Expansions in terms of heat polynomials and associated functions

Rosenbloom, P. C.; Widder, D. V.

doi:10.1090/s0002-9947-1959-0107118-2

Cited by 160 publications

(93 citation statements)

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“…This work is an extension of some results of Rosenbloom and Widder [7] on expansions in terms of heat polynomials and associated functions. Despite these works, no attention is given to the expansions of distributions in terms of heat polynomials and their Appell transforms.…”

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confidence: 74%

Expansions of distributions in terms of generalized heat polynominals and their Appell transforms

Pathak

Debnath

1980

International Journal of Mathematics and Mathematical Sciences

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ABSTRACT. This paper is concerned with expansions of distributions in terms of the generalized heat polynomials and of their Appell transforms. Two different techniques are used to prove theorems concerning expansions of distributions. A theorem which provides an orthogonal series expansion of generalized functions is also established. It is shown that this theorem gives an inversion formula for a certain generalized integral transformation.

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confidence: 74%

Expansions of distributions in terms of generalized heat polynominals and their Appell transforms

Pathak

Debnath

1980

International Journal of Mathematics and Mathematical Sciences

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“…The non-stationary problems are brought down to stationary by the discretization of the time. The first paper devoted to the Trefftz functions in which the time is considered as a continuous variable, discussed a one-dimensional (one spatial variable) heat conduction equation (Rosenbloom and Widder 1956). This aspect of the Trefftz functions method was developed for the heat conduction problems in the papers (Ciałkowski et al 1999(Ciałkowski et al , 2007Yano et al 1983) for the wave equation and thermoelasticity problems in the papers (Grysa and Maciag 2011;Maciag 2004Maciag , 2005Maciag , 2007Maciag and Wauer 2005a, b) and for the equation of a plate vibration in the paper .…”

Section: Introductionmentioning

confidence: 99%

Solution of the direct and inverse problems for beam

Maciąg

Pawińska

2014

Comp. Appl. Math.

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The article presents an approximate method of solving direct and inverse problems described by Bernoulli-Euler inhomogeneous equation of vibrations of a beam. A semianalytical solution is approximated by a linear combination of the Trefftz functions (T-functions, solving functions), which satisfies identically the homogenous equation describing the vibrations of a beam. In the paper, the properties of the solving functions have been investigated, theorems concerning their linear independence have been formulated and proved. A method of obtaining the particular solution of the inhomogeneous equation has been shown. To get this solution, recurrent formulas enabling us to determine the inverse operator for monomials have been derived. The paper discusses two kinds of inverse problems. The first one is a boundary inverse problem, in which the boundary conditions are to be determined, based on known displacements within the area. In the second one, the load on the beam needs to be found (identification of the source). The solving functions can be used as a finite element method base functions. This approach is tested for solving inverse problems. The paper includes examples which illustrate the usefulness of the method.

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“…The method originates from [3] but only for the case of one-dimensional heat-conduction problems. Heat polynomials were applied for solving unsteady heat conduction problems [4].…”

Section: Introductionmentioning

confidence: 99%

Solution of the two-dimensional wave equation by using wave polynomials

Macig

Wauer

2005

J Eng Math

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Abstract. The paper demonstrates a specific power-series-expansion technique to solve approximately the two-dimensional wave equation. As solving functions (Trefftz functions) so-called wave polynomials are used. The presented method is useful for a finite body of certain shape geometry. Recurrent formulas for the wave polynomials and their derivatives are obtained in the Cartesian and polar coordinate system. The accuracy of the method is discussed and some examples are shown.

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Expansions in terms of heat polynomials and associated functions

Cited by 160 publications

References 4 publications

Expansions of distributions in terms of generalized heat polynominals and their Appell transforms

Expansions of distributions in terms of generalized heat polynominals and their Appell transforms

Solution of the direct and inverse problems for beam

Solution of the two-dimensional wave equation by using wave polynomials

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