This thesis consists of six research papers.In ``Regularity of the extremal solution in a MEMS model with advection,'' we examine the equation given by $ -\Delta u + c(x) \cdot \nabla u = \lambda f(u) $ in $ \Omega$ with Dirichlet boundary conditions and where $ f(u) = (1-u)^{-2}$ or $ f(u) =e^u$. Our main result is that the associated extremal solution is smooth provided this is the case for the advection free case; $c(x)=0$. In ``Estimates on pull-in distances in MEMS models and other nonlinear eigenvalue problems'' we prove some results, which were observed numerically, regarding equations of the form $ -\Delta u = \lambda |x|^\alpha F(u) $ in $B$ where $B$ is the unit ball in $ \IR^N$. In addition we obtain upper and lower estimates on the extremal solutions associated with various nonlinear eigenvalue problems. In ``The critical dimension for a fourth order elliptic problem with singular nonlinearity,'' we examine the equation given by $ \Delta^2 u = \lambda (1-u)^{-2}$ in $ B$ with Dirichlet boundary conditions where $B$ is the unit ball in $ \IR^N$. Our main result is that the extremal solution $u^*$ is smooth if and only if $ N \le 8$. In ``Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains'' we examine the equation $ \Delta^2 u = \lambda f(u)$ in $ \Omega$ with Navier boundary conditions where $ \Omega$ is a general bounded domain in $ \IR^N$. We obtain various results concerning the regularity of the associated extremal solution. In ``Regularity of the extremal solutions in elliptic systems'' we examine the elliptic system given by $ -\Delta u = \lambda e^v$, \; $ -\Delta v = \gamma e^u$ in $ \Omega$ where $\lambda$ and $\gamma$ are positive constants and we obtain results concering the regularity of the extremal solutions. In ``Optimal Hardy inequalities for general elliptic operators with improvements'' we examine some very general Hardy inequalities. Optimal constants are obtained and we characterize the improvements of these general Hardy inequalities. In addition we prove various weighted versions of these inequalities with improvements and many other results.
11Kansa radial basis function method with fictitious centres for solving nonlinear boundary value problems
Kansa radial basis function method with fictitious centres for solving nonlinear boundary value problems
103Finite Volume Method for 2D Linear and Nonlinear Elliptic Problems with Discontinuities
In this paper we study the approximation of solutions to linear and nonlinear elliptic problems with discontinuous coefficients in the discrete duality finite volume (DDFV) framework. This family of schemes allows very general meshes and inherits the main properties of the continuous problem. In order to take into account the discontinuities and to prevent consistency defect in the scheme, we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs. We first illustrate our approach by the study of the 1D situation. Then, we show how to design our new scheme, called m-DDFV, and we propose its analysis. We also describe an iterative solver, whose convergence is proved, which can be used to solve the nonlinear discrete equations defining the finite volume scheme. Finally, we provide numerical results which confirm that the m-DDFV scheme significantly improves the convergence rate of the usual DDFV method for both linear and nonlinear problems.
Read moreComputational method for a fourth order nonlinear elliptic boundary value problem
In this paper we consider a boundary value problem for a fourth order nonlinear elliptic equation, which models a bending plate on nonlinear elastic foundation. Differently from other authors, here we propose a novel approach to investigation of solvability and numerical solution of the problem. Namely, we reduce it to an operator equation for the right hand side function and under some easily verified conditions we have established the existence and uniqueness of a solution. We have also constructed an iterative method for the solution of the problem. The theoretical results are illustrated on several examples, where the advantages in convergence rate of the method over other methods are shown.
Read more21A novel method for nonlinear singular fourth order four-point boundary value problems
A novel method for nonlinear singular fourth order four-point boundary value problems
3BUILDING ELLIPTIC PROBLEM SOLVERS WITH ELLPACK
BUILDING ELLIPTIC PROBLEM SOLVERS WITH ELLPACK
6Meshless analysis of an improved element-free Galerkin method for linear and nonlinear elliptic problems**Project supported by the National Natural Science Foundation of China (Grant No. 11471063), the Chongqing Research Program of Basic Research and Frontier Technology, China (Grant No. cstc2015jcyjBX0083), and the Educational Commission Foundation of Chongqing City, China (Grant No. KJ1600330).
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.
Read moreMultiblock mortar expanded mixed formulation for second order nonlinear elliptic problems
Multiblock mortar expanded mixed formulation for second order nonlinear elliptic problems
3The Numerical Solution of Third Order Boundary Value Problems Using Piecewise Bernstein polynomials by the Galerkin Weighted Residual Method
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.
Read more64Sign-changing solutions for some fourth-order nonlinear elliptic problems
Sign-changing solutions for some fourth-order nonlinear elliptic problems
3ERROR ESTIMATES OF MIXED FINITE ELEMENT APPROXIMATIONS FOR A CLASS OF FOURTH ORDER ELLIPTIC CONTROL PROBLEMS
In this paper, we consider the error estimates of the numerical solutions of a class of fourth order linear-quadratic elliptic optimal control problems by using mixed finite element methods. The state and co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise polynomials of order k (k � 1). L 2 and L ∞ -error estimates are derived for both the control and the state approximations. These results are seemed to be new in the literature of the mixed finite element methods for fourth order elliptic control problems.
Read more2Finite element approximation for fourth-order nonlinear problem in the plane
Finite element approximation for fourth-order nonlinear problem in the plane
68Haar wavelet collocation method for three-dimensional elliptic partial differential equations
Haar wavelet collocation method for three-dimensional elliptic partial differential equations
1The method of lower and upper solutions for second order periodic Stieltjes differential equations
As an application of the Schauder Fixed Point Theorem, an existence theory is developed for second order nonlinear differential equations with Stieltjes derivative related to a left-continuous, nondecreasing function. By the method of lower and upper solutions, under a Nagumo-type assumption we get a very general result which can be further applied to deduce the existence of solutions for second order nonlinear problems in the settings of impulsive differential equations, time scale analysis or generalized differential equations.
Read moreThe Hardy inequality and second order nonlinear eigenvalue problems
The Hardy inequality and second order nonlinear eigenvalue problems
10Solving of second order nonlinear PDE problems by using artificial controls with controlled error
In this paper, we find the approximate solution of a second order nonlinear partial differential equation on a simple connected region in R2. We transfer this problem to a new problem of second order non-linear partial differential equation on a rectangle. Then, we transformed the later one to an equivalent optimization problem. Then we consider the optimization problem as a distributed parameter system with artificial controls. Finally, by using the theory of measure, we obtain the approximate solution of the original problem. In this paper also the global error in L1 is controlled.
Read more
Copyright 2026 Cactus Communications. All rights reserved.
