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The method of fundamental solutions for a biharmonic inverse boundary determination problem

A. Zeb, D. B. Ingham, D. Lesnic
2008 Computational Mechanics  
In this paper, a nonlinear inverse boundary value problem associated to the biharmonic equation is investigated.  ...  It is shown that the MFS regularization numerical technique produces a stable and accurate numerical solution for an optimal choice of the regularization parameter.  ...  The comments and suggestions made by the referees are gratefully acknowledged.  ... 
doi:10.1007/s00466-008-0246-6 fatcat:soyod2lqendhvjbefrntmttr2m

The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations

Liviu Marin, Daniel Lesnic
2005 Computers & structures  
In this paper, the application of the method of fundamental solutions to the Cauchy problem associated with twodimensional Helmholtz-type equations is investigated.  ...  The resulting system of linear algebraic equations is ill-conditioned and therefore its solution is regularized by employing the first-order Tikhonov functional, while the choice of the regularization  ...  Acknowledgment Liviu Marin would like to acknowledge the financial support received from the EPSRC.  ... 
doi:10.1016/j.compstruc.2004.10.005 fatcat:no2ct5dk7fbuxef46gr6fu4pha

Some Remarks on the Method of Fundamental Solutions for Two-Dimensional Elasticity

M. R. Hematiyan, M. Arezou, N. Koochak Dezfouli, M. Khoshroo
2019 CMES - Computer Modeling in Engineering & Sciences  
In this paper, some remarks for more efficient analysis of two-dimensional elastostatic problems using the method of fundamental solutions are made.  ...  By solving two example problems with stress concentration, the effectiveness of the proposed methods is demonstrated.  ...  [Chen, Karageorghis and Li (2016) ] investigated on the configuration of source points in the MFS for 2D and 3D boundary value problems governed by the Laplace and biharmonic equations.  ... 
doi:10.32604/cmes.2019.08275 fatcat:kgbboovezvgaxgi6zrqr4nse3e

Adaptive error estimation technique of the Trefftz method for solving the over-specified boundary value problem

K.H. Chen, C.T. Chen, J.F. Lee
2009 Engineering analysis with boundary elements  
The main difficulty of the inverse problem is that divergent results occur when the boundary condition on over-specified boundary is contaminated by artificial random errors.  ...  In this paper, numerical solutions are investigated based on the Trefftz method for an over-specified boundary value problem contaminated with artificial noise.  ...  Acknowledgements The authors are grateful to Prof. H. Power at 29th World Conference on BEM/MRM at Wessex Institute of Technology in Southampton, UK, for helpful comments.  ... 
doi:10.1016/j.enganabound.2009.02.001 fatcat:axe4wkbwefh6pbumro2mlfjogi

A meshless method for the stable solution of singular inverse problems for two-dimensional Helmholtz-type equations

Liviu Marin
2010 Engineering analysis with boundary elements  
We investigate a meshless method for the stable and accurate solution of inverse problems associated with two-dimensional Helmholtz-type equations in the presence of boundary singularities.  ...  The governing equation and boundary conditions are discretized by the method of fundamental solutions (MFS).  ...  Acknowledgements The financial support received from the Romanian Ministry of Education, Research and Innovation through IDEI Programme, Exploratory Research Projects, Grant PN II-ID-PCE-1248/2008, is  ... 
doi:10.1016/j.enganabound.2009.03.009 fatcat:uvsfscbzkffafpqwhgjik7o2se

Localized Method of Fundamental Solutions for Two-Dimensional Inhomogeneous Inverse Cauchy Problems

Junli Zhang, Hui Zheng, Chia-Ming Fan, Ming-Fu Fu
2022 Mathematics  
Several numerical experiments are carried out to demonstrate the efficiency of the LMFS for the inhomogeneous inverse Cauchy problems.  ...  Since the inverse Cauchy problem is ill posed, a small disturbance will lead to great errors in the numerical simulations. More accurate numerical methods are needed in the inverse Cauchy problem.  ...  Conflicts of Interest: The authors declare no conflict of interest. Mathematics 2022, 10, 1464  ... 
doi:10.3390/math10091464 fatcat:hooikqkkbfhchcem7y6v2eww2e

A numerical study of the SVD-MFS solution of inverse boundary value problems in two-dimensional steady-state linear thermoelasticity

Liviu Marin, Andreas Karageorghis, Daniel Lesnic
2014 Numerical Methods for Partial Differential Equations  
[25, 29-31], Stokes problems [32], the biharmonic equation [33], etc. have all been successfully solved by the MFS.  ...  For a survey of applications of the MFS to inverse problems, we refer the reader to Karageorghis et al. [21].  ...  The financial support received from the Romanian National Authority for Scientific Research (CNCS-UEFISCDI), project number PN-II-ID-PCE-2011-3-0521, is gratefully acknowledged.  ... 
doi:10.1002/num.21898 fatcat:pzz6w4phgfb4ziopd2shluwm3q

The method of fundamental solutions for problems in static thermo-elasticity with incomplete boundary data

L. Marin, A. Karageorghis, D. Lesnic, B. T. Johansson
2016 Inverse Problems in Science and Engineering  
In the present paper, we consider a similar type but different inverse formulation of the previously investigated problem for which we have also been able to prove the uniqueness of solution in Section  ...  Recently, the authors have solved numerically the inverse boundary value problem in static thermo-elasticity proposed in [19] using the method of fundamental solutions (MFS) [16, 23, 24] .  ...  Acknowledgements The authors would like to thank the University of Cyprus for supporting this research. L.  ... 
doi:10.1080/17415977.2016.1191072 fatcat:y57r3a2mszg2bho6ffdxp2v2ga

Solving Laplace problems with corner singularities via rational functions [article]

Abinand Gopal, Lloyd N. Trefethen
2019 arXiv   pre-print
A new method is introduced for solving Laplace problems on 2D regions with corners by approximation of boundary data by the real part of a rational function with fixed poles exponentially clustered near  ...  Newman in 1964 in approximation theory, we first prove that such approximations can achieve root-exponential convergence for a wide range of problems, all the way up to the corner singularities.  ...  We are grateful for advice to Alex Barnett, Timo Betcke, Leslie Greengard, Dave Hewett, Daan Huybrechs, Yuji Nakatsukasa, Vladimir Rokhlin, Kirill Serkh, and André Weideman.  ... 
arXiv:1905.02960v2 fatcat:gflu2lteznhnvms3nfztfkj3te

Quadrature by fundamental solutions: kernel-independent layer potential evaluation for large collections of simple objects [article]

David B. Stein, Alex H. Barnett
2021 arXiv   pre-print
Well-conditioned boundary integral methods for the solution of elliptic boundary value problems (BVPs) are powerful tools for static and dynamic physical simulations.  ...  Our extensive numerical tests include 2D FMM-based Helmholtz and Stokes BVPs with up to 1000 objects (281000 unknowns), and a 3D Laplace BVP with 10 ellipsoids separated by 1/30 of a diameter.  ...  Acknowledgements We are grateful for discussions with Manas Rachh, and the use of his 2D biharmonic FMM code.  ... 
arXiv:2109.08802v1 fatcat:u4mpaos7l5edzlba67vvu6dmyi

Laplace's equation and the Dirichlet–Neumann map: a new mode for Mikhlin's method

Johan Helsing, Eddie Wadbro
2005 Journal of Computational Physics  
Mikhlin's method for solving Laplace's equation in domains exterior to a number of closed contours is discussed with particular emphasis on the Dirichlet-Neumann map.  ...  In the literature there already exist two computational modes for Mikhlin's method. Here a new mode is presented. The new mode is at least as stable as the previous modes.  ...  When the boundary data correspond to solutions with sources and sinks inside the contours, standard methods for Laplace's and the biharmonic equations seem to be those of Mikhlin and Sherman [19, 20  ... 
doi:10.1016/ fatcat:l6uy4tburbasda7drwbqmmxyxm

The April meeting in New York

L. W. Cohen
1953 Bulletin of the American Mathematical Society  
Davis: On the Cauchy problem for the Euler-Poisson-Darboux equation. Let u(x, t) denote u(x h Xi, • • • , X mt /).  ...  The limits of integration have to be adjusted to conditions of a problem. The well known Whittaker's form of solutions of Laplace-and wave equation are particular cases of (5).  ...  Hidehiko Yamabe: On the conjecture of Iwasawa and Gleason. Let G be a connected locally compact group, and U be a compact neighborhood of the identity element.  ... 
doi:10.1090/s0002-9904-1953-09720-8 fatcat:z2kkyk3llzfn5jrssy25klqef4

Transformation Cloaking in Elastic Plates [article]

Ashkan Golgoon, Arash Yavari
2020 arXiv   pre-print
In this paper we formulate the problem of elastodynamic transformation cloaking for Kirchhoff-Love plates and elastic plates with both the in-plane and out-of-plane displacements.  ...  In this case, there are two sets of governing equations that need to be simultaneously transformed under the cloaking map.  ...  A.G. benefited from discussions with Fabio Sozio, Arzhang Angoshtari, Amirhossein Tajdini, and Souhayl Sadik.  ... 
arXiv:2005.05078v2 fatcat:4dbtkub355bhjcl6whfpfjmq7y

Polyharmonic functions on trees

Joel M. Cohen, Flavia Colonna, Kohur Gowrisankaran, David Singman
2002 American Journal of Mathematics  
In this section we solve the discrete counterpart of the classical Riquier problem for biharmonic functions on general trees.  ...  Next, we show that the discrete version of a characterization of harmonic functions due to Globevnik and Rudin holds for biharmonic functions on trees.  ...  The authors wish to thank Ibtesam Bajunaid for bringing the topic of biharmonic functions to their attention.  ... 
doi:10.1353/ajm.2002.0027 fatcat:rpiocwgkhfgl5azunadsyun7ji

The April meeting in Brooklyn

R. D. Schafer, L. A. MacColl
1955 Bulletin of the American Mathematical Society  
Albert and C. B. Morrey, Jr., respectively. AMERICAN MATHEMATICAL SOCIETY (July Sessions for contributed papers were held at 3:15 P.M. on Friday, and at 10:00 A.M. and 3:15 P.M. on Saturday.  ...  Lax: Differentiability of solutions of partial differential equations. I. Preliminary report.The Cauchy problem for symmetric hyperbolic systems Au~v in the sense of Friedrichs (Comm.  ...  If the chain X n or the process X(f) is temporally homogeneous and V depends only on x, these integral equations reduce to Fredholm equations by using generating functions or Laplace transforms.  ... 
doi:10.1090/s0002-9904-1955-09928-2 fatcat:cbglza66offornzw3yha4wdezu
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