Computational method for a fourth order nonlinear elliptic boundary value problem

In this work, a Second order non-linear elliptic boundary value problem with discontinuous coefficients arising in elastohydrodynamic lubrication is studied. Physically, it corresponds to a two dimensional model of the contact between an elastie rolling ball and a plane lubricated by a thin fluid film, when the viscosity is assumed to be constant. The model of Elrod-Adams is introduced in order to take into account the phenomenon of cavitation. A result of existence is obtained by means of regularization and fixed point techniques.

A survey on various computational techniques for nonlinear elliptic boundary value problems

Previous article Next article On the Numerical Solution of Degenerate Linear and Nonlinear Elliptic Boundary Value ProblemsHarold J. KushnerHarold J. Kushnerhttps://doi.org/10.1137/0705053PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Richard S. Varga, Matrix iterative analysis, Prentice-Hall Inc., Englewood Cliffs, N.J., 1962xiii+322 MR0158502 Google Scholar[2] Ronald A. Howard, Dynamic programming and Markov processes, The Technology Press of M.I.T., Cambridge, Mass., 1960viii+136 MR0118514 0091.16001 Google Scholar[3] J. H. Eaton and , L. A. Zadeh, Optimal pursuit strategies in discrete-state probabilistic systems, Trans. ASME Ser. D. J. Basic Engrg., 84 (1962), 23–29 MR0153510 CrossrefGoogle Scholar[4] William Feller, An Introduction to Probability Theory and Its Applications. Vol. I, John Wiley & Sons Inc., New York, N.Y., 1950xii+419 MR0038583 0039.13201 Google Scholar[5] H. J. Kushner, Methods for the numerical solution of degenerate linear and non-linear elliptic boundary value problems, Tech. Rep., 67-7, Center for Dynamical Systems, Brown University, Providence, 1967 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails The Numerical Solution of Singular Perturbations of Boundary Value ProblemsFred W. DorrSIAM Journal on Numerical Analysis, Vol. 7, No. 2 | 14 July 2006AbstractPDF (2164 KB)Vacancy Transients During Impurity Diffusion in SemiconductorsPhysical Review B, Vol. 1, No. 6 | 15 March 1970 Cross Ref BIBLIOGRAPHYIterative Solution of Nonlinear Equations in Several Variables | 1 Jan 1970 Cross Ref Numerical methods for the solution of the degenerate nonlinear elliptic equations arising in optimal stochastic control theoryIEEE Transactions on Automatic Control, Vol. 13, No. 4 | 1 Aug 1968 Cross Ref Volume 5, Issue 4| 1968SIAM Journal on Numerical Analysis647-889 History Submitted:15 January 1968Published online:14 July 2006 InformationCopyright © 1968 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0705053Article page range:pp. 664-679ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

In this paper, we are concerned with the solvability for a class of second order nonlinear impulsive boundary value problem. New criteria are established based on Schaefer's fixed-point theorem. An example is presented to illustrate our main result. Our results essentially extend and complement some previous known results.

There are few numerical techniques available to solve third order boundary value problems. In this paper, we present Galerkin weighted residual method for constructing the numerical solution of third order linear and nonlinear boundary value problems with two point boundary conditions. The method is formulated as a rigorous matrix form. Several numerical examples of both linear and nonlinear boundary value problems in the literature are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods in the literature. Keywords: Galerkin method, Third order linear and nonlinear BVPs, Bernstein Polynomials.

Via the linking theorem, the existence of nontrivial solutions for a nonlinear elliptic Dirichlet boundary value problem with an inverse square potential is proved.

The subharmonicity of a function, which is defined on solutions of a fourth order nonlinear elliptic differential equation, leads to estimates on the solution and the gradient and Laplacian of the solution at interior points in terms of values on the boundary.

A flexible rod is rotated from one end. The equilibrium equation is a fourth order nonlinear two‐point boundary value problem which depends on two parameters λ and α representing the importance of centrifugal effects to flexural rigidity and the angle between the rotation axis and the clamped end, respectively. Previous studies on the existence and uniqueness of solution of the equilibrium equation assumed α = 0. Among the findings of these studies is the existence of a critical value λc beyond which the uniqueness of the trivial solution is lost. The computations of λc required the solution of a nonlinear bifurcation problem. On the other hand, this work is concerned with the existence and uniqueness of solution of the equilibrium equation when α ≠ 0 and in particular in the computations of a critical value λc such that the equilibrium equation has a unique solution for each α ≠ 0 provided λ < λc. For small α ≠ 0 this requires the solution of a nonlinear perturbed bifurcation problem.

The paper is devoted to prove maximum principles for a certain func- tionals defined for solution of the fourth order nonlinear elliptic equation. The max- imum principle so obtained is used to prove the nonexistence of nontrivial solutions of the fourth order nonlinear elliptic equation with some zero boundary conditions. Hopf's maximum principle is the main ingredient.

The classical mesh independence principle (MIP) describes a desirable property for Newton's method, its main feature being that the discrete iterations exhibit the same quadratic convergence behavior for any mesh size, i.e., uniformly as the mesh is refined. We study the latter property for a general class of second order nonlinear elliptic boundary value problems solved by finite element discretization. For this, a more specific principle, the mesh independence principle for quadratic convergence (MIPQC), is introduced. It is proved that the MIPQC holds if and only if the elliptic equation is semilinear.

We present three criteria for bifurcation from infinity of solu- tions of general boundary value problems for nonlinear elliptic systems of partial differential equations. Our sufficient conditions for bifurcation are computable, via the Atiyah-Singer family index theorem, from the coefficients of derivatives of leading order of the linearized differential operators and do not involve the analysis of the asymptotic derivative at infinity.

We first give a stabilized improved moving least squares (IMLS) approximation, which has better computational stability and precision than the IMLS approximation. Then, analysis of the improved element-free Galerkin method is provided theoretically for both linear and nonlinear elliptic boundary value problems. Finally, numerical examples are given to verify the theoretical analysis.

We consider symmetry properties of solutions to nonlinear elliptic boundary value problems defined on bounded symmetric domains of \({\mathbb R^n}\) . The solutions take values in ordered Banach spaces E, e.g. \({E=\mathbb R^N}\) ordered by a suitable cone. The nonlinearity is supposed to be quasimonotone increasing. By considering cones that are different from the standard cone of componentwise nonnegative elements we can prove symmetry of solutions to nonlinear elliptic systems which are not covered by previous results. We use the method of moving planes suitably adapted to cover the case of solutions of nonlinear elliptic problems with values in ordered Banach spaces.

Higher Order Nonlinear Multi-Point Fractional Boundary Value Problems

E. R. El-Zahar ∗ and Y. S. Hamed † ∗ Department of Basic Engineering Science, Faculty of Engineering, Shebin El-Kom, Menofia University, Egypt. † Department of Engineering Mathematics, Faculty of Electronic Engineering, Menouf 32952, Menofia University, Egypt. ∗ Present address: Department of Mathematics, Faculty of science, Al-Kharj University, Al-Kharj, Saudi Arabia. † Present address: Department of Mathematics and Statistics, Faculty of Science, Taif University, Taif, Saudi Arabia.