The Experts below are selected from a list of 33285 Experts worldwide ranked by ideXlab platform
Cheinshan Liu - One of the best experts on this subject based on the ideXlab platform.
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adaptive multilayer method of fundamental solutions using a weighted greedy qr decomposition for the Laplace Equation
Journal of Computational Physics, 2012Co-Authors: Takemi Shigeta, D L Young, Cheinshan LiuAbstract:The mixed boundary value problem of the Laplace Equation is considered. The method of fundamental solutions (MFS) approximates the exact solution to the Laplace Equation by a linear combination of independent fundamental solutions with different source points. The accuracy of the numerical solution depends on the distribution of source points. In this paper, a weighted greedy QR decomposition (GQRD) is proposed to choose significant source points by introducing a weighting parameter. An index called an average degree of approximation is defined to show the efficiency of the proposed method. From numerical experiments, it is concluded that the numerical solution tends to be more accurate when the average degree of approximation is larger, and that the proposed method can yield more accurate solutions with a less number of source points than the conventional GQRD.
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a highly accurate mctm for inverse cauchy problems of Laplace Equation in arbitrary plane domains
Cmes-computer Modeling in Engineering & Sciences, 2008Co-Authors: Cheinshan LiuAbstract:We consider the inverse Cauchy problems for Laplace Equation in simply and doubly connected plane domains by recoverning the unknown bound- ary value on an inaccessible part of a noncircular contour from overspecified data. A modified Trefftz method is used directly to solve those problems with a simple collocation technique to determine unknown coefficients, which is named a mod- ified collocation Trefftz method (MCTM). Because the condition number is small for the MCTM, we can apply it to numerically solve the inverse Cauchy problems without needing of an extra regularization, as that used in the solutions of direct problems for Laplace Equation. So, the computational cost of MCTM is very sav- ing. Numerical examples show the effectiveness of the new method in providing an excellent estimate of unknown boundary data, even by subjecting the given data to a large noise. Keyword: Inverse Cauchy problem, Modified Trefftz method, Laplace Equation, Modified collocation Trefftz method (MCTM)
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a modified collocation trefftz method for the inverse cauchy problem of Laplace Equation
Engineering Analysis With Boundary Elements, 2008Co-Authors: Cheinshan LiuAbstract:We consider an inverse problem for Laplace Equation by recovering the boundary value on an inaccessible part of a circle from an overdetermined data on an accessible part of that circle. The available data are assumed to have a Fourier expansion, and thus the finite terms truncation plays a role of regularization to perturb the ill-posedness of this inverse problem into a well-posed one. Hence, we can apply a modified indirect Trefftz method to solve this problem and then a simple collocation technique is used to determine the unknown coefficients, which is named a modified collocation Trefftz method. The results may be useful to detect the corrosion inside a pipe through the measurements on a partial boundary. Numerical examples show the effectiveness of the new method in providing an excellent estimate of unknown data from the given data under noise.
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a modified trefftz method for two dimensional Laplace Equation considering the domain s characteristic length
Cmes-computer Modeling in Engineering & Sciences, 2007Co-Authors: Cheinshan LiuAbstract:A newly modified Trefftz method is developed to solve the exterior and interior Dirichlet problems for two-dimensional Laplace Equation, which takes the characteristic length of problem domain into account. After introducing a circular artificial boundary which is uniquely determined by the physical problem domain, we can derive a Dirichlet to Dirichlet mapping equa- tion, which is an exact boundary condition. By truncating the Fourier series expansion one can match the physical boundary condition as accu- rate as one desired. Then, we use the colloca- tion method and the Galerkin method to derive linear Equations system to determine the Fourier coefficients. Here, the factor of characteristic length ensures that the modified Trefftz method is stable. We use a numerical example to ex- plore why the conventional Trefftz method is fail- ure and the modified onestill survives. Numerical examples with smooth boundaries reveal that the present method can offer very accurate numeri- cal results with absoluteerrors about in the orders from 10 −10 to 10 −16 . The new method is pow- erful even for problems with complex boundary shapes, with discontinuous boundary conditions or with corners on boundary. Keyword: Laplace Equation, Artificial bound- ary condition, Modified Trefftz method, Char- acteristic length, Collocation method, Galerkin method, DtD mapping
Xiaojun Yang - One of the best experts on this subject based on the ideXlab platform.
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a local fractional variational iteration method for Laplace Equation within local fractional operators
Abstract and Applied Analysis, 2013Co-Authors: Yongju Yang, Dumitru Baleanu, Xiaojun YangAbstract:The local fractional variational iteration method for local fractional Laplace Equation is investigated in this paper. The operators are described in the sense of local fractional operators. The obtained results reveal that the method is very effective.
E A Volkov - One of the best experts on this subject based on the ideXlab platform.
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modified combined grid method for solving the dirichlet problem for the Laplace Equation on a rectangular parallelepiped
Computational Mathematics and Mathematical Physics, 2010Co-Authors: E A VolkovAbstract:A modified combined grid method is proposed for solving the Dirichlet problem for the Laplace Equation on a rectangular parallelepiped. The six-point averaging operator is applied at next-to-the-boundary grid points, while the 18-point averaging operator is used instead of the 26-point one at the remaining grid points. Assuming that the boundary values given on the faces have fourth derivatives satisfying the Holder condition, the boundary values on the edges are continuous, and their second derivatives obey a matching condition implied by the Laplace Equation, the grid solution is proved to converge uniformly with the fourth order with respect to the mesh size.
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a two stage difference method for solving the dirichlet problem for the Laplace Equation on a rectangular parallelepiped
Computational Mathematics and Mathematical Physics, 2009Co-Authors: E A VolkovAbstract:A novel two-stage difference method is proposed for solving the Dirichlet problem for the Laplace Equation on a rectangular parallelepiped. At the first stage, approximate values of the sum of the pure fourth derivatives of the desired solution are sought on a cubic grid. At the second stage, the system of difference Equations approximating the Dirichlet problem is corrected by introducing the quantities determined at the first stage. The difference Equations at the first and second stages are formulated using the simplest six-point averaging operator. Under the assumptions that the given boundary values are six times differentiable at the faces of the parallelepiped, those derivatives satisfy the Holder condition, and the boundary values are continuous at the edges and their second derivatives satisfy a matching condition implied by the Laplace Equation, it is proved that the difference solution to the Dirichlet problem converges uniformly as O(h4lnh−1), where h is the mesh size.
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on a combined grid method for solving the dirichlet problem for the Laplace Equation in a rectangular parallelepiped
Computational Mathematics and Mathematical Physics, 2007Co-Authors: E A VolkovAbstract:A combined grid method for solving the Dirichlet problem for the Laplace Equation in a rectangular parallelepiped is proposed. At the grid points that are at the distance equal to the grid size from the boundary, the 6-point averaging operator is used. At the other grid points, the 26-point averaging operator is used. It is assumed that the boundary values have the third derivatives satisfying the Lipschitz condition on the faces; on the edges, they are continuous and their second derivatives satisfy the compatibility condition implied by the Laplace Equation. The uniform convergence of the grid solution with the fourth order with respect to the grid size is proved
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grid approximation of the first derivatives of the solution to the dirichlet problem for the Laplace Equation on a polygon
Proceedings of the Steklov Institute of Mathematics, 2006Co-Authors: E A VolkovAbstract:Using the method of composite square and polar grids, we construct approximations of the first derivatives of the solution to the Dirichlet problem for the Laplace Equation on a polygon and find error estimates for such approximations.
A S Fokas - One of the best experts on this subject based on the ideXlab platform.
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the Laplace Equation in the exterior of the hankel contour and novel identities for hypergeometric functions
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2013Co-Authors: A S Fokas, M L GlasserAbstract:By using conformal mappings, it is possible to express the solution of certain boundary-value problems for the Laplace Equation in terms of a single integral involving the given boundary data. We s...
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the Laplace Equation for the exterior of the hankel contour and novel identities for hypergeometric functions
arXiv: Mathematical Physics, 2012Co-Authors: A S Fokas, M L GlasserAbstract:By employing conformal mappings, it is possible to express the solution of certain boundary value problems for the Laplace Equation in terms of a single integral involving the given boundary data. We show that such explicit formulae can be used to obtain novel identities for special functions. A convenient tool for deriving this type of identities is the so-called \emph{global relation}, which has appeared recently in a wide range of boundary value problems. As a concrete application, we analyze the Neumann boundary value problem for the Laplace Equation in the exterior of the so-called Hankel contour, which is the contour that appears in the definition of both the gamma and the Riemann zeta functions. By utilizing the explicit solution of this problem, we derive a plethora of novel identities involving the hypergeometric function.
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on a transform method for the Laplace Equation in a polygon
Ima Journal of Applied Mathematics, 2003Co-Authors: A S Fokas, A A KapaevAbstract:Let q(x, y) satisfy a boundary value problem for the Laplace Equation in an arbitrary convex polygon with n sides. An integral representation in the complex k-plane is given for q(x, y) in terms of n functions ρ j (k), j = 1,...,n. The function ρ j consists of an integral over the jth side involving both q x and q y , thus each ρ j involves one unknown boundary value. The functions ρ j are not independent but they satisfy the important global relation that their sum vanishes. The solution of a given boundary value problem reduces to the analysis of this single relation for the n unknown ρ j . For a general polygon with general Poincare boundary conditions, this gives rise to a matrix Riemann-Hilbert problem. In this paper it is shown that for simple polygons and for a large class of boundary conditions, the above Riemann-Hilbert problem (a) can either be reduced to a triangular RH problem which can be solved in closed form or (b) can be bypassed, and the ρ j can be obtained using only algebraic manipulations. As an illustration of these 'triangular' and 'algebraic' cases we solve the Laplace Equation in the quarter-plane, the semi-infinite strip and the right isosceles triangle with certain Poincare boundary conditions. These boundary value problems, which include the Dirichlet and the Neumann problems as particular cases, cannot be solved by conformal mappings.
Tongkeun Chang - One of the best experts on this subject based on the ideXlab platform.
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a dirichlet problem of the fractional Laplace Equation in the bounded lipschitz domain
arXiv: Analysis of PDEs, 2012Co-Authors: Tongkeun ChangAbstract:In this paper, we study a Dirichlet problem of a fractional Laplace Equation in a bounded Lipschitz domain in $ \R, n \geq 2$. Our main result is that for the given data $F \in \dot H^s(\Om^c), 0 < s<1$, we find the function which satisfies that $\De^s u =0$ in $\Om$, $u|_{\Om^c} =F$ and $|u|_{\dot{H}^s(\R)} \leq c |F|_{\dot H^s(\Om^c)}$. Furthermore, we represent the solution with an integral operator.
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boundary integral operator for the fractional Laplace Equation in a bounded lipschitz domain
arXiv: Analysis of PDEs, 2011Co-Authors: Tongkeun ChangAbstract:We study the boundary integral operator induced from fractional Laplace Equation in a bounded Lipschitz domain. As an application, we study the boundary value problem of a fractional Laplace Equation.
