Chebyshev Polynomials for Solving a Class of Singular Integral Equations

Dardery, Samah M.; Allan, Mohamed M.

doi:10.4236/am.2014.54072

Cited by 12 publications

(6 citation statements)

References 13 publications

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“…We will now solve these equations using the Chebyshev polynomials as follows. Since the sheet current density is unbounded at ξ = ±1 the solution can be found, see [3][4][5], as:…”

Section: Solution For a Strip In The Meissner Statementioning

confidence: 99%

“…where b and J 0 are the unknowns. To satisfy the endpoint conditions (26), we set, following [3][4][5], J 0 (bη) = J c 1 − η 2 ψ (η) and, since J 0 should be an even function,…”

Section: The Bean Model Solution For a Strip Carrying Transport Currentmentioning

confidence: 99%

“…Useful by themselves, these solutions are often used also to test the accuracy of various numerical methods developed for more complicated applied superconductivity problems for which no analytical solution is known. Numerical solution to equation ( 1) can be sought also using expansions in Chebyshev polynomials (the Chebyshev spectral method) [3][4][5]. These expansions converge uniformly for any function of bounded variation (i.e.…”

Section: Introductionmentioning

confidence: 99%

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Chebyshev spectral method for superconductivity problems

Prigozhin

Kozyrev

2020

Supercond. Sci. Technol.

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We solve several applied superconductivity problems using the series in Chebyshev polynomials. Although this method may be less general than the usually employed finite element methods, accuracy of the obtained solutions is often much higher due to the very fast convergence of the Chebyshev expansions. First, as an introduction, we consider the thin strip magnetization and transport current problems for which the analytical solutions are known. Then, assuming the Bean critical-state model, we apply this method to two-dimensional problems for stacks and pancake coils made of coated conductors. Finally, we extend this approach to evolutionary stack problems with an arbitrary current-voltage relation. Simulation of coils with a large number of turns is also discussed.

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“…We will now solve these equations using the Chebyshev polynomials as follows. Since the sheet current density is unbounded at ξ = ±1 the solution can be found, see [3][4][5], as:…”

Section: Solution For a Strip In The Meissner Statementioning

confidence: 99%

“…where b and J 0 are the unknowns. To satisfy the endpoint conditions (26), we set, following [3][4][5], J 0 (bη) = J c 1 − η 2 ψ (η) and, since J 0 should be an even function,…”

Section: The Bean Model Solution For a Strip Carrying Transport Currentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Chebyshev spectral method for superconductivity problems

Prigozhin

Kozyrev

2020

Supercond. Sci. Technol.

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“…Darbery and Allan [12] summarized three term relations of four kind of Chebyshev polynomials which is given in Mason and Handscomb [21].…”

Section: Preliminariesmentioning

confidence: 99%

“…For singular integral equations many efficient methods are derived and proved convergence of the method as well as showed illustrative examples (see Chakrabarti at al. [10]- [11], Dardery and Allan [12], Capobianco et al [13], Elliot [14]- [15] and so on).…”

Section: Introductionmentioning

confidence: 99%

Semi Bounded Solution of Hypersingular Integral Equations of the First Kind on the Rectangle

Eshkuvatov¹,

Kommuji²,

Aloev³

et al. 2020

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A hypersingular integral equations (HSIEs) of the first kind on the interval [−1, 1] with the assumption that kernel of the hypersingular integral is constant on the diagonal of the domain is considered. Truncated series of Chebyshev polynomials of the third and fourth kinds are used to find semi bounded (unbounded on the left and bounded on the right and vice versa) solutions of HSIEs of first kind. Exact calculations of singular and hypersingular integrals with respect to Chebyshev polynomials of third and forth kind with corresponding weights allows us to obtain high accurate approximate solution. Gauss-Chebyshev quadrature formula is extended for regular kernel integrals. Three examples are provided to verify the validity and accuracy of the proposed method. Numerical examples reveal that approximate solutions are exact if solution of HSIEs is of the polynomial forms with corresponding weights.

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