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A numerical technique for linear elliptic partial differential equations in polygonal domains.

P Hashemzadeh1, A S Fokas2, S A Smitheman1

  • 1Department of Applied Mathematics and Theoretical Physics , University of Cambridge , Cambridge CB3 0WA, UK.

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Summary

The unified transform method offers a novel approach to solving partial differential equations (PDEs) by reformulating boundary value problems in the complex Fourier plane. This method provides guidelines for numerical implementation, improving efficiency for complex problems.

Keywords:
Fokas methodFokas transformHelmholtz equationLaplaceelliptic partial differential equationmodified Helmholtz equation

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Area of Science:

  • Applied Mathematics
  • Numerical Analysis
  • Mathematical Physics

Background:

  • Integral representations for elliptic PDEs rely on Green's theorem but require unknown boundary values.
  • Boundary value problems (BVPs) in PDEs often involve challenges with determining both Dirichlet and Neumann data.

Purpose of the Study:

  • To present guidelines for the numerical implementation of the unified transform method for linear elliptic PDEs.
  • To offer concrete rules for selecting basis functions and collocation points to ensure a low condition number in the numerical solution.

Main Methods:

  • The unified transform method formulates BVPs in the complex Fourier plane, analogous to Green's function methods in the physical plane.
  • Two global relations in the Fourier plane couple Dirichlet and Neumann boundary values, enabling the determination of the Dirichlet to Neumann map.
  • Numerical implementation involves choosing basis functions and complex collocation points to evaluate global relations.

Main Results:

  • The study provides practical guidelines for selecting appropriate bases and collocation points for the unified transform.
  • Concrete rules are presented for choosing collocation points to minimize the condition number of the resulting linear system.
  • The unified transform offers an elegant approach to determining the Dirichlet to Neumann map for linear elliptic PDEs.

Conclusions:

  • The unified transform method provides an effective and numerically stable approach for solving BVPs of linear elliptic PDEs.
  • The guidelines and rules presented facilitate the practical application and efficient computation using the unified transform.