Domain Decomposition Spectral Method for Mixed Inhomogeneous Boundary Value Problems of High Order Differential Equations on Unbounded Domains

Zhang, Chao; Guo, Bing

doi:10.1007/s10915-012-9581-z

Cited by 12 publications

(8 citation statements)

References 17 publications

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“…For the same level of solution accuracy, the Laguerre spectral scheme requires more regularities of W(v, t) than the Legendre spectral scheme in bounded domains; see Remark 4.1 of this work, or Remark 4.1 of [17]. Nevertheless, when the solutions of the underlying problems are oscillatory in a certain bounded subdomain or the solutions may vary rapidly near the fixed boundaries, Legendre polynomials are better than Laguerre functions; see [17,44]. On the other hand, the solutions of the Fokker-Planck equation in statistical physics and the Black-Scholes equation in financial mathematics usually present the Gaussian distribution.…”

Section: Introductionmentioning

confidence: 94%

“…In other words, the choice of the scaling parameters is meticulous for Laguerre and Hermite methods; see [35]. For cases of certain nonlinear problems defined on unbounded domains; see [6,10,11,17,18,23,31,34,36,40,43,44]. Wang [37] considered spectral methods for the nonlinear Fokker-Planck equation defined on the whole line using the scaled generalized Laguerre functions coupled with the domain decomposition.…”

Section: Introductionmentioning

confidence: 99%

“…The scaling parameter offers great flexibility to match the asymptotic behaviors of exact solutions for differential equations of degenerate type at |v| → ∞ . Whereas the solutions may vary rapidly near the fixed boundaries or in the corresponding bounded subdomains, the above framework is no longer the most appropriate; see [17,44].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multi-Domain Decomposition Pseudospectral Method for Nonlinear Fokker–Planck Equations

Sun

Wang

2019

Commun. Appl. Math. Comput.

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Results on the composite generalized Laguerre-Legendre interpolation in unbounded domains are established. As an application, a composite Laguerre-Legendre pseudospectral scheme is presented for nonlinear Fokker-Planck equations on the whole line. The convergence and the stability of the proposed scheme are proved. Numerical results show the efficiency of the scheme and conform well to theoretical analysis.

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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

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Multi-Domain Decomposition Pseudospectral Method for Nonlinear Fokker–Planck Equations

Sun

Wang

2019

Commun. Appl. Math. Comput.

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“…Actually, some numerical techniques have been used to apply SM for unbounded domains; but, with some persistent challenges. Among the commonly used techniques are the domain truncation, implementation of basis functions that are intrinsically unbounded, and coordinate transformation (mapping) [12,14,16,17,21,23].…”

Section: Introductionmentioning

confidence: 99%

“…The concept of mapping in not new. It has been implemented successfully to convert unbounded domain to bounded one and vice versa [12,14,16,17,21,23,[30][31][32]. For example, Grosch and Orszag, in 1977, used mapping to transform semi-infinite domain [0,∞) to (0,1) using exponential and algebraic mapping, and the resulting transformation functions are used to solve PDEs by approximating the solution using Chebyshev polynomials [30].…”

Section: Introductionmentioning

confidence: 99%

Efficient Mapping of High Order Basis Sets for Unbounded Domains

Mumtaz¹,

Alharbi²

2018

CiCP

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Many physical problems involve unbounded domains where the physical quantities vanish at infinities. Numerically, this has been handled using different techniques such as domain truncation, approximations using infinitely extended and vanishing basis sets, and mapping bounded basis sets using some coordinate transformations. Each technique has its own advantages and disadvantages. Yet, approximating simultaneously and efficiently a wide range of decaying rates has persisted as major challenge. Also, coordinate transformation, if not carefully implemented, can result in non-orthogonal mapped basis sets. In this work, we revisited this issue with an emphasize on designing appropriate transformations using sine series as basis set. The transformations maintain both the orthogonality and the efficiency. Furthermore, using simple basis set (sine function) help avoid the expensive numerical integrations. In the calculations, four types of physically recurring decaying behaviors are considered, which are: non-oscillating and oscillating exponential decays, and non-oscillating and oscillating algebraic decays. The results and the analyses show that properly designed high-order mapped basis sets can be efficient tools to handle challenging physical problems on unbounded domains. Decay rate ranges as large of 6 orders of magnitudes can be approximated efficiently and concurrently.

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Composite generalized Laguerre spectral method for nonlinear Fokker–Planck equations on the whole line

Wang

2016

Math Methods in App Sciences

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In this paper, we propose a composite Laguerre spectral method for the nonlinear Fokker-Planck equations modelling the relaxation of fermion and boson gases. A composite Laguerre spectral scheme is constructed. Its convergence is proved. Numerical results show the efficiency of this approach and coincide well with theoretical analysis. Some results on the Laguerre approximation and techniques used in this paper are also applicable to other nonlinear problems on the whole line.

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Domain Decomposition Spectral Method for Mixed Inhomogeneous Boundary Value Problems of High Order Differential Equations on Unbounded Domains

Cited by 12 publications

References 17 publications

Multi-Domain Decomposition Pseudospectral Method for Nonlinear Fokker–Planck Equations

Multi-Domain Decomposition Pseudospectral Method for Nonlinear Fokker–Planck Equations

Efficient Mapping of High Order Basis Sets for Unbounded Domains

Composite generalized Laguerre spectral method for nonlinear Fokker–Planck equations on the whole line

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