Nonlinear Fourth Order Research Articles

Analysis of the Local Discontinuous Galerkin Method with Generalized Numerical Fluxes for One-Dimensional Nonlinear Fourth-Order Time-Dependent Problems

Analysis of the Local Discontinuous Galerkin Method with Generalized Numerical Fluxes for One-Dimensional Nonlinear Fourth-Order Time-Dependent Problems

Existence of Solutions for Generalized Nonlinear Fourth-Order Differential Equations

In this article, we studied the existence of solutions for a more general form of nonlinear fourth-order differential equations by using a new generalization of the Arzelá–Ascoli theorem and Schauder fixed theorem under easier and general conditions. Moreover, we provided some sufficient conditions on the nonlinear function that allowed us to deduce the nonexistence results. Finally, we outlined an example to illustrate our main results.

Existence and Asymptotic Behaviors to a Nonlinear Fourth-order Parabolic Equation with a General Source

Existence and Asymptotic Behaviors to a Nonlinear Fourth-order Parabolic Equation with a General Source

On the Existence and Uniqueness of a Positive Solution to a Boundary Value Problem with Symmetric Boundary Conditions for One Nonlinear Fourth-Order Ordinary Differential Equation

On the Existence and Uniqueness of a Positive Solution to a Boundary Value Problem with Symmetric Boundary Conditions for One Nonlinear Fourth-Order Ordinary Differential Equation

A sixth order kernel functions approach for nonlinear fourth order boundary value problems

A sixth order kernel functions approach for nonlinear fourth order boundary value problems

Monotone Iterative Technique for a Kind of Nonlinear Fourth-Order Integro-Differential Equations and Its Application

In this paper, we consider the existence and iterative approximation of solutions for a class of nonlinear fourth-order integro-differential equations (IDEs) with Navier boundary conditions. We first prove the existence and uniqueness of analytical solutions for a linear fourth-order IDE, which has rich applications in engineering and physics, and then we establish a maximum principle for the corresponding operator. Based upon the maximum principle, we develop a monotone iterative technique in the presence of lower and upper solutions to obtain iterative solutions for the nonlocal nonlinear problem under certain conditions. Some examples are presented to illustrate the main results.

Eleventh Degree Polynomial Series Solution Approach of Special Non-linear Fourth Order Boundary Value Problems

In this paper a new eleventh degree polynomial series solution approach is developed for the solution of special non-linear fourth order boundary value problems. The method consist first of obtaining a linear differentials system of twelve equations from the boundary conditions, governing equation and its three successive derivatives which were evaluated at boundary points. Secondly we assume the approximate solution in the form of a polynomial of degree eleventh with twelve unknown coefficients. To determine the unknown coefficients we incorporate the assumed solution into linear differentials systems of twelve equations which results into a linear systems of twelve equations with twelve unknown and which can be solve uniquely. It is clear from the tables and figures that the method is in good agreement with the exact and with some existing results in the literatures. Also it can be seen from example 3.3 that the exact solution is reproduced which is an added advantage of the method.

On the Existence and Uniqueness of a Positive Solution to a Boundary Value Problem with Symmetric Boundary Conditions for One Nonlinear Fourth-order Ordinary Differential Equation

The paper considers a boundary value problem with symmetric boundary conditions for one nonlinear fourth-order ordinary differential equation on the segment $[0,1]$, which describes the deformation of an elastic beam. Using special topological means in semi-ordered spaces with a cone, based on the fixed point principle, sufficient conditions for the existence and uniqueness of a positive solution to the problem under study are established. The proof of the existence of at least one positive solution to the boundary value problem was carried out using the index of the operator's fixed point. To prove the uniqueness of the solution, the theorem on fixed points of alpha-concave operators - was accordingly used.

The Galerkin method and hinged beam dynamics

In this paper we prove large-time existence and uniqueness of high regularity weak solutions to some initial/boundary value problems involving a nonlinear fourth order wave equation. These sorts of problems arise naturally in the study of vibrations in beams that are hinged at both ends. The method used to prove large-time existence is the Galerkin approximation method.

Travelling wave solutions of the beam equation with jumping nonlinearity

We study the existence of travelling wave solutions of a nonlinear fourth–order partial differential equation which can be considered as a model of an asymmetrically supported beam. We use a variational approach and show that there exist infinitely many travelling wave solutions under considerably weakened assumptions than those previously used in the literature. On the other hand, allowing the presence of sign preserving nonlinearities results in a limitation of the possible values of the wave speed. We pose several questions that still remain open, particularly regarding the range of wave speeds and the number of solutions. Moreover, some forms of classical solutions are presented together with some numerical experiments

Existence of Solutions to Nonlinear Fourth-Order Beam Equation

This paper studies the boundary value problem for a fourth-order difference equation with three quasidifferences. The new existence criterion of at least one solution to the issues considered is obtained using the theory of variational methods. The main result is illustrated in some examples.

The reproducing kernel method for nonlinear fourth-order BVPs

<abstract><p>Based on the reproducing kernel theory, we solve the nonlinear fourth order boundary value problem in the reproducing kernel space $ W_{2}^{5}[0, 1] $. Its approximate solution is obtained by truncating the n-term of the exact solution and using the $ \varepsilon $-best approximate method. Meanwhile, the approximate solution $ u^{(i)}_{n}(x) $ converges uniformly to the exact solution $ u^{(i)}(x), (i, 0, 1, 2, 3, 4) $. The validity and accuracy of this method are verified by some examples.</p></abstract>

一类非线性四阶抛物型方程解的存在性

一类非线性四阶抛物型方程解的存在性

EFFECT OF MARANGONI CONVECTION ON UNSTEADY SQUEEZED FLOW OF WATER BASE CNTS NANOFLUID IN THE PRESENCE OF MAGNETIC FIELD AND VARIABLE THERMAL CONDUCTIVITY OVER A STRETCHING SURFACE

This research paper explains the effect of Marangoni convection on unsteady squeezing flow of water base CNTs for both MWCNT and SWCNT in the presence of magnetic field and variable thermal conductivity over a stretching surface. By using the similarity transformation, the partial differential equation is converted to nonlinear fourth order ordinary differential equations. The analytical method, namely Optimal Homotopic Analysis Method used to find the analytical solution of the nonlinear problem which analyze the problem. The result of important parameters for both velocity and temperature profiles are plotted and discussed. The BVPh 2.0 package is used to obtain the convergence of the problem up to 25 iteration. The skin friction coefficient and Nusselt number is explained in table form.

On the Classical Solutions for the Kuramoto-Sivashinsky Equation with Ehrilch-Schwoebel Effects

The Kuramoto-Sivashinsky equation with Ehrilch-Schwoebel effects models the evolution of surface morphology during Molecular Beam Epitaxy growth, provoked by an interplay between deposition of atoms onto the surface and the relaxation of the surface profile through surface diffusion. It consists of a nonlinear fourth order partial differential equation. Using energy methods we prove the well-posedness (i.e., existence, uniqueness and stability with respect to the initial data) of the classical solutions for the Cauchy problem, associated with this equation.

Three-dimensional application of the meshless generalized finite difference method for solving the extended Fisher–Kolmogorov equation

In this article, a novel meshless local collocation method is proposed for the numerical solution of the three-dimensional (3D) extended Fisher–Kolmogorov (EFK) equation. The second-order Crank–Nicolson scheme and the meshless generalized finite difference method (GFDM) are respectively adopted to discretize the time and spatial derivatives of the EFK equation. A different setting of collocation nodes is introduced to the meshless GFDM for solving the nonlinear fourth order system resulting after the time discretization process. Two numerical experiments are carried out to verify the accuracy and the convergence of the developed numerical meshless algorithm.

A C0 Interior Penalty Discontinuous Galerkin Method and an equilibrated a posteriori error estimator for a nonlinear fourth order elliptic boundary value problem of p-biharmonic type

We consider a C0 Interior Penalty Discontinuous Galerkin (C0IPDG) approximation of a nonlinear fourth order elliptic boundary value problem of p-biharmonic type and an equilibrated a posteriori error estimator. The C0IPDG method can be derived from a discretization of the corresponding minimization problem involving a suitably defined reconstruction operator. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W2,p norm in terms of the associated primal and dual energy functionals. It requires the construction of an equilibrated flux and an equilibrated moment tensor based on a three-field formulation of the C0IPDG approximation. The relationship with a residual-type a posteriori error estimator is studied as well. Numerical results illustrate the performance of the suggested approach.

Snap-through instability of helicoidal composite imperfect beams surrounded by nonlinear elastic foundation

Bernstein polynomials are exploited, for the first time, in investigation of the nonlinear bending and snap-through instability phenomena of geometrically imperfect bioinspired composite beams with helicoidal orientation scheme. The curved bioinspired beam subjected to different types of lateral loading and embedded on three parameters elastic foundations is considered in formulation and analysis. The nonlinear equilibrium equation of the system based on Euler-Bernoulli beam hypothesis, von Kármán nonlinear strain, and initial imperfect curvature is developed. The equilibrium equation portrayed by nonlinear fourth order integro-differential equation is solved by Bernstein polynomials method and closed form solution of the load-deflection relationship is obtained. The accuracy of the present model and solution procedure is examined by comparing our results with previous results. The developed model is exploited to present the influence of lamination scheme, amplitude of initial imperfection and elastic foundation constants on the nonlinear bending and snap-through/reverse snap-through behaviors in detail. The numerical results show that the Bernstein polynomials approach can capture successfully the nonlinear equilibrium path and overcoming limit (turning) points, which usually appear in the nonlinear response of curved structures. The developed model can be used in design and control the snap-through response and open opportunities for this type of structures in many real applications.

Oscillation Result for Nonlinear Fourth-Order Homogeneous Neutral Delay Dynamic Equations

We introduce an oscillatory result for fourth order homogeneous neutral delay dynamic equations on time scales, which deals with a unification and extension of the differential and difference equations depending upon the time scale defines on a continuous set and a discrete set respectively.

Reducing Nonlinear Partial Differential Equation Using Lie Infinitesimals Method

Theoretical underpinnings of differential equations have advanced greatly, particularly in the twentieth century. This expansion is due to the rapid and effective development of supporting mathematical fields (such as functional analysis, measure theory, and function spaces), as well as an ever-increasing need for applications, particularly in engineering, science, and medicine. The Lie infinitesimals method was employed to reduce the nonlinear fourth order PDE into an ordinary differential equation then the resulting ODE solved by the finite difference method. The Lie infinitesimals method is used to solve significantly more complex problems which used in manufacture.