A unified approach for the solution of the Fokker-Planck equation

Wei, Guannan

doi:10.1088/0305-4470/33/27/311

Cited by 83 publications

(101 citation statements)

References 42 publications

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“…The second inequality indicates that if ratio r is chosen, then an appropriate computational support can be used to ensure a given accuracy 10 Àg . This theoretical estimation is in good agreement with previous numerical tests (Wei, 2000a). The implementation of the DSC algorithm for structural analysis is described below.…”

Section: Discrete Singular Convolutionsupporting

confidence: 88%

“…For integrity and convenience, the DSC algorithm is briefly reviewed in this section. However, the reader is referred to the original work for more detailed information (Wei, 1999a(Wei, , 2000a(Wei, , 2001a. The implementation of the DSC algorithm to plate analysis is described.…”

Section: The Problem and Methods Of Solutionmentioning

confidence: 99%

“…The DSC concerns with the computer realization of singular convolutions (SCs) (Wei, 1999a(Wei, , 2000a(Wei, , 2001a. Its mathematical foundation is the theory of distributions (Schwarz, 1951) and wavelet analysis.…”

Section: Introductionmentioning

confidence: 99%

“…By appropriately selecting parameters of a DSC kernel, the DSC approach exhibits controllable accuracy for approximations and shows excellent flexibility in handling complex geometries, boundary and internal support conditions. It was demonstrated (Wei, 2000a) that different implementations of the DSC algorithm, such as global, local, Galerkin, collocation, and finite difference, can be deduced from a single starting point. Thus, the DSC algorithm provides a unified representation to these numerical methods.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, the DSC algorithm provides a unified representation to these numerical methods. The DSC algorithm has found its success in solving the FokkerPlanck equation (Wei, 1999a(Wei, , 2000a, the Schr€ o odinger equation (Wei, 2000b), the Navier-Stokes equation for incompressible fluid flow (Wei, 2001a;Wan et al, 2001). In the context of image processing, DSC kernels were used to facilitate a new anisotropic diffusion operator for image restoration from noise corruption (Wei, 1999b).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Plate vibration under irregular internal supports

Zhao

Wang

Xiang

2002

International Journal of Solids and Structures

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This paper studies the problem of plate vibration under complex and irregular internal support conditions. Such a problem has its widely spread industrial applications and has not been addressed in the literature yet, partially due to the numerical difficulties. A novel computational method, discrete singular convolution (DSC), is introduced for solving this problem. The DSC algorithm exhibits controllable accuracy for approximations and shows excellent flexibility in handling complex geometries, boundary conditions and internal support conditions. Convergence and comparison studies are carried out to check the validity and accuracy of the DSC method. Case studies are considered to the combination of a few different boundary conditions and irregular internal supports. The latter are generated by using an image processing algorithm. Completely independent verifications are conducted by using the established pb-2 Ritz method, which is available for two relatively simpler support patterns. The morphology of the first few eigenmodes is found to be localized to largest support-free spatial regions. Ó

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Section: Discrete Singular Convolutionsupporting

confidence: 88%

Section: The Problem and Methods Of Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Plate vibration under irregular internal supports

Zhao

Wang

Xiang

2002

International Journal of Solids and Structures

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Discrete singular convolution methodology for free vibration and stability analyses of arbitrary straight‐sided quadrilateral plates

Cívalek

2007

Commun. Numer. Meth. Engng.

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SUMMARYA new discrete singular convolution (DSC) method is developed for vibration, buckling and static analyses of arbitrary straight-sided quadrilateral plates. The straight-sided quadrilateral domain is mapped into a square domain in the computational space using a four-node element. By using the geometric transformation, the governing equations and boundary conditions of the plate are transformed from the physical domain into a square computational domain. Numerical examples illustrating the accuracy and convergence of the DSC method for straight-sided quadrilateral thin plates such as rectangular, skew, trapezoidal and rhombic plates are presented. The results obtained by the DSC method were compared with those obtained by the other numerical and analytical methods.

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Parameter optimization in the regularized Shannon's kernels of higher‐order discrete singular convolutions

Xiong

Zhao

2003

Commun. Numer. Meth. Engng.

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SUMMARYThe -type discrete singular convolution (DSC) algorithm has recently been proposed and applied to solve kinds of partial di erential equations (PDEs). With appropriate parameters, particularly the key parameter r in its regularized Shannon's kernel, the DSC algorithm can be more accurate than the pseudospectral method. However, it was previously selected empirically or under constrained inequalities without optimization. In this paper, we present a new energy-minimization method to optimize r for higher-order DSC algorithms. Objective functions are proposed for the DSC algorithm for numerical di erentiators of any di erential order with any discrete convolution width. Typical optimal parameters are also shown. The validity of the proposed method as well as the resulted optimal parameters have been veriÿed by extensive examples.

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A unified approach for the solution of the Fokker-Planck equation

Cited by 83 publications

References 42 publications

Plate vibration under irregular internal supports

Plate vibration under irregular internal supports

Discrete singular convolution methodology for free vibration and stability analyses of arbitrary straight‐sided quadrilateral plates

Parameter optimization in the regularized Shannon's kernels of higher‐order discrete singular convolutions

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