×

zbMATH — the first resource for mathematics

Convergence error estimate in solving free boundary diffusion problem by radial basis functions method. (English) Zbl 1040.91058
Summary: This paper gives an order of convergence in applying the radial basis functions as a meshless method for solving diffusion type problems under free boundary condition. For illustration, the numerical solution of the Black-Scholes equation for pricing American options, which is a classical heat diffusion equation under free boundary value condition, is obtained and compared with the traditional binomial method for numerical verification.

MSC:
91G60 Numerical methods (including Monte Carlo methods)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
91G20 Derivative securities (option pricing, hedging, etc.)
60G40 Stopping times; optimal stopping problems; gambling theory
PDF BibTeX Cite
Full Text: DOI
References:
[1] Chen, W.; Tanaka, M., A meshless, exponential convergence, integration-free, and boundary-only RBF technique, Comput math appl, 43, 379-391, (2002) · Zbl 0999.65142
[2] Chen, W.; Tanaka, M., New insights into boundary-only and domain-type RBF methods, Int J nonlinear sci numer simul, 1, 3, 145-151, (2000) · Zbl 0954.65084
[3] Franke, C.; Schaback, R., Convergence order estimates of meshless collocation methods using radial basis functions, Adv comput math, 8, 381-399, (1998) · Zbl 0909.65088
[4] Golberg, M.A.; Chen, C.S.; Karur, S.R., Improved multiquadric approximation for partial differential equations, Engng anal bound elem, 18, 9-17, (1996)
[5] Hardy, R.L., Multiquadric equations of topography and other irregular surfaces, J geophys res, 176, 1905-1915, (1971)
[6] Hon, Y.C.; Mao, X.Z., A radial basis function method for solving options pricing model, J financial engng, 8, 1, 1-24, (1999)
[7] Hon, Y.C.; Mao, X.Z., An efficient numerical scheme for Burgers equation, Appl math comput, 95, 37-50, (1998) · Zbl 0943.65101
[8] Hon, Y.C.; Cheung, K.F.; Mao, X.Z.; Kansa, E.J., A multiquadric solution for the shallow water equations, ASCE J hydraul engng, 125, 524-533, (1999)
[9] Hon, Y.C.; Lu, M.W.; Xue, W.M.; Zhu, Y.M., Multiquadric method for the numerical solution of a biphasic mixture model, Appl math comput, 88, 153-175, (1997) · Zbl 0910.76059
[10] Hon, Y.C.; Lu, M.W.; Xue, W.M.; Zhou, X., A new formulation and computation of the TRIPHASIC model for mechano-electrochemical mixtures, Comput mech, 24, 155-165, (1999) · Zbl 0946.76097
[11] Kansa, E.J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics. II. solution to parabolic, hyperbolic and elliptic partial differential equations, Comput math appl, 19, 147-161, (1990) · Zbl 0850.76048
[12] Madych, W.R.; Nelson, S.A., Multivariate interpolation and conditionally positive definite functions. II, Math comput, 54, 189, 211-230, (1990) · Zbl 0859.41004
[13] Micchelli, C.A., Interpolation of scattered data: distance matrix and conditionally positive definite functions, Construct approx, 2, 11-22, (1986) · Zbl 0625.41005
[14] Powell, M.D.J., (), 223-241
[15] Wendland, H., Meshless Galerkin approximation using radial basis functions, Math comp, 68, 1521-1531, (1999) · Zbl 1020.65084
[16] Wilmott, P.; Howison, S.; Dewynne, J., The mathematics of financial derivatives—a student introduction, (1995), Cambridge University Press Cambridge · Zbl 0842.90008
[17] Wu, L.; Kwok, Y.K., A front-fixing finite difference method for the valuation of American options, J financial engng, 6, 83-97, (1997)
[18] Wu, Z., Hermite – birkhoff interpolation of scattered data by radial basis interpolation, Approx theor appl, 8, 2, 1-10, (1992) · Zbl 0757.41009
[19] Wu, Z.; Schaback, R., Local error estimates for radial basis function interpolation of scattered data, IMA J numer anal, 13, 13-27, (1993) · Zbl 0762.41006
[20] Wu, Z.; Schaback, R., Shape preserving properties and convergence of univariate multiquadric quasi-interpolation, ACTA math appl sinica, 10, 4, 441-446, (1994) · Zbl 0822.41025
[21] Wu, Z.; Schaback, R., Shape preserving interpolation with radial basis function, ACTA math appl sinica, 10, 4, 443-448, (1994)
[22] Wu, Z., Compactly supported radial function and the strang – fix conditions, Appl math comput, 84, 2-3, 115-124, (1997) · Zbl 0882.41007
[23] Wu, Z., Solving PDE with radial basis function and the error estimation, (), GuangZhou
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.