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Analytical Solution of Laplace and Poisson Equations Using Conformal Mapping and Kronecker Products


In this paper, using the eigenvalues and eigenvectors of symmetric block diagonal matrices with infinite dimension and numerical method of finite difference, a closed-form solution for exact solution of Laplace equation is presented. The method of this paper has applications in different states of boundary conditions like Neumann, Dirichlet, and other mixed boundary conditions. Using the method of this paper, a mathematical model for the exact solution of the Poisson equation is derived. Since these equations have many applications in engineering problems, in each part of this paper, examples, like water seepage problem through the soil and torsion of prismatic bars, are presented. Finally, a method is provided for torsion problem of prismatic bars with non-circular and non-rectangular cross-sections utilizing conformal mapping.

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The second author is grateful to the University of Tehran for financial support under grant number 27938/1/15.

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Correspondence to H. Rahami.

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Mirhosseini, S.M., Rahami, H. & Kaveh, A. Analytical Solution of Laplace and Poisson Equations Using Conformal Mapping and Kronecker Products. Int J Civ Eng 14, 369–377 (2016).

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  • Analytical solution
  • Laplace equation and Poisson equation
  • Block diagonal matrices
  • Water seepage through soil
  • Torsion of non-circular and non-rectangular cross-sections