numerical solution of the biharmonic problem. by Ross Douglas MacBride

Cover of: numerical solution of the biharmonic problem. | Ross Douglas MacBride

Published in [Toronto] .

Written in English

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  • Harmonic functions,
  • Differential equations -- Numerical solutions

Edition Notes

Thesis (M.Sc.)--University of Toronto, 1968.

Book details

ContributionsToronto, Ont. University. Theses (M.Sc.)
LC ClassificationsLE3 T525 MSC 1968 M32
The Physical Object
Pagination[72 leaves]
Number of Pages72
ID Numbers
Open LibraryOL20256522M

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() The boundary element solution of the Laplace and biharmonic equations subjected to noisy boundary data. International Journal for Numerical Methods in Engineering() A alternating boundary element method for solving cauchy problems for the biharmonic by: Spectral methods can be adequate to solve problems in fluid dynamics numerically.

A comprehensive treatment of these methods can be found in the book of Canuto, Hussaini, Quarteroni and Zang [1]. The Numerical Solution of the Biharmonic Equation, Using a Spectral Multigrid Method | SpringerLinkCited by: 5.

All applications are illustrated with numerical examples. This unique reference work offers a method of deriving exact solutions to the biharmonic equation in the context of elasticity problems. The authors propose a number of new solutions, the like of which have never before been outlined in Western literature.

Read more Read lessFormat: Hardcover. () A quadratic spline collocation method for the Dirichlet biharmonic problem. Numerical AlgorithmsCited by: 1. INTRODUCTION CONSIDER the problem of determining a function u(xi, x^,x of n independent variables which satisfies a linear partial differential equation L(u) = 0 in G, and certain linear boundary conditions M(u) = 0 on B.

G is a. given region of the (Xi,JC space and B is the boundary Cited by: SOLUTION OF A BIHARMONIC EQUATION biharmonic equation based on point or point stencils with the second- or the fourth-order accuracy, numerical solution of the biharmonic problem.

book, are well-known and can be found in [2]. The schemes require the use of fictitious points outside of the computational domain.

Fourth-order 9-point stencil schemes. The method of Muskhelishvili for solving the biharmonic equation using conformal mapping is investigated. In [R.H. Chan, T.K. DeLillo, and M.A. Horn, SIAM J.

Sci. Comput., 18 (), pp. Furthermore numerical results for the full-multigrid method (FMG) are presented. As in the case of Poisson’s equation, FMG turns out to be an efficient approximate direct solver for the biharmonic equation with complexity O(N 2) when solving the problem on a NxN grid, yielding solutions up to discretization by: 7.

The aim of this paper is to analyze mathematically the method of fundamental solutions applied to the biharmonic problem. The key idea is to use Almansi-type decomposition of biharmonic functions, which enables us to represent the biharmonic function in terms of two harmonic functions.

Based on this decomposition, we prove that an approximate solution exists uniquely and that the Cited by: 3. PDF | In this paper, we study a system of biharmonic equations coupled by the boundary conditions.

These boundary conditions contain some combinations | Find, read and cite all the research you. Numerical solution of a second biharmonic boundary value problem | SpringerLink The second boundary value problem for the biharmonic equation is equivalent to the Dirichlet problems for two Poisson equations.

Several finite difference approximations are defined to solve these Dirichlet problems and discretization error estimates are by: 2. () A new coupled approach high accuracy numerical method for the solution of 3D non-linear biharmonic equations. Applied Mathematics and Computation() Ciarlet–Raviart mixed finite element approximation for an optimal control problem governed by the first bi-harmonic Cited by:   This reference work offers a method of deriving exact solutions to the biharmonic equation in the context of elasticity problems, and proposes a number of new solutions.

Beginning with an in-depth presentation of a general mathematical model, this text proceeds to outline specific applications, extending the developed method to special harmonic. The fundamental solution of the biharmonic equation is a singular particular solution to equation.

()D∇4v = δ(Q − P) where δ(Q − P) is the Dirac delta function representing the load density at a point Q:{ξ,n} due to a concentrated transverse unit force at point P: {x, y}. () A new coupled approach high accuracy numerical method for the solution of 3D non-linear biharmonic equations. Applied Mathematics and Computation() On the first eigenvalue of a fourth order Steklov by: Numerical methods vary in their behavior, and the many different types of differ-ential equation problems affect the performanceof numerical methods in a variety of ways.

An excellent book for “real world” examples of solving differential equations is that of Shampine, Gladwell, and Thompson [74].File Size: 1MB.

MFS FOR BIHARMONIC PROBLEMS The efficiency of the above scheme for the BMFS is demonstrated in the solution of a simple problem, previously considered by Bogomolny [5], namely: Solve V4^ = 0 on the unit circle, given \ {/ and cnl//8n on the boundary, when the problem has the solution >/' =x2 + y2 + x+ by: The biharmonic equation is first split into two Poisson equations and two classes of finite difference schemes are defined for obtaining the numerical solution.

These classes correspond to the type of difference approximation defined for the missing boundary by: Numerical examples reveal the efficiency that the new method can provide a highly accurate numerical solution even the problem domain might have a corner singularity, and the given boundary data.

the first biharmonic equation as a system of coupled harmonic equations, but some of the methods discussed here are completely new, including a conjugate gradient type algorithm. In the last part of this report we discuss the extension of the above methods to the numerical solution of the two dimensional Stokes problem in p- connected domains.

Setting the Reynolds number equal to zero, in a method for solving the Navier-Stokes equations numerically, results in a fast numerical method for biharmonic problems.

The equation is treated as a system of two second order equations and a simple smoothing process is essential for by: Johnson, C., On the convergence of a mixed finiteelement method for plate bending problems, Numer.

Math. 21 (), Mercier, B., Numerical solution of the biharmonic problem by mixed finite elements of class CO, Report, Laboratorio di Analisi Numerics del C. R., Pavia, by: Mathematical Aspects of Finite Elements in Partial Differential Equations addresses the mathematical questions raised by the use of finite elements in the numerical solution of partial differential equations.

This book covers a variety of topics, including finite element method, hyperbolic partial differential equation, and problems with. A C0 Linear Finite Element Method for Biharmonic Problems More importantly, it is shown that the numerical solution of the proposed method converges to the exact one with optimal orders both under L2 and dis-crete H2 norms, while the recovered numerical gradient converges to the exact one with aFile Size: KB.

The numerical solution of Poisson equations and biharmonic equations is an important problem in numerical analysis. A vast arrangement of investigating effort has been published on the development of numerical solution of Poisson equations and biharmonic by: 8.

ISBN: X OCLC Number: Description: viii, pages: illustrations ; 26 cm: Contents: Homogeneous solutions for the biharmonic problem; method of solution for the biharmonic problem of mathematical physics; plane problem of the theory of elasticity in cartesian co-ordinates; plane problem of the theory of elasticity in polar co-ordinates; biharmonic problem of.

the non-linear bending problem of elasto-plastic plate by using monotone operator theory. In this work existence of the weak solution of the non-linear problem in 2(Ω)Sobolev space is given and by using finite difference method numerical solution for linear bending problems with various boundary conditions is by: 2.

Harmonic and biharmonic boundary value problems (BVP) arising in physical situations in fluid mechanics are, in general, intractable by analytic techniques.

In the last decade there has been a rapid increase in the application of integral equation techniques for the numerical solution of such problems. MULTIPLICITY OF SOLUTIONS FOR EQUATIONS INVOLVING A NONLOCAL TERM AND THE BIHARMONIC OPERATOR GIOVANY M.

FIGUEIREDO, RUBIA G. NASCIMENTO Abstract. In this work we study the existence and multiplicity result of so-lutions to the equation 2u M Z jruj2 dx u= jujq 2u+ juj2 u in ; u= u= 0 on @; where is a bounded smooth domain of RN, N 5, 1.

Biharmonic equation The biharmonic quation e is the \square of Laplace equation", u 2 = 0; (1) where = @ 2 [email protected] x 1 + n is the Laplacian op erator.

e Lik Laplace equation, biharmonic equation is elliptic, but, b eing of order four rather than o, w t it requires o w t b oundary conditions rather than one to de ne a unique solution. In 2D, it is File Size: KB. In this work we present the fully nonconforming virtual element method for the approximation of biharmonic problems.

Our method works on un-structured polygonal meshes, provides arbitrary approximation order and does not require any global C0 regularity for the numerical solution. The. Numerical Solution of Thermoporoelasticity Problems F or thermoporo elasticity problems, it is essential to construct sc hemes with decoupling into physical processes, when the transition to a.

Test 3a: A mixed finite element approximations to an ∞-Biharmonic function using p-Biharmonic functions for various p for the problem given by () and () with m = 1.

Notice that as p. In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes ically, it is used in the modeling of thin structures that react elastically to external forces.

Approximate solution of boundary integral equations for biharmonic problems in non-smooth domains successive numerical solutions produced via uniformly overresolved meshes. Example In the domain Dconsider the biharmonic problem (1) with the following boundary conditions.

On the Numerical Solutions of Harmonic, Biharmonic and Similar Equations by the Difference Method not Through Successive Approximations By Hatsuo ISHIZAKI Contents Page 1.

Introduction 2 2. Laplace's and Poisson's equations 2 3. A Legendre spectral collocation method is presented for the solution of the biharmonic Dirichlet problem on a square. The solution and its Laplacian are approximated using the set of basis functions suggested by Shen, which are linear combinations of Legendre polynomials.

A Schur complement approach is used to reduce the resulting linear system Cited by: In this article, we present two new novel finite difference approximations of order two and four, respectively, for the three dimensional non-linear triharmonic partial differential equations on a compact stencil where the values of u, ∂ 2 u / ∂n 2 and ∂ 4 u / ∂ n 4 are prescribed on the boundary.

We introduce new ideas to handle the boundary conditions and there is no need to Cited by: 3. Get this from a library. Biharmonic problem in the theory of elasticity. [SERGEY A LURIE] -- This reference work offers a method of deriving exact solutions to the biharmonic equation in the context of elasticity problems, and proposes a number of new solutions.

Beginning with an in-depth. Looking for Biharmonic equation. Find out information about Biharmonic equation. A solution to the partial differential equation Δ2 u = 0, where Δ is the Laplacian operator; occurs frequently in problems in electrostatics Explanation of Biharmonic equation.

biharmonic equations • numerical solution INTRODUCTION form [15 Homotopy Analysis Method(HAM) initially proposed by Liao in [1, 2] is a powerful method to obtain series solution of various linear and nonlinear problems. The method has been shown to be a reliable technique for solving effectively, easily and accurately.The dual reciprocity method is now established as a suitable approach to the boundary element method solution of non-homogeneous field problems.

The Cited by: 3.I just used it to verify that my numerical simulation is correct for very short time scales. This is a nice sanity check. I will also try m = 0 to get a solution that does not decay as fast, but is circularly symmetric.

Thanks! $\endgroup$ – alext87 Sep 30 '15 at

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