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Representation Formulas (representation + formula)

Distribution by Scientific Domains
Mathematics and Statistics100%

Kinds of Representation Formulas

  • integral representation formula
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    Selected Abstracts

    On the integral representation formula for a two-component elastic composite

    MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2006
    Miao-Jung Ou
    Abstract The aim of this paper is to derive, in the Hilbert space setting, an integral representation formula for the effective elasticity tensor for a two-component composite of elastic materials, not necessarily well-ordered. This integral representation formula implies a relation which links the effective elastic moduli to the N -point correlation functions of the microstructure. Such relation not only facilitates a powerful scheme for systematic incorporation of microstructural information into bounds on the effective elastic moduli but also provides a theoretical foundation for inverse-homogenization. The analysis presented in this paper can be generalized to an n -component composite of elastic materials. The relations developed here can be applied to the inverse-homogenization for a special class of linear viscoelastic composites. The results will be presented in another paper. Copyright © 2005 John Wiley & Sons, Ltd. [source]

    Some Riemann boundary value problems in Clifford analysis

    MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2010
    Klaus Gürlebeck
    Abstract In this paper, we mainly study the Rm (m>0) Riemann boundary value problems for functions with values in a Clifford algebra C,(V3, 3). We prove a generalized Liouville-type theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k -monogenic functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions are presented. Combining the Plemelj formula and the integral representation formulas with the above generalized Liouville theorem, we prove that the Rm (m>0) Riemann boundary value problems for monogenic functions, harmonic functions and biharmonic functions are solvable. Explicit representation formulas of the solutions are given. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    Multi-periodic eigensolutions to the Dirac operator and applications to the generalized Helmholtz equation on flat cylinders and on the n -torus

    MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2009
    Denis Constales
    Abstract In this paper, we study the solutions to the generalized Helmholtz equation with complex parameter on some conformally flat cylinders and on the n -torus. Using the Clifford algebra calculus, the solutions can be expressed as multi-periodic eigensolutions to the Dirac operator associated with a complex parameter ,,,. Physically, these can be interpreted as the solutions to the time-harmonic Maxwell equations on these manifolds. We study their fundamental properties and give an explicit representation theorem of all these solutions and develop some integral representation formulas. In particular, we set up Green-type formulas for the cylindrical and toroidal Helmholtz operator. As a concrete application, we explicitly solve the Dirichlet problem for the cylindrical Helmholtz operator on the half cylinder. Finally, we introduce hypercomplex integral operators on these manifolds, which allow us to represent the solutions to the inhomogeneous Helmholtz equation with given boundary data on cylinders and on the n -torus. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    Elliptic and parabolic problems in unbounded domains

    MATHEMATISCHE NACHRICHTEN, Issue 1 2004
    Patrick Guidotti
    Abstract We consider elliptic and parabolic problems in unbounded domains. We give general existence and regularity results in Besov spaces and semi-explicit representation formulas via operator-valued fundamental solutions which turn out to be a powerful tool to derive a series of qualitative results about the solutions. We give a sample of possible applications including asymptotic behavior in the large, singular perturbations, exact boundary conditions on artificial boundaries and validity of maximum principles. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

    Numerical solution of thermal convection problems using the multidomain boundary element method

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2002
    W. F. Florez
    Abstract The multidomain dual reciprocity method (MD-DRM) has been effectively applied to the solution of two-dimensional thermal convection problems where the momentum and energy equations govern the motion of a viscous fluid. In the proposed boundary integral method the domain integrals are transformed into equivalent boundary integrals by the dual reciprocity approach applied in a subdomain basis. On each subregion or domain element the integral representation formulas for the velocity and temperature are applied and discretised using linear continuous boundary elements, and the equations from adjacent subregions are matched by additional continuity conditions. Some examples showing the accuracy, the efficiency and flexibility of the proposed method are presented. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 469,489, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10016 [source]