Extending the unified transform: curvilinear polygons and variable coefficient PDEs

2018 ◽  
Vol 40 (2) ◽  
pp. 976-1004 ◽  
Author(s):  
Matthew J Colbrook
Keyword(s):  
Fourier Transforms ◽  
Elliptic Boundary ◽  
Neumann Boundary ◽  
Integral Transforms ◽  
Variable Coefficients ◽  
Central Component ◽  
Unknown Boundary ◽  
Boundary Values ◽  
Global Relation ◽  
Chebyshev Interpolation

Abstract We provide the first significant extension of the unified transform (also known as the Fokas method) applied to elliptic boundary value problems, namely, we extend the method to curvilinear polygons and partial differential equations (PDEs) with variable coefficients. This is used to solve the generalized Dirichlet-to-Neumann map. The central component of the unified transform is the coupling of certain integral transforms of the given boundary data and of the unknown boundary values. This has become known as the global relation and, in the case of constant coefficient PDEs, simply links the Fourier transforms of the Dirichlet and Neumann boundary values. We extend the global relation to PDEs with variable coefficients and to domains with curved boundaries. Furthermore, we provide a natural choice of global relations for separable PDEs. These generalizations are numerically implemented using a method based on Chebyshev interpolation for efficient and accurate computation of the integral transforms that appear in the global relation. Extensive numerical examples are provided, demonstrating that the method presented in this paper is both accurate and fast, yielding exponential convergence for sufficiently smooth solutions. Furthermore, the method is straightforward to use, involving just the construction of a simple linear system from the integral transforms, and does not require knowledge of Green’s functions of the PDE. Details on the implementation are discussed at length.

scholarly journals The unified transform for mixed boundary condition problems in unbounded domains

2019 ◽  
Vol 475 (2222) ◽  
pp. 20180605 ◽  
Author(s):  
Matthew J. Colbrook ◽  
Lorna J. Ayton ◽  
Athanassios S. Fokas
Keyword(s):  
Boundary Condition ◽  
Mixed Boundary Condition ◽  
Mixed Boundary ◽  
Acoustic Scattering ◽  
Unbounded Domains ◽  
Unknown Boundary ◽  
Normal Velocity ◽  
Boundary Values ◽  
Helmholtz Equations ◽  
Global Relation

This paper implements the unified transform to problems in unbounded domains with solutions having corner singularities. Consequently, a wide variety of mixed boundary condition problems can be solved without the need for the Wiener–Hopf technique. Such problems arise frequently in acoustic scattering or in the calculation of electric fields in geometries involving finite and/or multiple plates. The new approach constructs a global relation that relates known boundary data, such as the scattered normal velocity on a rigid plate, to unknown boundary values, such as the jump in pressure upstream of the plate. By approximating the known data and the unknown boundary values by suitable functions and evaluating the global relation at collocation points, one can accurately obtain the expansion coefficients of the unknown boundary values. The method is illustrated for the modified Helmholtz and Helmholtz equations. In each case, comparisons between the traditional Wiener–Hopf approach, other spectral or boundary methods and the unified transform approach are discussed.

scholarly journals Evolution equations on time-dependent intervals

2019 ◽  
Vol 84 (5) ◽  
pp. 1044-1060
Author(s):  
Athanasios S Fokas ◽  
Beatrice Pelloni ◽  
Baoqiang Xia
Keyword(s):  
Evolution Equations ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Time Dependent ◽  
Unknown Boundary ◽  
Boundary Values ◽  
Linear Evolution ◽  
First Case ◽  
Linear Integral ◽  
Initial Boundary Value

Abstract We study initial boundary value problems for linear evolution partial differential equations posed on a time-dependent interval $l_1(t)<x<l_2(t)$, $0<t<T$, where $l_1(t)$ and $l_2(t)$ are given, real, differentiable functions, and $T$ is an arbitrary constant. For such problems, we show how to characterize the unknown boundary values in terms of the given initial and boundary conditions. As illustrative examples we consider the heat equation and the linear Schrödinger equation. In the first case, the unknown Neumann boundary values are expressed in terms of the Dirichlet boundary values and of the initial value through the unique solution of a system of two linear integral equations with explicit kernels. In the second case, a similar result can be proved but only for a more restrictive class of boundary curves.

A Hybrid Method for Solving Inhomogeneous Elliptic PDEs Based on Fokas Method

2018 ◽  
Vol 18 (4) ◽  
pp. 653-672 ◽  
Author(s):  
Eleftherios-Nektarios G. Grylonakis ◽  
Christos K. Filelis-Papadopoulos ◽  
George A. Gravvanis
Keyword(s):  
Hybrid Method ◽  
Integral Transforms ◽  
Numerical Results ◽  
Elliptic Pdes ◽  
Unknown Boundary ◽  
Convex Polygons ◽  
Helmholtz Equations ◽  
Global Relation ◽  
Fokas Method ◽  
Full Solution

AbstractIn this paper we propose a hybrid method for solving inhomogeneous elliptic PDEs based on the unified transform. This approach relies on the derivation of the global relation, containing certain integral transforms of the given boundary data as well as of the unknown boundary values. Herewith, the approximate global relation for the Poisson equation is solved numerically using a collocation method on the complex λ-plane, based on Legendre expansions. The corresponding numerical results are presented using closed-form expressions and numerical approximations for different types of boundary and source data, indicating the applicability of the considered approach. Additionally, the full solution is computed in a recursive manner by splitting the domain into smaller concentric polygons, and by using a spatial-stepping scheme followed by an interpolation step. Furthermore, numerical results are also given for the solution of the Poisson and the inhomogeneous Helmholtz equations on several convex polygons. Additional results are provided for the case of nonconvex polygons as well as for the case of a problem with discontinuities across an interface. The proposed approach provides a framework for solving inhomogeneous elliptic PDEs using the unified transform.

Wavelet packets associated with linear canonical transform on spectrum

2021 ◽  
pp. 2150030
Author(s):  
M. Younus Bhat ◽  
Aamir H. Dar
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Integral Transforms ◽  
Wavelet Packets ◽  
Linear Canonical Transform ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Vector Valued

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.

Derivation by a Transform Method of Integral Equations of Unsteady Lifting Surface Theory in Subsonic and Supersonic Flow

1979 ◽  
Vol 30 (4) ◽  
pp. 529-543
Author(s):  
Shigenori Ando ◽  
Akio Ichikawa
Keyword(s):  
Fourier Transforms ◽  
Integral Transforms ◽  
Physical Models ◽  
Streamwise Direction ◽  
Fourier Inversion ◽  
Transform Method ◽  
Inviscid Flows ◽  
Surface Theory ◽  
Method Of Integral Equations ◽  
Lifting Surface

SummaryApplications of “integral transforms of in-plane coordinate variables” in order to formulate unsteady planar lifting surface theories are demonstrated for both sub- and supersonic inviscid flows. It is concise and pithy. Fourier transforms are exclusively used, except for only Laplace transform in the supersonic streamwise direction. It is found that the streamwise Fourier inversion in the subsonic case requires some caution. Concepts based on the theory of distributions seem to be essential, in order to solve the convergence difficulties of integrals. Apart from this caution, the method of integral transforms of in-plane coordinate variables makes it be pure-mathematical to formulate the lifting surface problems, and makes aerodynamicist’s experiences and physical models such as vortices or doublets be useless.

scholarly journals Response Due To Impulsive Force In Generalized Thermomicrostretch Elastic Solid

2015 ◽  
Vol 20 (3) ◽  
pp. 487-502
Author(s):  
V. Kumar ◽  
R. Singh
Keyword(s):  
Fourier Transforms ◽  
Normal Force ◽  
Elastic Solid ◽  
Integral Transforms ◽  
Normal Displacement ◽  
Impulsive Force ◽  
Field Equations ◽  
Laplace And Fourier Transforms ◽  
Eigen Value ◽  
Plain Strain

Abstract A two dimensional Cartesian model of a generalized thermo-microstretch elastic solid subjected to impulsive force has been studied. The eigen value approach is employed after applying the Laplace and Fourier transforms on the field equations for L-S and G-L model of the plain strain problem. The integral transforms have been inverted into physical domain numerically and components of normal displacement, normal force stress, couple stress and microstress have been illustrated graphically.

The Heat Equation for the -Neumann Problem, II

1988 ◽  
Vol 40 (2) ◽  
pp. 502-512 ◽  
Author(s):  
Richard Beals ◽  
Nancy K. Stanton
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Neumann Problem ◽  
Elliptic Boundary ◽  
Neumann Boundary ◽  
Hermitian Manifold ◽  
Neumann Boundary Conditions ◽  
Compact Manifolds ◽  
Elliptic Boundary Value ◽  
Image Position

Let Ω be a compact complex n + 1-dimensional Hermitian manifold with smooth boundary M. In [2] we proved the following.THEOREM 1. Suppose satisfies condition Z(q) with 0 ≦ q ≦ n. Let □p,q denote the -Laplacian on (p, q) forms onwhich satisfy the -Neumann boundary conditions. Then as t → 0;,(0.1)(If q = n + 1, the -Neumann boundary condition is the Dirichlet boundary condition and the corresponding result is classical.)Theorem 1 is a version for the -Neumann problem of results initiated by Minakshisundaram and Pleijel [8] for the Laplacian on compact manifolds and extended by McKean and Singer [7] to the Laplacian with Dirichlet or Neumann boundary conditions and by Greiner [5] and Seeley [9] to elliptic boundary value problems on compact manifolds with boundary. McKean and Singer go on to show that the coefficients in the trace expansion are integrals of local geometric invariants.

scholarly journals Supraconvergence of a Finite Difference Scheme for Elliptic Boundary Value Problems of the Third Kind in Fractional Order Sobolev Spaces

2006 ◽  
Vol 6 (2) ◽  
pp. 154-177 ◽  
Author(s):  
E. Emmrich ◽  
R.D. Grigorieff
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Elliptic Boundary ◽  
Special Kind ◽  
Variable Coefficients ◽  
Second Order ◽  
Nonuniform Grids ◽  
Close Relationship

AbstractIn this paper, we study the convergence of the finite difference discretization of a second order elliptic equation with variable coefficients subject to general boundary conditions. We prove that the scheme exhibits the phenomenon of supraconvergence on nonuniform grids, i.e., although the truncation error is in general of the first order alone, one has second order convergence. All error estimates are strictly local. Another result of the paper is a close relationship between finite difference scheme and linear finite element methods combined with a special kind of quadrature. As a consequence, the results of the paper can be viewed as the introduction of a fully discrete finite element method for which the gradient is superclose. A numerical example is given.

scholarly journals Gradient estimates via rearrangements for solutions of some Schrödinger equations

2018 ◽  
Vol 16 (03) ◽  
pp. 339-361 ◽  
Author(s):  
Sibei Yang ◽  
Der-Chen Chang ◽  
Dachun Yang ◽  
Zunwei Fu
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Gradient Estimates ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Elliptic Boundary Value ◽  
Neumann Boundary Value Problems ◽  
Laplace Type

In this paper, by applying the well-known method for dealing with [Formula: see text]-Laplace type elliptic boundary value problems, the authors establish a sharp estimate for the decreasing rearrangement of the gradient of solutions to the Dirichlet and the Neumann boundary value problems of a class of Schrödinger equations, under the weak regularity assumption on the boundary of domains. As applications, the gradient estimates of these solutions in Lebesgue spaces and Lorentz spaces are obtained.

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