Pseudospectral Solution of the Fokker–Planck Equation with Equilibrium Bistable States: the Eigenvalue Spectrum and the Approach to Equilibrium

Shizgal, Bernie D.

doi:10.1007/s10955-016-1594-9

Cited by 14 publications

(15 citation statements)

References 52 publications

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“…This potential has been used to solve the Fokker-Planck equation [25]. The Schrödingerlike equation associated to this specific problem implies an analytical solution of the Riccati equation for the ground state.…”

Section: Second Type Of Polynomial Potentialmentioning

confidence: 99%

Variational eigenfunctions for excited states inspired by supersymmetric quantum mechanics and the Gram–Schmidt process

Batael,

Drigo Filho

2023

J. Phys. A: Math. Theor.

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Factorization methods such as the Hamiltonian hierarchy have been useful to find eigenfunctions for Schrödinger equations, in particular, for potentials that are partially or approximately solvable. In this paper, an alternative approach is proposed to study excited states via the variational method. The trial functions are built from the exact or approximate superpotential for the ground state combined with the Gram–Schmidt process to ensure orthogonalization between the functions. The results found variationally for one dimensional potentials are compared with previous results from the literature. The energy eigenvalues obtained agree with previous ones and, for most of the results, the percentage difference between the proposed approach and others in the literature is less than 0.1%. The method introduced is an effective and intuitive approach to determine trial wave functions for the excited states. This approach can be useful in studying the Schrödinger equation and related problems which can be mapped onto a Schrödinger type-equation as, for example, the Fokker–Planck equation.

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Section: Second Type Of Polynomial Potentialmentioning

confidence: 99%

Variational eigenfunctions for excited states inspired by supersymmetric quantum mechanics and the Gram–Schmidt process

Batael,

Drigo Filho

2023

J. Phys. A: Math. Theor.

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“…The eigenvalues obtained using Chebfun are compared with those obtained in [24] when the interval of integration has length l := 4. They are reported in Table 4 and are in an excellent agreement.…”

Section: Fokker-planck Eigenproblemmentioning

confidence: 99%

Spectral Collocation Solutions to Second Order Singular Sturm-Liouville Eigenproblems

Gheorghiu

2020

Preprint

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We comparatively use some classical spectral collocation methods as well as highly performing Chebfun algorithms in order to compute the eigenpairs of second order singular Sturm-Liouville problems with separated self-adjoint boundary conditions. For both the limit-circle non oscillatory and oscillatory cases we pay a particular attention. Some "hard" benchmark problems, for which usual numerical methods (f. d., f. e. m., etc.) fail, are analysed. For the very challenging Bessel eigenproblem we will try to find out the source and the meaning of the singularity in the origin. For a double singular eigenproblem due to Dunford and Schwartz we we try to find out the precise meaning of the notion of continuous spectrum. For some singular problems only a tandem approach of the two classes of methods produces credible results.

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“…and satisfy the boundary conditions (10). In (11) the nodes x k are equidistant with spacing h and symmetric with respect to the origin.…”

Section: A Singular Schrödinger Eigenproblem On the Real Linementioning

confidence: 99%

“…In contrast, we will use an analogous transformation but which depends on only one parameter. Also, very recently spectral methods based on non-classical orthogonal polynomials have been used in [11] in order to solve some Schrödinger problems connected with Fokker-Planck operator. A particular attention will be paid in this paper to the challenging issue of continuous spectra vs. discrete (numerical) eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations

Gheorghiu¹

2020

Preprint

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We are concerned with the study of some classical spectral collocation methods as well as with the new software system Chebfun in computing high order eigenpairs of singular and regular Schrödinger eigenproblems. We want to highlight both the qualities as well as the shortcomings of these methods and evaluate them in conjunction with the usual ones. In order to resolve a boundary singularity we use Chebfun with domain truncation. Although it is applicable with spectral collocation, a special technique to introduce boundary conditions as well as a coordinate transform, which maps an unbounded domain to a finite one, are the special ingredients. A challenging set of "hard"benchmark problems, for which usual numerical methods (f. d., f. e. m., shooting etc.) fail, are analyzed. In order to separate "good"and "bad"eigenvalues we estimate the drift of the set of eigenvalues of interest with respect to the order of approximation and/or scaling of domain parameter. It automatically provides us with a measure of the error within which the eigenvalues are computed and a hint on numerical stability. We pay a particular attention to problems with almost multiple eigenvalues as well as to problems with a mixed spectrum.

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Pseudospectral Solution of the Fokker–Planck Equation with Equilibrium Bistable States: the Eigenvalue Spectrum and the Approach to Equilibrium

Cited by 14 publications

References 52 publications

Variational eigenfunctions for excited states inspired by supersymmetric quantum mechanics and the Gram–Schmidt process

Variational eigenfunctions for excited states inspired by supersymmetric quantum mechanics and the Gram–Schmidt process

Spectral Collocation Solutions to Second Order Singular Sturm-Liouville Eigenproblems

Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations

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