A new generalized Laguerre spectral approximation and its applications. (English) Zbl 1080.65097
A new family of generalized Laguerre polynomials is introduced. Various orthogonal projections are investigated. Some approximation results are established. As an example of their important applications, the mixed spherical harmonic-generalized Laguerre approximation is developed. A mixed spectral scheme is proposed for a three-dimensional model problem. Its convergence is proved. Numerical results demonstrate the high accuracy of this new spectral method.
The theory can be used for the solution of some partial differential equations on the unbounded domain, which can be applied in some economical problems and financial mathematic as a simulation tool. Similar problems arise also in quantum mechanics.
The theory can be used for the solution of some partial differential equations on the unbounded domain, which can be applied in some economical problems and financial mathematic as a simulation tool. Similar problems arise also in quantum mechanics.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 91G60 | Numerical methods (including Monte Carlo methods) |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35Q40 | PDEs in connection with quantum mechanics |
| 35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |
Keywords:
generalized Laguerre approximation; mixed spherical harmonic-generalized Laguerre spectral method; Black-Scholes equation; convergence; numerical results; financial mathematics; quantum mechanicsReferences:
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