A fast algorithm for Prandtl's integro-differential equation

Capobianco, Maria Rosaria; Criscuolo, G.; Junghanns, Peter

doi:10.1016/s0377-0427(96)00124-0

Cited by 32 publications

(36 citation statements)

References 11 publications

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“…relation (3.7)). Further, this reformulation allows us to use standard mathematical techniques (see [2,3]) for the stability and the convergence analysis.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

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Numerical solution of a singular integral equation with Cauchy kernel in the plane contact problem

Capobianco

Criscuolo

2010

Quart. Appl. Math.

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Abstract. This paper describes a collocation method for solving numerically a singular integral equation with Cauchy and Volterra operators, associated with a proper constraint condition. The numerical method is based on the transformation of the given integral problem into a hypersingular integral equation and then applying a collocation method to solve the latter equation. Convergence of the resulting method is then discussed, and optimal convergence rates for the collocation and discrete collocation methods are given in suitable weighted Sobolev spaces. Numerical examples are solved using the proposed numerical technique.

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“…relation (3.7)). Further, this reformulation allows us to use standard mathematical techniques (see [2,3]) for the stability and the convergence analysis.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…Collocation and quadrature methods for solving numerically integral equations such as (2.3) are presented in [2]. In that paper the authors develop widely the necessary theory for establishing the convergence results in weighted Sobolev spaces.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Numerical solution of a singular integral equation with Cauchy kernel in the plane contact problem

Capobianco

Criscuolo

2010

Quart. Appl. Math.

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“…Kalandiya [6]. The rigorous theoretical study of Prandtl's IDE in a rather general form was given in the paper of M. R. Capobianco, G. Criscualo, P. A. Junghanns [7] alongside with an effective algorithm for solutions. By Prandtl's IDE numerous problems of contact interaction of stringers with massive deformable bodies of different geometric shapes are described.…”

Section: Introductionmentioning

confidence: 99%

On a method for solving Prandtl's integro-differential equation applied to problems of continuum mechanics using polynomial approximations

Mkhitaryan

Mkrtchyan

Kanetsyan

2017

Z. Angew. Math. Mech.

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In the present paper, using the ideas of the well-known method for solving singular integral equations (SIE), based on Gaussian quadrature formulas for ordinary integrals and singular integral with Cauchy kernel, a somewhat different approach is proposed for solving the Prandtl integro-differential equation (IDE). This approach in accordance with a polynomial method includes different approximation methods for a continuous component of Prandtl's IDE solution, namely an approximation in the form of Bernstein's polynomials, Chebyshev's polynomials of the first kind and approximation in the form of Lagrange interpolation polynomial at the Chebyshev knots. As a result, the Prandtl IDE is reduced to the system of linear algebraic equations (SLAE) of the rather simple structure. These different approximation methods will be illustrated by the example of the problem on contact interaction between a stringer of finite length, having a variable along the length rigidity in tension-compression or a variable cross-section, and an elastic semi-infinite plate. This interaction is described by the Prandtl IDE. The comparative numerical analysis of the results obtained by different methods is carried out.

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“…Em [19], Nagamine & Cuminato Um dos primeiros trabalhos que investigam a convergência uniforme ponderada de métodos de aproximação polinomial, para resolver EIS, foi apresentado por Capobianco, Junghanns, Luther & Mastroianni em [3], e uma extensão foi apresentada em [12]. Em [4], [5] e [14] a análise de convergência do MCP, para resolver EIDS, foi realizada considerando condições de fronteira homogêneas, e em [4] a convergência desse método é provada na norma L 2 ponderada. Em [14], a mesma análise é realizada na norma L 1 ponderada, e em [5], na norma uniforme ponderada.…”

Section: Contextualizaçãounclassified

“…Em [4], a equação (4.1) para a 1 = 0 é analisada em espaços L 2 ponderados, obtendo-se uma estimativa para a taxa de convergência dependente de restrições bastante parecidas com as apresentadas no parágrafo anterior. Em [14], a limitação dos operadores integrais é discutida em espaços L 1 ponderados, tendo restrições sobre as funções f e l diminuídas.…”

Section: Capítulo 4 Outros Métodos Numéricos Do Tipo Colocação Polinunclassified

Método de colocação polinomial para equações integro-diferenciais singulares: convergência

Rosa¹

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A fast algorithm for Prandtl's integro-differential equation

Abstract: Collocation and quadrature methods for singular integro-differential equations of Prandtl's type are studied in weighted Sobolev spaces. A fast algorithm basing on the quadrature method is proposed. Convergence results and error estimates are given

Cited by 32 publications

References 11 publications

Numerical solution of a singular integral equation with Cauchy kernel in the plane contact problem

Numerical solution of a singular integral equation with Cauchy kernel in the plane contact problem

On a method for solving Prandtl's integro-differential equation applied to problems of continuum mechanics using polynomial approximations

Método de colocação polinomial para equações integro-diferenciais singulares: convergência

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