Fourth-order Nonlinear Equation Research Articles

A mixed virtual element method for the two-dimensional Navier-Stokes equations in stream-function formulation

This work presents the formulation and analysis of a H1− conforming mixed virtual element method (VEM) for the two-dimensional stationary incompressible Navier-Stokes (NS) equations in stream-function formulation. By representing the velocity field as the curl of a stream function, we recast the second-order NS system into a fourth-order nonlinear equation for the scalar stream function, inherently satisfying the incompressibility constraint. Introducing a vorticity variable enables construction of H1− conforming VEM spaces for both stream function and vorticity and circumventing stringent C1− continuity constraints. The proposed method provides an initial exploration of stream function-vorticity discretizations on general polygonal meshes using the flexible VEM of arbitrary order. Existence and uniqueness of discrete solutions are established theoretically under a small data assumption. Optimal error estimates are then derived in the energy norm for the stream function, H1− norm for the stream function and L2− norm for the vorticity, rigorously demonstrating convergence. Numerical results validate the error analysis and illustrate the accuracy and robustness of the mixed VEM for simulation of incompressible flows on complex geometries.

Multi-level physics informed deep learning for solving partial differential equations in computational structural mechanics

Physics-informed neural network has emerged as a promising approach for solving partial differential equations. However, it is still a challenge for the computation of structural mechanics problems since it involves solving higher-order partial differential equations as the governing equations are fourth-order nonlinear equations. Here we develop a multi-level physics-informed neural network framework where an aggregation model is developed by combining multiple neural networks, with each one involving only first-order or second-order partial differential equations representing different physics information such as geometrical, constitutive, and equilibrium relations of the structure. The proposed framework demonstrates a remarkable advancement over the classical neural networks in terms of the accuracy and computation time. The proposed method holds the potential to become a promising paradigm for structural mechanics computation and facilitate the intelligent computation of digital twin systems.

Novel complexiton, rational wave, multi-lumps and the kink solitary wave solutions to the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

In this exploration, we aim to seek a number of new exact solutions to the new (3+1)-dimensional integrable fourth-order nonlinear equation, which is widely used to describe the shallow water waves. Employing the Cole-Hopf transformation, we develop its bilinear form. Then, taking advantage of the ansatz function method, a new functional form is utilized to probe the singular complexiton solutions. Based on which, the non-singular complexiton solutions are derived by imposing the constraint conditions. In addition, we find the rational wave solutions and multi-lumps solutions wielding the rational function method and new homoclinic method respectively. At the end, we investigate the kink solitary wave solutions using the variational approach that is based on the variational principle and Ritz method. Meanwhile, the Hamiltonian of the system is also elaborated. Correspondingly, the graphic descriptions of the extracted results are presented to unfold their dynamic behaviors through Maple. As we all know, the findings of this paper are firstly reported and can enlarge the exact solutions of the considered PDE.

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.

A fourth-order nonlinear equation studied by using a multivariate bilinear neural network method

A fourth-order nonlinear equation studied by using a multivariate bilinear neural network method

Unconditionally optimal time two-mesh mixed finite element algorithm for a nonlinear fourth-order distributed-order time fractional diffusion equation

In this article, a fast second-order time two-mesh mixed finite element (TT-MMFE) algorithm is considered to numerically solve the nonlinear fourth-order distributed-order time fractional diffusion equation, where the generated formula for the fractional BDF2 and fractional trapezoidal rule through a free parameter θ (FBT-θ formula) combined with the composite trapezoid formula for the distributed-order derivative is used to discretize the time direction, and the MFE method is utilized to approximate the space direction, respectively. Further, to improve the computing efficiency of our numerical method, the TT-M algorithm is applied to deal with the nonlinear system. For the case of nonsmooth solution, the corrected TT-MMFE scheme by adding starting parts is designed. The existence and uniqueness of the numerical solution are proved in detail. Different from the traditional finite element analysis, the error between the exact solution and the finite element solution is split into temporal error and spatial error by introducing a so-called the error splitting technique, which helps us obtain the spatial error independent of the time step. Finally, numerical examples are given to validate that the TT-MMFE method can reduce computing time, and the corrected TT-MMFE scheme by adding starting parts can recover the convergence rate.

Novel dynamical behaviors of interaction solutions of the new (3+1)-dimensional integrable fourth-order nonlinear equation

The breather, rogue and interaction waves of the (3+1)-dimensional integrable fourth-order nonlinear equation are reviewed. The breather and rouge waves are attained with the aid of the extended homoclinic test. Thereafter, the interaction solutions between a lump wave and a 1-kink or 2-kink soliton are researched. Additionally, four kinds of interaction solutions between lump, kink, and periodic waves through a ” rational-cosh-cos” test function are established. Furthermore, the dynamic attributions of the attained solutions are presented utilizing the graphical analysis.

A novel two-grid Crank-Nicolson mixed finite element method for nonlinear fourth-order sin-Gordon equation

<p>A new nonlinear fourth-order sin-Gordon equation with actual physical background is first created. Then, by introducing an auxiliary function, the nonlinear fourth-order sin-Gordon equation is decomposed into the nonlinear system of equations of second-order derivatives of spatial variables. Subsequently, the time derivative is discretized by using the Crank-Nicolson (CN) scheme to construct a new time semi-discretized mixed CN (TSDMCN) scheme. Thereafter, the spatial variables in the TSDMCN scheme are discretized by using a two-grid mixed finite element (MFE) method to construct a new two-grid CN MFE (TGCNMFE) method with unconditional stability and second-order time precision, which consists of a system of nonlinear MFE equations defined on coarser grids and a system of linear MFE equations defined on finer grids with sufficiently high precision, and is very easy to solve. The existence, stability, and error estimates of the TSDMCN and TGCNMFE solutions are strictly proved theoretically, and the superiorities of the TGCNMFE method and the correctness of theoretical results are verified by two sets of numerical experiments.</p>

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.

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.

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."

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)

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.

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

PurposeThe purpose of this paper is to study the new (3 + 1)-dimensional integrable fourth-order nonlinear equation which is used to model the shallow water waves.Design/methodology/approachBy means of the Cole–Hopf transform, the bilinear form of the studied equation is extracted. Then the ansatz function method combined with the symbolic computation is implemented to construct the breather, multiwave and the interaction wave solutions. In addition, the subequation method tis also used to search for the diverse travelling wave solutions.FindingsThe breather, multiwave and the interaction wave solutions and other wave solutions like the singular periodic wave structure and dark wave structure are obtained. To the author’s knowledge, the solutions obtained are all new and have never been reported before.Originality/valueThe solutions obtained in this work have never appeared in other literature and can be regarded as an extension of the solutions for the new (3 + 1)-dimensional integrable fourth-order nonlinear equation.

Bilinear auto-Bäcklund transformations, breather and mixed lump–kink solutions for a (3+1)-dimensional integrable fourth-order nonlinear equation in a fluid

In this paper, we investigate a (3+1)-dimensional integrable fourth-order nonlinear equation that can model both the right- and left-going waves in a fluid. Based on the Hirota method, we derive three sets of the bilinear auto-Bäcklund transformations along with some analytic solutions. Through the extended homoclinic test approach, we obtain some breather solutions. We find that the breather propagates steadily along a straight line, with one hole and one peak in each period. Graphical investigation indicates that the coefficients in that equation affect the location and shape of the breather. Moreover, we construct some mixed lump–kink solutions. Fusion and fission between a lump wave and a kink soliton are analyzed graphically. The solutions addressed in this paper may be applied to mimic some complex waves in fluids.

Efficient high-order physical property-preserving difference methods for nonlinear fourth-order wave equation with damping

Efficient high-order physical property-preserving difference methods for nonlinear fourth-order wave equation with damping

Soliton solutions to generalized (2+1)-dimensional Hietarinta-type equation and resonant NLSE along with stability analysis

This paper discusses the generalized [Formula: see text]-dimensional Hietarinta-type fourth-order nonlinear equation (GHTE) and Resonant nonlinear Schrödinger equation (RNLSE). We will study the soliton solutions of the GHTE as well as RNLSE by the renowned, efficient and successful method called sub-ODE technique. The solutions are categorized as dark soliton, periodic, rational, positive, hyperbolic and Weierstrass elliptic solutions (WES). Stability analysis of the solutions is performed for both the models. Later on, these results will be displayed graphically by 3D, 2D, contour, density and streamline plots which describe the behavior of the waves.

Nontrivial solutions of modified nonlinear fourth-order elliptic equation in [formula omitted

This paper is concerned with the following fourth-order elliptic equation of the form: Δ2u−Δu−uΔ(u2)+u=|u|p−2u,x∈RN,where Δ2u≔Δ(Δu), 4≤N≤6 and p∈2,2∗. By making use of the constrained minimization on a special manifold, we prove that the existence of nontrivial solutions of the above problem for any p∈2,2∗. Our result can be regarded as a partial extension of some related literature with 4<p<2∗.

Energy stability of a temporal variable-step difference scheme for time-fractional nonlinear fourth-order reaction-diffusion equation

In this paper, we propose a nonuniform numerical formula for the Caputo fractional derivative at the half-grid based on the piecewise linear interpolation and construct a difference scheme for the time-fractional nonlinear fourth-order reaction-diffusion equation. By virtue of two discrete tools: the discrete orthogonal convolution kernels and the discrete complementary convolution kernels, we obtain the positive definiteness of the discrete time-fractional derivative. Then a discrete variational energy dissipation law of the proposed difference scheme is established for the time-fractional nonlinear fourth-order reaction-diffusion equation, which is asymptotically compatible with the associated energy dissipation law for the classical equation as the value of fractional order approaches to one. Numerical experiments demonstrate the effectiveness and the energy dissipation of the proposed difference scheme with an adaptive time-stepping strategy.

Compact difference scheme for time-fractional nonlinear fourth-order diffusion equation with time delay

We construct a compact difference scheme for two-dimensional time-fractional nonlinear fourth-order diffusion equation with time delay. By choosing the second-order spatial derivative of the original unknown as an additional variable, the fourth-order problem is transformed into a second-order system. Then the linearized compact difference scheme is formulated by employing the L2−1σ formula and the compact difference operator to approximate temporal Caputo derivative and spatial second-order derivatives, respectively. The unique solvability of the proposed scheme is proved. Then the scheme is proved to be stable and convergent with the rate of second order in time and fourth order in space. The efficiency of the scheme is supported by some numerical experiments.