for third-order differential equations , 1992 .

[35]  W. Merryfield,et al.  Properties of Collocation Third-Derivative Operators , 1993 .

[36]  B. Guo,et al.  Spectral Methods and Their Applications , 1998 .

[37]  Jie Shen,et al.  Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials , 2006, J. Sci. Comput..

[38]  Edward H. Twizell,et al.  Numerical methods for the solution of special sixth-order boundary-value problems , 1992 .