Fourth-order Equation Research Articles

Soliton molecules, interaction and other wave solutions of the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

In the present work, we aim to explore the new (3+1)-dimensional integrable fourth-order nonlinear equation(IFNE) for describing the shallow water waves. First, we study its N-soliton solutions via the bilinear form which is constructed by applying the Cole-Hopf transform. The resonance conditions of the soliton molecular are extracted and the soliton molecules are obtained. Second, the ansatz function method together with the symbolic computation, is implemented to develop the interaction wave solutions(IWSs). Finally, we take advantage of the Bernoulli sub-equation function method(BSFM) to look into the travelling wave solutions(TWSs). Different kinds of the TWSs like the singular-kink and kink solitary wave solutions are found. Correspondingly, the dynamic performances of the solutions are depicted graphically to present the physical interpretations. And for all we know, the solutions got in this work are all new and can be regarded as an extension of the solutions for the new (3+1) dimensional IFNE, which are expected to have practical significance for the application of these equations in physics.

Mathematical analysis of a fourth-order degenerate equation with a nonlinear second-order term

A fourth-order partial differential model in this paper is called the thin-film equation in the field of fluid theory. The existence of the weak solution is obtained by solving two approximation problems. In order to perform the limit for small parameters in the approximation problems, the Galerkin method and the entropy functional method are both used. By means of some classical compactness results we can give the existence of nonnegative weak solutions. Finally, by redefining the entropy functional, the long-time exponential decay is given in the sense of L1-norm.

Resonant Y-Type solutions, N-Lump waves, and hybrid solutions to a Ma-type model: a study of lump wave trajectories in superposition

In this paper, we incorporate new constrained conditions into N-soliton solutions for a (2+1)-dimensional fourth-order nonlinear equation recently developed by Ma, resulting in the derivation of resonant Y-type solitons, lump waves, soliton lines and breather waves. We utilize the velocity-module resonance method to mix resonant waves with line waves and breather solutions. To investigate the interaction between higher-order lumps and resonant waves, soliton lines, and breather waves, we use the long wave limit method. We analyze the motion trajectory equations before and after the collision of lumps and other waves. To illustrate the physical behavior of these solutions, several figures are included. We also analyze the Painlevé integrability and explore the existence of multi-soliton solutions for the Ma equation in general. We demonstrate that our specific Ma-type equation is not Painlevé integrable; however, it does exhibit multi-soliton solutions.

An inverse problem of recovering the heat source coefficient in a fourth-order time-fractional pseudo-parabolic equation

In this paper, we have considered the problem of recovering the time dependent source term for the fourth order time fractional pseudoparabolic equation theoretically and numerically, for the first time, by considering an additional measurement at the left boundary of the space interval. This is very challenging and interesting inverse problem with many important applications in various fields of physics and mechanics. The existence of unique solution to the problem has been discussed by means of the contraction principle and Banach Fixed-point theorem on a small time interval and the unique solvability theorem is proved. The stability results for the inverse problem have also been presented. However, since the governing equation is yet ill-posed (very slight errors in the additional input may cause relatively significant errors in the output force), the regularization of the solution is needed. Therefore, to get a stable solution, a regularized objective function is to be minimized for retrieval of the unknown coefficient of the forcing term. The proposed problem is discretized using the quintic B-spline (QnB-spline) collocation technique and has been reshaped as a non-linear least-squares optimization of the Tikhonov regularization function. The stability analysis of the direct numerical scheme has also been presented. The MATLAB subroutine lsqnonlin tool has been used to expedite the numerical computations. Both analytical and perturbed data are inverted and the numerical outcomes for two benchmark test examples are reported and discussed.

On a two-dimensional Dirichlet type problem for a linear hyperbolic equation of fourth order

Abstract For a linear hyperbolic equation of fourth order, a Dirichlet type boundary problem in an orthogonally convex domain is investigated. Sharp sufficient conditions guaranteeing solvability and well-posedness of the problem under consideration are established.

Nonlinear fourth-order dynamic equations on unbounded time scales

"In this paper, we investigate nonlinear fourth-order dynamic equations on unbounded time scales. The existence and uniqueness of the solutions for these problems are obtained."

Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient

Fractional and high-order PDEs have become prominent in theory and in the modeling of many phenomena. In this article, we study the temporal fractal nature for fourth-order time-fractional stochastic partial integro-differential equations (TFSPIDEs) and their gradients, which are driven in one-to-three dimensional spaces by space–time white noise. By using the underlying explicit kernels, we prove the exact global temporal continuity moduli and temporal laws of the iterated logarithm for the TFSPIDEs and their gradients, as well as prove that the sets of temporal fast points (where the remarkable oscillation of the TFSPIDEs and their gradients happen infinitely often) are random fractals. In addition, we evaluate their Hausdorff dimensions and their hitting probabilities. It has been confirmed that these points of the TFSPIDEs and their gradients, in time, are most likely one everywhere, and are dense with the power of the continuum. Moreover, their hitting probabilities are determined by the target set B’s packing dimension dimp(B). On the one hand, this work reinforces the temporal moduli of the continuity and temporal LILs obtained in relevant literature, which were achieved by obtaining the exact values of their normalized constants; on the other hand, this work obtains the size of the set of fast points, as well as a potential theory of TFSPIDEs and their gradients.

Fourth-order moments analysis for partially coherent electromagnetic beams in random media

A theory for the characterization of the fourth-order moment of electromagnetic wave beams is presented in the case when the source is partially coherent. A Gaussian–Schell model is used for the partially coherent random source. The white-noise paraxial regime is considered, which holds when the wavelength is much smaller than the correlation radius of the source, the beam radius of the source, and the correlation length of the medium, which are themselves much smaller than the propagation distance. The complex wave amplitude field can then be described by the Itô-Schrödinger equation. This equation gives closed evolution equations for the wave field moments at all orders and here the fourth-order moment equations are considered. The general fourth-order moment equations are solved explicitly in the scintillation regime (when the correlation radius of the source is of the same order as the correlation radius of the medium, but the beam radius is much larger) and the result gives a characterization of the intensity covariance function. The form of the intensity covariance function derives from the solution of the transport equation for the Wigner distribution associated with the second-order wave moment. The fourth-order moment results for polarized waves are used in an application for imaging of partially coherent sources.

Lifespan estimates and asymptotic stability for a class of fourth-order damped p-Laplacian wave equations with logarithmic nonlinearity

Lifespan estimates and asymptotic stability for a class of fourth-order damped p-Laplacian wave equations with logarithmic nonlinearity

On the accumulation points of non-periodic orbits of a difference equation of fourth order

In this paper, we are interested in analyzing the dynamics of the fourth-order difference equation xn+4=max⁡{xn+3,xn+2,xn+1,0}−xn, with arbitrary real initial conditions. We fully determine the accumulation point sets of the non-periodic solutions that, in fact, are configured as proper compact intervals of the real line. This study complements the previous knowledge of the dynamics of the difference equation already achieved in Csörnyei and Laczkovich (2001) [5] and Linero Bas and Nieves Roldán (2021) [10].

Letter to the Editor: Singular-manifold view on a (3+1)-dimensional fourth-order nonlinear equation in a fluid via HFF 32, 1664 (2022)

Letter to the Editor: Singular-manifold view on a (3+1)-dimensional fourth-order nonlinear equation in a fluid via HFF 32, 1664 (2022)

On the completeness of a system of decreasing elementary solutions for one class of fourth-order operator-differential equations on the semi-axis

In this work, we study a homogeneous fourth-order elliptic operator-differential equation with an unbounded operator under non-homogeneous boundary conditions on the semi-axis. We obtain sufficient conditions for the completeness of decreasing elementary solutions of the equation under study in the space of all its solutions from a fourth-order Sobolev type space.

Numerical solution of the cavity scattering problem for flexural waves on thin plates: Linear finite element methods

Flexural wave scattering plays a crucial role in optimizing and designing structures for various engineering applications. Mathematically, the flexural wave scattering problem on an infinite thin plate is described by a fourth-order plate wave equation on an unbounded domain, making it challenging to solve directly using the regular linear finite element method (FEM). In this paper, we propose two numerical methods, the interior penalty FEM (IP-FEM) and the boundary penalty FEM (BP-FEM) with a transparent boundary condition (TBC), to study flexural wave scattering by an arbitrary-shaped cavity on an infinite thin plate. Both methods decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary. A TBC is then constructed based on the analytical solutions of the Helmholtz and modified Helmholtz equations in the exterior domain, effectively truncating the unbounded domain into a bounded one. Using linear triangular elements, the IP-FEM and BP-FEM successfully suppress the oscillation of the bending moment of the solution on the cavity boundary, demonstrating superior stability and accuracy compared to the regular linear FEM when applied to this problem.

On the Dynamics of Aeolian Sand Ripples

AbstractThe dynamics of aeolian sand ripples is described by a 1D non-linear evolutive fourth order equation. In this paper, we prove the well-posedness of the classical solutions of the Cauchy problem, associated with this equation.

A local energy-based discontinuous Galerkin method for fourth-order semilinear wave equations

Abstract This paper proposes an energy-based discontinuous Galerkin scheme for fourth-order semilinear wave equations, which we rewrite as a system of second-order spatial derivatives. Compared to the local discontinuous Galerkin methods, the proposed scheme uses fewer auxiliary variables and is more computationally efficient. We prove several properties of the scheme. For example, we show that the scheme is unconditionally stable and that it achieves optimal convergence in $L^2$ norm for both the solution and the auxiliary variables without imposing penalty terms. A key part of the proof of the stability and convergence analysis is the special choice of the test function for the auxiliary equation involving the time derivative of the displacement variable, which leads to a linear system for the time evolution of the unknowns. Then we can use standard mathematical techniques in discontinuous Galerkin methods to obtain stability and optimal error estimates. We also obtain energy dissipation and/or conservation of the scheme by choosing simple and mesh-independent interelement fluxes. Several numerical experiments are presented to illustrate and support our theoretical results.

Continuation of global solution curves using global parameters

This paper provides both the theoretical results and numerical calculations of global solution curves, by continuation in global parameters. Each point on the solution curves is computed directly as the global parameter is varied, so that all of the turns that the solution curves make, as well as their different branches, appear automatically on the computer screen. For radial p-Laplace equations we present a simplified derivation of the regularizing transformation from P. Korman (2015), and use this transformation for more accurate numerical computations. While for p>2 the solutions are not of class C2, we show that they are of the form w(rp2(p−1)), where w(z) is of class C2. Bifurcation diagrams are also calculated for non-autonomous problems, and for the fourth order equations modeling elastic beams. We show that the first harmonic of the solution can also serve as a global parameter.

A three level linearized compact difference scheme for a fourth-order reaction-diffusion equation

A high-order accuracy finite difference scheme is investigated to solve the one-dimensional extended Fisher-Kolmogorov (EFK) equation. A three level linearized compact finite difference scheme is proposed. Priori estimates and unique solvability are discussed in detail by the discrete energy method. The unconditional stability and convergence of the difference solution are proved. The new compact difference scheme has second-order accuracy in time and fourth-order accuracy in space in maximum norm. Numerical experiments demonstrate the accuracy, efficiency of our proposed technique.

Solvability of the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with variable viscosity coefficient

We show the existence and the regularity properties of (a class of) weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with density-dependent viscosity coefficients, by analyzing a fourth-order nonlinear elliptic equation for the stream function. For some stationary symmetric flows, we reformulate the Navier–Stokes equations as ordinary differential equations and give explicit examples of weak solutions. We present some further (ir-)regularity results in the case of piecewise-constant viscosity coefficients.

Energy-stable finite element method for a class of nonlinear fourth-order parabolic equations

In this paper, an energy-stable finite element method with the Crank–Nicolson type of temporal discretization scheme is developed and analyzed for a class of nonlinear fourth-order parabolic equations (including the Swift–Hohenberg (SH) equation and the extended Fisher–Kolmogorov (EFK) equation). In addition to the energy stability properties, the optimal spatial convergence properties in both L∞(L2)- and L∞(H1)-norm and the second-order temporal approximation rate are also obtained for numerical solutions approximating to the real solution and its Laplacian for the developed energy-stable finite element method in both semi- and fully discrete schemes. Numerical experiments are carried out to validate all attained theoretical results.

A new recovery based [formula omitted] element method for fourth-order equations

In this paper, we study a new Hessian recovery based method for fourth-order differential equations by combining finite element methods (FEMs) and finite difference methods (FDMs). The new C0 finite element method is established under the framework of Petrov–Galerkin methods. Meanwhile, some unsuccessful higher-order methods based on the Hessian recovery operator are also discussed. Finally, we extend our method to solve Cahn-Hilliard equations. Numerical results validate that our proposed C0 method works well for Cahn-Hilliard, with good convergence property and easy implementation.