×

zbMATH — the first resource for mathematics

Jacobi rational approximation and spectral method for differential equations of degenerate type. (English) Zbl 1132.41315
Summary: We introduce an orthogonal system on the half line, induced by Jacobi polynomials. Some results on the Jacobi rational approximation are established, which play important roles in designing and analyzing the Jacobi rational spectral method for various differential equations, with the coefficients degenerating at certain points and growing up at infinity. The Jacobi rational spectral method is proposed for a model problem appearing frequently in finance. Its convergence is proved. Numerical results demonstrate the efficiency of this new approach.

MSC:
41A20 Approximation by rational functions
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35K65 Degenerate parabolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Richard Askey, Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. · Zbl 0298.33008
[2] Robin K. Bullough and Philip J. Caudrey , Solitons, Topics in Current Physics, vol. 17, Springer-Verlag, Berlin-New York, 1980. · Zbl 0428.00010
[3] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. · Zbl 0344.46071
[4] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. of Political Economy, 81(1973), 637-654. · Zbl 1092.91524
[5] F. Black, E. Derman and W. Toy, A one factor model of interest rates and its application to treasury bond options, Financial Analysts Journal, 46(1990), 33-39.
[6] John P. Boyd, Spectral methods using rational basis functions on an infinite interval, J. Comput. Phys. 69 (1987), no. 1, 112 – 142. · Zbl 0615.65090 · doi:10.1016/0021-9991(87)90158-6 · doi.org
[7] John P. Boyd, Orthogonal rational functions on a semi-infinite interval, J. Comput. Phys. 70 (1987), no. 1, 63 – 88. · Zbl 0614.42013 · doi:10.1016/0021-9991(87)90002-7 · doi.org
[8] John C. Cox, Jonathan E. Ingersoll Jr., and Stephen A. Ross, A theory of the term structure of interest rates, Econometrica 53 (1985), no. 2, 385 – 407. · Zbl 1274.91447 · doi:10.2307/1911242 · doi.org
[9] L. U. Dothan, On the term structure of interest rates, J. of Financial Economics, 6(1978), 59-69.
[10] Ben-yu Guo, Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations, J. Math. Anal. Appl. 243 (2000), no. 2, 373 – 408. · Zbl 0951.41006 · doi:10.1006/jmaa.1999.6677 · doi.org
[11] Ben-Yu Guo and Jie Shen, On spectral approximations using modified Legendre rational functions: application to the Korteweg-de Vries equation on the half line, Indiana Univ. Math. J. 50 (2001), no. Special Issue, 181 – 204. Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000). · Zbl 0992.65111 · doi:10.1512/iumj.2001.50.2090 · doi.org
[12] Ben-yu Guo and Jie Shen, Irrational approximations and their applications to partial differential equations in exterior domains, Adv. in Comp. Math., DOI 10.1007/s10444-006-9020-5. · Zbl 1140.65087
[13] Ben-Yu Guo, Jie Shen, and Zhong-Qing Wang, A rational approximation and its applications to differential equations on the half line, J. Sci. Comput. 15 (2000), no. 2, 117 – 147. · Zbl 0984.65104 · doi:10.1023/A:1007698525506 · doi.org
[14] Ben-yu Guo, Jie Shen, and Zhong-qing Wang, Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval, Internat. J. Numer. Methods Engrg. 53 (2002), no. 1, 65 – 84. \? and \?\? finite element methods: mathematics and engineering practice (St. Louis, MO, 2000). · Zbl 1001.65129 · doi:10.1002/nme.392 · doi.org
[15] Guo Ben-yu and Wang Li-lian, Jacobi interpolation approximations and their applications to singular differential equations, Adv. Comput. Math. 14 (2001), no. 3, 227 – 276. · Zbl 0984.41004 · doi:10.1023/A:1016681018268 · doi.org
[16] Ben-yu Guo and Li-lian Wang, Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Approx. Theory 128 (2004), no. 1, 1 – 41. · Zbl 1057.41003 · doi:10.1016/j.jat.2004.03.008 · doi.org
[17] Ben-Yu Guo and Zhong-Qing Wang, Modified Chebyshev rational spectral method for the whole line, Discrete Contin. Dyn. Syst. suppl. (2003), 365 – 374. Dynamical systems and differential equations (Wilmington, NC, 2002). · Zbl 1060.65112
[18] Benyu Guo and Zhongqing Wang, Legendre rational approximation on the whole line, Sci. China Ser. A 47 (2004), no. suppl., 155 – 164. · Zbl 1084.41006 · doi:10.1360/04za0014 · doi.org
[19] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. · Zbl 0047.05302
[20] O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. · Zbl 0184.52603
[21] Patrick J. Roache, Computational fluid dynamics, Hermosa Publishers, Albuquerque, N.M., 1976. With an appendix (”On artificial viscosity”) reprinted from J. Computational Phys. 10 (1972), no. 2, 169 – 184; Revised printing. · Zbl 0247.76035
[22] Zhong-Qing Wang and Ben-Yu Guo, A rational approximation and its applications to nonlinear partial differential equations on the whole line, J. Math. Anal. Appl. 274 (2002), no. 1, 374 – 403. · Zbl 1121.41303 · doi:10.1016/S0022-247X(02)00334-7 · doi.org
[23] Zhong-qing Wang and Ben-yu Guo, Modified Legendre rational spectral method for the whole line, J. Comput. Math. 22 (2004), no. 3, 457 – 474. · Zbl 1071.65158
[24] Zhong-Qing Wang and Ben-Yu Guo, A mixed spectral method for incompressible viscous fluid flow in an infinite strip, Comput. Appl. Math. 24 (2005), no. 3, 343 – 364. · Zbl 1213.65135 · doi:10.1590/S0101-82052005000300002 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.