scholarly journals Characterization of balls via solutions of the modified Helmholtz equation

2021 ◽  
Vol 359 (8) ◽  
pp. 945-948
Author(s):  
Nikolay Kuznetsov
Keyword(s):  
Helmholtz Equation ◽  
Modified Helmholtz Equation

scholarly journals A Tikhonov-Type Regularization Method for Identifying the Unknown Source in the Modified Helmholtz Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Jinghuai Gao ◽  
Dehua Wang ◽  
Jigen Peng
Keyword(s):  
Helmholtz Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Inverse Source Problem ◽  
A Posteriori ◽  
Modified Helmholtz Equation ◽  
Numerical Tests ◽  
Theoretical Frame ◽  
Set Up

An inverse source problem in the modified Helmholtz equation is considered. We give a Tikhonov-type regularization method and set up a theoretical frame to analyze the convergence of such method. A priori and a posteriori choice rules to find the regularization parameter are given. Numerical tests are presented to illustrate the effectiveness and stability of our proposed method.

scholarly journals Moving Heat Source Reconstruction from Cauchy Boundary Data: The Cartesian Coordinates Case

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Nilson C. Roberty ◽  
Denis M. de Sousa ◽  
Marcelo L. S. Rainha
Keyword(s):  
Helmholtz Equation ◽  
Boundary Data ◽  
Representation Formula ◽  
Transient Heat Conduction ◽  
Thermal Source ◽  
Source Reconstruction ◽  
Modified Helmholtz Equation ◽  
Time Space ◽  
Definition Of ◽  
Characteristic Interval

We consider the problem of reconstruction of an unknown characteristic interval and block transient thermal source inside a domain. By exploring the definition of an Extended Dirichlet to Neumann map in the time space cylinder that has been introduced in Roberty and Rainha (2010a), we can treat the problem with methods similar to that used in the analysis of the stationary source reconstruction problem. Further, the finite differenceθ-scheme applied to the transient heat conduction equation leads to a model based on a sequence of modified Helmholtz equation solutions. For each modified Helmholtz equation the characteristic interval and parallelepiped source function may be reconstructed uniquely from the Cauchy boundary data. Using representation formula we establish reciprocity functional mapping functions that are solutions of the modified Helmholtz equation to their integral in the unknown characteristic support. Numerical experiment for capture of an interval and an rectangular parallelepiped characteristic source inside a cubic box domain from boundary data are presented in threedimensional and one-dimensional implementations. The problem of centroid determination is addressed and questions are discussed from an computational points of view.

scholarly journals An integral equation–based numerical method for the forced heat equation on complex domains

2020 ◽  
Vol 46 (5) ◽  
Author(s):  
Fredrik Fryklund ◽  
Mary Catherine A. Kropinski ◽  
Anna-Karin Tornberg
Keyword(s):  
Integral Equation ◽  
Heat Equation ◽  
Helmholtz Equation ◽  
Singular Integrals ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Elliptic Pdes ◽  
Time Step ◽  
Modified Helmholtz Equation ◽  
High Regularity

Abstract Integral equation–based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.

Fast Solution for Solving the Modified Helmholtz Equation withthe Method of Fundamental Solutions

2015 ◽  
Vol 17 (3) ◽  
pp. 867-886 ◽  
Author(s):  
C. S. Chen ◽  
Xinrong Jiang ◽  
Wen Chen ◽  
Guangming Yao
Keyword(s):  
Helmholtz Equation ◽  
Large Scale ◽  
Solution Process ◽  
Homogeneous Solution ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Dense Matrix ◽  
Modified Helmholtz Equation ◽  
Large Scale Problems ◽  
Particular Solution

AbstractThe method of fundamentalsolutions (MFS)is known as aneffective boundary meshless method. However, the formulation of the MFS results in a dense and extremely ill-conditioned matrix. In this paper we investigate the MFS for solving large-scale problems for the nonhomogeneous modified Helmholtz equation. The key idea is to exploit the exponential decay of the fundamental solution of the modified Helmholtz equation, and consider a sparse or diagonal matrix instead of the original dense matrix. Hence, the homogeneous solution can be obtained efficiently and accurately. A standard two-step solution process which consists of evaluating the particular solution and the homogeneous solution is applied. Polyharmonic spline radial basis functions are employed to evaluate the particular solution. Five numerical examples in irregular domains and a large number of boundary collocation points are presented to show the simplicity and effectiveness of our approach for solving large-scale problems.

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