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A second order finite difference analog of the first biharmonic boundary value problem 总被引:7,自引:0,他引:7
J. H. Bramble 《Numerische Mathematik》1966,9(3):236-249
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Serhii V. Gryshchuk Sergiy A. Plaksa 《Mathematical Methods in the Applied Sciences》2016,39(11):2939-2952
We consider a commutative algebra over the field of complex numbers with a basis {e1,e2} satisfying the conditions , . Let D be a bounded domain in the Cartesian plane xOy and Dζ={xe1+ye2:(x,y)∈D}. Components of every monogenic function Φ(xe1+ye2) = U1(x,y)e1+U2(x,y)ie1+U3(x,y)e2+U4(x,y)ie2 having the classic derivative in Dζ are biharmonic functions in D, that is, Δ2Uj(x,y) = 0 for j = 1,2,3,4. We consider a Schwarz‐type boundary value problem for monogenic functions in a simply connected domain Dζ. This problem is associated with the following biharmonic problem: to find a biharmonic function V(x,y) in the domain D when boundary values of its partial derivatives ?V/?x, ?V/?y are given on the boundary ?D. Using a hypercomplex analog of the Cauchy‐type integral, we reduce the mentioned Schwarz‐type boundary value problem to a system of integral equations on the real axes and establish sufficient conditions under which this system has the Fredholm property. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
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O. A. Matevosyan 《Differential Equations》2016,52(10):1379-1383
We study the uniqueness of the solution of a boundary value problem for the biharmonic equation in unbounded domains under the assumption that the generalized solution of this problem has a bounded Dirichlet integral with weight |x|a. Depending on the value of the parameter a, we prove uniqueness theorems or present exact formulas for the dimension of the solution space of this problem in the exterior of a compact set and in a half-space. 相似文献
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Zhijun Zhang 《Journal of Mathematical Analysis and Applications》2011,381(2):922-934
In this paper we analyze the second expansion of the unique solution near the boundary to the singular Dirichlet problem −Δu=b(x)g(u), u>0, x∈Ω, u|∂Ω=0, where Ω is a bounded domain with smooth boundary in RN, g∈C1((0,∞),(0,∞)), g is decreasing on (0,∞) with and g is normalised regularly varying at zero with index −γ (γ>1), , is positive in Ω, may be vanishing on the boundary. 相似文献
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V. V. Karachik 《Differential Equations》2017,53(6):756-765
We study the existence and uniqueness of solutions of a generalized third boundary value problem for the inhomogeneous biharmonic equation in the unit ball. 相似文献
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Summary This paper provides a fast and storage-saving method for the solution of the first biharmonic boundary value problem (b.v.p.). The b.v.p. is approximated via a special variational finite difference technique suggested earlier by V.G. Korneev. It is shown theoretically that our method produces an approximate solution to the finite difference equations inO(NlnNln–1) arithmetical operations, whereN is the number of unknowns and (0<<1) denotes the relative accuracy required. The numerical results obtained by our computer code CGMFC decisively substantiate the theoretical estimates given. 相似文献
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On the solution a of second order singularly-perturbed boundary value problem by the Sinc-Galerkin method 总被引:1,自引:0,他引:1
Mohamed El-Gamel John R. Cannon 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(1):45-58
Sinc methods are now recognized as an efficient numerical method for problems whose solutions may have singularities, or infinite domains, or boundary layers. This work deals with the Sinc-Galerkin method for solving second order singularly perturbed boundary value problems. The method is then tested on linear and nonlinear examples and a comparison with spline method and finite element scheme is made. It is shown that the Sinc-Galerkin method yields better results.Received: January 3, 2003; revised: July 14, 2003 相似文献
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Mohamed El-Gamel John R. Cannon 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(8):45-58
Sinc methods are now recognized as an efficient numerical method for problems whose solutions may have singularities, or infinite domains, or boundary layers. This work deals with the Sinc-Galerkin method for solving second order singularly perturbed boundary value problems. The method is then tested on linear and nonlinear examples and a comparison with spline method and finite element scheme is made. It is shown that the Sinc-Galerkin method yields better results. 相似文献
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We will consider Adomain decomposition method and the homotopy method to solve a fifth order singularly perturbed BVP arising in viscoelastic flows. The success and pitfalls of the methods will be investigated. Numerical testing will be provided to show the efficiency of the methods proposed. Comparison with the work of others will also be done. 相似文献
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《Journal of Computational and Applied Mathematics》1986,14(3):279-288
An algorithm is described for the numerical solution of a free boundary problem using continuation. The problem considered is a quasi-steady problem arising in electrochemical machining. The domain of the problem is mapped onto a square and the governing nonlinear equations are discretised using finite differences. A Newton-like iteration is employed for the solution of the nonlinear algebraic system and global convergence is achieved by means of continuation. Numerical results are included. 相似文献
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This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems. 相似文献
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Vassilis G. Papanicolaou 《Probability Theory and Related Fields》1990,87(1):27-77
Using standard reflected Brownian motion (SRBM) and martingales we define (in the spirit of Stroock and Varadhan-see [S-V]) the probabilistic solution of the boundary value problem
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In this paper, the existence of positive solutions for a singular second-order three-point boundary value problem is investigated. By using Krasnoselskii??s fixed point theorem, several sufficient conditions for the existence of positive solutions and the eigenvalue intervals on which there exist positive solutions are obtained. Finally, two examples are given to illustrate the importance of results obtained. 相似文献
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The existence, nonexistence, and multiplicity of nonnegative solutions are established for the three-point boundary value problem
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Shing-Liang Lu 《Annali dell'Universita di Ferrara》1999,45(1):57-73
We study, in the rectangle Ω=(0,a)× (0,b), the Dirichlet boundary value problem for the elliptic partial differential equation
Sunto Noi studiamo nel rettangolo Ω=(0,a)×(0,b), il problema di Dirichlet con condizioni al contorno per l’equazione differenziale alle derivate parziali相似文献 |