Nonlinear Biharmonic Equation Research Articles

Solving a nonlinear biharmonic boundary value problem

In this paper we study a boundary value problem for a nonlinear biharmonic equation, which models a bending plate on nonlinear elastic foundation. We propose a new approach to existence and uniqueness and numerical solution of the problem. It is based on the reduction of the problem to finding fixed point of a nonlinear operator for the nonlinear term. The result is that under some easily verified conditions we have established the existence and uniqueness of a solution and the convergence of an iterative method for the solution. The positivity of the solution and the monotony of iterations are also considered. Some examples demonstrate the applicability of the obtained theoretical results and the efficiency of the iterative method.

Existence results and iterative method for solving a nonlinear biharmonic equation of Kirchhoff type

In this paper, for investigating a boundary value problem for a nonlinear biharmonic equation of Kirchhoff type, we propose an efficient approach by the reduction of the problem to an operator equation for the nonlinear part of the equation. The result is that we have established the existence and uniqueness of a solution under some easily verified conditions. Also, we propose an iterative method for finding the solution. Some examples demonstrate the applicability of the theoretical results and the efficiency of the iterative method.

Error analysis of a meshless weak form method based on radial point interpolation technique for Sivashinsky equation arising in the alloy solidification problem

In this paper, meshless weak form techniques are applied to find the numerical solution of nonlinear biharmonic Sivashinsky equation arising in the alloy solidification problem. Stability and convergence analysis of time-discrete scheme are proved. An error analysis of meshless global weak form method based on radial point interpolation technique is proposed for this nonlinear biharmonic equation. In addition, a comparison between meshless global and local weak form methods is done from the perspective of accuracy and efficiency. The main purpose of this paper is to show that the meshless weak form techniques can be used for solving the nonlinear biharmonic partial differential equations especially Sivashinsky equation. The numerical results confirm the good efficiency of the proposed methods for solving this nonlinear biharmonic model.

Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian

We investigate a class of nonlinear biharmonic equations with p-Laplacian{Δ2u−βΔpu+λV(x)u=f(x,u)in RN,u∈H2(RN), where N≥1, β∈R, λ>0 is a parameter and Δpu=div(|∇u|p−2∇u) with p≥2. Unlike most other papers on this problem, we replace Laplacian with p-Laplacian and allow β to be negative. Under some suitable assumptions on V(x) and f(x,u), we obtain the existence and multiplicity of nontrivial solutions for λ large enough. The proof is based on variational methods as well as Gagliardo–Nirenberg inequality.

A Fourth-Order Compact Finite Difference Scheme for Higher-Order PDE-Based Image Registration

AbstractImage registration is an ill-posed problem that has been studied widely in recent years. The so-called curvature-based image registration method is one of the most effective and well-known approaches, as it produces smooth solutions and allows an automatic rigid alignment. An important outstanding issue is the accurate and efficient numerical solution of the Euler-Lagrange system of two coupled nonlinear biharmonic equations, addressed in this article. We propose a fourth-order compact (FOC) finite difference scheme using a splitting operator on a 9-point stencil, and discuss how the resulting nonlinear discrete system can be solved efficiently by a nonlinear multi-grid (NMG) method. Thus after measuring the h-ellipticity of the nonlinear discrete operator involved by a local Fourier analysis (LFA), we show that our FOC finite difference method is amenable to multi-grid (MG) methods and an appropriate point-wise smoothing procedure. A high potential point-wise smoother using an outer-inner iteration method is shown to be effective by the LFA and numerical experiments. Real medical images are used to compare the accuracy and efficiency of our approach and the standard second-order central (SSOC) finite difference scheme in the same NMG framework. As expected for a higher-order finite difference scheme, the images generated by our FOC finite difference scheme prove significantly more accurate than those computed using the SSOC finite difference scheme. Our numerical results are consistent with the LFA analysis, and also demonstrate that the NMG method converges within a few steps.

NONLINEAR BIHARMONIC EQUATION WITH POLYNOMIAL GROWTH NONLINEAR TERM

We investigate the existence of solutions of the nonlinear biharmonic equation with variable coefficient polynomial growth nonlinear term and Dirichlet boundary condition. We get a theorem which shows that there exists a bounded solution and a large norm solution depending on the variable coefficient. We obtain this result by variational method, generalized mountain pass geometry and critical point theory.

Infinitely Many Spiky Solutions for a Hénon Type Biharmonic Equation with Critical Growth

Abstract Following the constructive method of Wei and Yan [29], with new ingredients to take care of high-space dimensions, we prove the existence of infinitely many solutions of the nonlinear biharmonic equation in the unit ball of ℝn (n ≥ 6, α > 0) with the Navier conditions v = Δv = 0 on the boundary.

Nonexistence of Positive Supersolutions of Nonlinear Biharmonic Equations without the Maximum Principle

We study classical positive solutions of the biharmonic inequality in exterior domains in ℝ n where f: (0, ∞) → (0, ∞) is continuous function. We give lower bounds on the growth of f(s) at s = 0 and/or s = ∞ such that inequality (0.1) has no C 4 positive solution in any exterior domain of ℝ n . Similar results were obtained by Armstrong and Sirakov for − Δv ≥ f(v) using a method which depends only on properties related to the maximum principle. Since the maximum principle does not hold for the biharmonic operator, we adopt a different approach which relies on a new representation formula and an a priori pointwise bound for nonnegative solutions of − Δ2 u ≥ 0 in a punctured neighborhood of the origin in ℝ n .

Numerical solution of non-linear biharmonic equation for elasto-plastic bending plate

A numerical solution for the non-linear boundary value problem of elasto-plastic bending plate by means of finite difference method is proposed. Test functions, as defined along this work, are employed for verifying the applicability of the computer program. Accuracy of the approximate solutions is demonstrated through the numerical examples.

A ROBUST HIGH-ORDER COMPACT METHOD FOR THE THREE DIMENSIONAL NONLINEAR BIHARMONIC EQUATIONS

In this paper, a new family of fourth-order compact finite difference schemes are considered using coupled approach for numerical solutions of the three-dimensional (3D) linear biharmonic problems. A new fourth-order accurate algorithm is developed through the different composition of these schemes for 3D nonlinear biharmonic equations. And an optimal combination is found in numerical experiments. The main advantage of this algorithm is that it avoids the difficulties of constructing high order compact difference schemes for 3D nonlinear biharmonic equations. The numerical solutions of unknown variable and its first derivative and Laplacian are obtained. Finally, numerical experiments are conducted to show the solution accuracy and verify the validity of our new method, including the steady Navier–Stokes equation and Cahn–Hilliard equation.

A new proof of I. Guerra's results concerning nonlinear biharmonic equations with negative exponents

Here we give a self-contained new proof of the asymptotic behavior of the radially symmetric entire solution ofΔ2u=−u−p,in R3. These results were obtained by I. Guerra in [4]. Our proof is much more direct and simpler.

A new coupled high-order compact method for the three-dimensional nonlinear biharmonic equations

This paper presents a new family of fourth- compact finite difference schemes for the numerical solution of three-dimensional nonlinear biharmonic equations using coupled approach. The numerical solutions of unknown variable and its first- derivatives as well as v(=Δ u) are obtained not only in the interior but also at the boundary. A prominent contribution of this work is that the boundary conditions for the variable v are approximated more accurately, which plays an important role for the efficiency of calculation. Finally, numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these schemes, including the steady Navier–Stokes equation in terms of vorticity-stream function formulation.

Nonlinear biharmonic boundary value problem

We consider the nonlinear biharmonic equation with variable coefficient and polynomial growth nonlinearity and Dirichlet boundary condition. We get two theorems. One theorem says that there exists at least one bounded solution under some condition. The other one says that there exist at least two solutions, one of which is a bounded solution and the other of which has a large norm under some condition. We obtain this result by the variational method, generalized mountain pass geometry and the critical point theory of the associated functional.

The behavior of solutions of the nonlinear biharmonic equation in an unbounded domain

Periodic (in one variable) solutions in the half-plane of the two-dimensional nonlinear biharmonic equation with exponential nonlinearity on the right-hand side are considered. The power-law and logarithmic asymptotics of the solutions at infinity are obtained.

A note on nonlinear biharmonic equations with negative exponents

In this paper we study radially symmetric entire solutions ofΔ2u=−u−q,u>0 in R3. We find the asymptotic behavior of these solutions for any q>1 and prove that for any q>3 there exists a solution with exactly linear growth.

Asymptotic expansion of radial solutions for supercritical biharmonic equations

Consider the positive, radial solutions of the nonlinear biharmonic equation Δ2 φ = φ p . There is a critical power p c such that solutions are linearly stable if and only if p ≥ p c . We obtain their asymptotic expansion at infinity in the case that p ≥ p c .

A New Fourth Order Difference Approximation for the Solution of Three-dimensional Non-linear Biharmonic Equations Using Coupled Approach

This paper deals with a new higher order compact difference scheme, which is, O(h4) using coupled approach on the 19-point 3D stencil for the solution of three dimensional nonlinear biharmonic equations. At each internal grid point, the solution u(x,y,z) and its Laplacian Δ4u are obtained. The resulting stencil algo-rithm is presented and hence this new algorithm can be easily incorporated to solve many problems. The present discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. Convergence analysis for a model problem is briefly discussed. The method is tested on three problems and compares very favourably with the corresponding second order approximation which we also discuss using coupled approach.

A new coupled approach high accuracy numerical method for the solution of 3D non-linear biharmonic equations

In this paper, we derive a new fourth order finite difference approximation based on arithmetic average discretization for the solution of three-dimensional non-linear biharmonic partial differential equations on a 19-point compact stencil using coupled approach. The numerical solutions of unknown variable u ( x , y , z ) and its Laplacian ∇ 2 u are obtained at each internal grid point. The resulting stencil algorithm is presented which can be used to solve many physical problems. The proposed method allows us to use the Dirichlet boundary conditions directly and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. The new method is tested on three problems and the results are compared with the corresponding second order approximation, which we also discuss using coupled approach.

Regularity of the solutions for nonlinear biharmonic equations in ℝ N

The purpose of this article is to establish the regularity of the weak solutions for the nonlinear biharmonic equation { Δ 2 u + a ( x ) u = g ( x , u ) , u ɛ H 2 ( ℝ N ) , where the condition u ∈ H 2(ℝ N ) plays the role of a boundary value condition, and as well expresses explicitly that the differential equation is to be satisfied in the weak sense.

Stability and intersection properties of solutions to the nonlinear biharmonic equation

We study the positive, regular, radially symmetric solutions to the nonlinear biharmonic equation Δ2φ = φp. First, we show that there exists a critical value pc, depending on the space dimension, such that the linearized operator has a negative eigenvalue, if p < pc, but no negative spectrum, if p ⩾ pc. Focusing on the supercritical case p ⩾ pc, we then show that the graphs of no two solutions intersect one another.