Approximate solution of a singular integral equation by means of jacobi polynomials

Karpenko, L. N.

doi:10.1016/0021-8928(67)90103-7

Cited by 39 publications

(15 citation statements)

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“…Now the set (4.14) and (4.16) can be solved numerically. In recent years an expanding number of methods has appeared for doing this (see for instance [30][31][32][33][34][35][36][37]). Perhaps the simplest but still very effective technique is the one introduced by Erdogan and Gupta [34], supplemented by Krenk's interpolation [36]: where the intensity values Y(1), Y(-1) at the crack tips can be found by For the stress intensity factor the interpolation values (4.22) must also be used.…”

Section: ) It Is Easy To Observe That ~Ba(za) Behaves Like [Za+(vi/mentioning

confidence: 99%

An integral equation approach to self-similar plane-elastodynamic crack problems

Georgiadis¹

1991

J Elasticity

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Akstraet. The elastodynamic problem of an expanding crack under homogeneous polynomialform loading was reduced to the solution of a Cauchy singular integral equation. In this manner the solution of the original problem can be obtained by using well-known numerical treatments available for Cauchy SIEs. The procedure was accomplished by means of the Busemann-Chaplygin similarity technique and complex variable methods. The analysis has been restricted to the subsonic case.

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Section: ) It Is Easy To Observe That ~Ba(za) Behaves Like [Za+(vi/mentioning

confidence: 99%

An integral equation approach to self-similar plane-elastodynamic crack problems

Georgiadis¹

1991

J Elasticity

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“…More precisely, in order to obtain a high order of convergence of approximate solutions, it is necessary to use orthogonal polynomials of Chebyshev, or more generally, of Jacobi. In the case where the functions a and b in (1.1) are constant on [−1, 1] Jacobi polynomials were applied to solve integral equations in [9].…”

Section: B(t) = M(t T) and K(x T) = M(x T) − M(t T) T − X mentioning

confidence: 99%

Orthogonal Approximate Solution of Cauchy-type Singular Integral Equations

Smarzewski

Sheshko

2003

Comput. Methods Appl. Math.

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“…The problem has also been considered in [3] in rather general terms. More notable and somewhat more detailed applications using the properties of Jacobi polynomials appeared in recent papers [4] and [5]. In [5], a special case of (1) is considered in integrated form.…”

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“…Hence, by using the properties of the Jacobi polynomials [4], [7], [9] and following a procedure similar to that of 2 and 3 ofthis paper, a second approximate method to solve the system ofequations (26) may also be developed. To do this, we first premultiply (26) by A-(assuming that A is nonsingular) and diagonalize the matrix A-1B D (dij).…”

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confidence: 99%

Approximate Solutions of Systems of Singular Integral Equations

Erdoğan¹

1969

SIAM J. Appl. Math.

175

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Using the properties of the related orthogonal polynomials, approximate solutions of systems of simultaneous singular integral equations are obtained, in which the essential features of the singularity of the unknown functions are preserved. In the system of integral equations of the first kind, the fundamental solution is the weight function of the Chebyshev polynomials of first or second kind. In the system of singular integral equations of the second kind with constant coefficients, the elements of the fundamental matrix are the weights of Jacobi polynomials. A direct method is introduced to obtain the fundamental matrix of the system. The approximate solution is then expressed as the fundamental function, representing the singular behavior of the unknown functions, multiplied by a series of proper orthogonal polynomials with unknown coefficients. The techniques of deriving the system of algebraic equations to determine these coefficients are described. In order to have an idea about the effectiveness of the method and the convergence problems arising from the truncation of the series, numerical results of an elastostatic problem are included.

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Approximate solution of a singular integral equation by means of jacobi polynomials

Cited by 39 publications

References 0 publications

An integral equation approach to self-similar plane-elastodynamic crack problems

An integral equation approach to self-similar plane-elastodynamic crack problems

Orthogonal Approximate Solution of Cauchy-type Singular Integral Equations

Approximate Solutions of Systems of Singular Integral Equations

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