Fourth-order Nonlinear Partial Differential Equation Research Articles

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

Hermitian numerical manifold method for large deflection of irregular Föppl-von Kármán plates

We proposed the Hermitian manifold numerical method (HNMM) fitted for finite strain analysis of thin plates with irregular domains. The large deflection of elastic thin plates is generally characterized by Föppl-von Kármán (FvK) equations, coupled with nonlinear fourth-order partial differential equations. The corresponding primal variational formulation requires the solution function to be globally C1 continuous but piecewise C2 continuous (namely H2 regular). Hermitian numerical manifold method (HNMM) can easily construct an approximation to solutions that satisfy the H2 regular requirements with structured meshes. Furthermore, those classical plate elements in finite element history can be embedded in the framework of HNMM and be more flexible and adaptable to solve plates with irregular domains. Thus, HNMM is formulated based on the Föppl-von Kármán (FvK) model using the dual cover system and a triplet attributes group on the physical patches. The numerical results demonstrate the accuracy of HNMM in large deflection of Föppl-von Kármán plate with complex domain shape.

A meshless generalized finite difference scheme for the stream function formulation of the Naiver-Stokes equations

In this study, a meshless numerical scheme, which is the combination of the generalized finite difference method, the fictitious-nodes technique, and the two-step Newton-Raphson method, was proposed to solve the stream function formulation of the Navier-Stokes equations. To reduce the numbers of unknowns and partial differential equations for the two-dimensional flow field of a viscous incompressible fluid, the stream function formulation, which is a fourth-order nonlinear partial differential equation with only one unknown variable, can be derived by introducing the definition of vorticity and stream function. By applying the generalized finite difference method, the derivatives in the stream function formulation can be simply expressed as a linear combination of functional data and weighting values at several nearest nodes. To deal with the two boundary conditions for the fourth-order partial differential equations, the fictitious-nodes technique was utilized to set up the fictitious nodes outside the computational domain, and cooperate with the generalized finite difference method to form a system of nonlinear algebraic equations. The aforementioned numerical procedure can yield a square Jacobian matrix, so the Newton-Raphson method can be adopted to efficiently acquire the numerical solutions. Several numerical examples were tested to verify the practicability of the proposed meshless numerical scheme.

Equilibrium validation for triblock copolymers via inverse norm bounds for fourth-order elliptic operators

Block copolymers play an important role in materials sciences and have found widespread use in many applications. From a mathematical perspective, they are governed by a nonlinear fourth-order partial differential equation which is a suitable gradient of the Ohta–Kawasaki energy. While the equilibrium states associated with this equation are of central importance for the description of the dynamics of block copolymers, their mathematical study remains challenging. In the current paper, we develop computer-assisted proof methods which can be used to study equilibrium solutions in block copolymers consisting of more than two monomer chains, with a focus on triblock copolymers. This is achieved by establishing a computer-assisted proof technique for bounding the norm of the inverses of certain fourth-order elliptic operators, in combination with an application of a constructive version of the implicit function theorem. While these results are only applied to the triblock copolymer case, the obtained norm estimates can also be directly used in other contexts such as the rigorous verification of bifurcation points, or pseudo-arclength continuation in fourth-order parabolic problems.

New symmetric bidirectional progressive surface-wave solutions to a generalized fourth-order nonlinear partial differential equation involving second-order time-derivative

New nonlinear fourth-order integrable equation of second-order in time has been established by Wazwaz via collaborating some recursion operators and the sense of Boussinesq equation. The proposed new equation has several real-life applications in the field of water-waves, such as gravity-capillary waves, waves in shallow water, and ocean engineering. The aim of the current study is threefold. First, we show that the propagation of Wazwaz equation is simultaneously moving symmetric bidirectional waves. Second, we extract new novel solutions such as bidirectional lumps and bidirectional kinks, singular-kinks, and periodic-kinks. Finally, by using graphical analysis, we show the physical structures of the recovery solutions and study the significance of the model’s parameters.

Second-Order Time Stepping Scheme Combined with a Mixed Element Method for a 2D Nonlinear Fourth-Order Fractional Integro-Differential Equations

In this article, we study a class of two-dimensional nonlinear fourth-order partial differential equation models with the Riemann–Liouville fractional integral term by using a mixed element method in space and the second-order backward difference formula (BDF2) with the weighted and shifted Grünwald integral (WSGI) formula in time. We introduce an auxiliary variable to transform the nonlinear fourth-order model into a low-order coupled system including two second-order equations and then discretize the resulting equations by the combined method between the BDF2 with the WSGI formula and the mixed finite element method. Further, we derive stability and error results for the fully discrete scheme. Finally, we develop two numerical examples to verify the theoretical results.

Exact solutions and dynamic properties of a nonlinear fourth-order time-fractional partial differential equation

This paper considers a fourth-order time-fractional partial differential equation with Riemann–Liouville definition. We first use the general method of separation of variables to transform the original equation into an ordinary differential equation and subsequently apply the trial equation method to obtain its integral form. The complete discrimination system for polynomial method(CDSPM) is also adopted herein. By applying this method, dynamic properties such as phase portraits are determined. The results suggest that the soliton solution coexists with the periodic solution as long as the homoclinic orbits exist. Moreover, to directly show our conclusions, the corresponding exact solutions to this equation are presented using this method.

Nonlocal and surface effects on nonlinear vibration response of a graded Timoshenko nanobeam

The free and forced vibration of a graded geometrically nonlinear Timoshenko nanobeam supported by on a nonlinear foundation is considered in this paper. The main contribution of this study is to propose a new formulation for the dynamic response of this beam by combining nonlocal and surface elasticity in addition to employing the physical neutral axis method which eliminates the quadratic nonlinearity from the equation of motion. Using the principle of virtual work, a fourth-order nonlinear partial differential equation is formulated and Galerkin technique is employed to yield a fourth-order ordinary differential equation with cubic nonlinearity in the temporal domain. The method of multiple scales is employed to obtain the analytical expression of the nonlinear frequency of the beam and its frequency response curve from a primary resonance analysis. To assess the accuracy of this analytical solution, it is compared with a numerical solution obtained using the differential quadrature method. The obtained analytical results are successfully validated for particular cases of the considered problem with results published by other authors. The effects of surface elasticity, nonlocality, the physical neutral axis, the beam aspect ratio, the power-law index and the elastic foundation coefficients on the free and forced vibration response of the graded Timoshenko nanobeam are thoroughly investigated for different types of boundary conditions .

Traveling wave structures of some fourth-order nonlinear partial differential equations

This study presents a large family of the traveling wave solutions to the two fourth-order nonlinear partial differential equations utilizing the Riccati-Bernoulli sub-ODE method. In this method, utilizing a traveling wave transformation with the aid of the Riccati-Bernoulli equation, the fourth-order equation can be transformed into a set of algebraic equations. Solving the set of algebraic equations, we acquire the novel exact solutions of the integrable fourth-order equations presented in this research paper. The physical interpretation of the nonlinear models are also detailed through the exact solutions, which demonstrate the effectiveness of the presented method.The Bäcklund transformation can produce an infinite sequence of solutions of the given two fourth-order nonlinear partial differential equations. Finally, 3D graphs of some derived solutions in this paper are depicted through suitable parameter values.

Painlevé analysis and traveling wave solutions of the fourth-order differential equation for pulse with non-local nonlinearity

Nonlinear fourth-order partial differential equation with non-local nonlinearity for describing pulses in optical fiber is considered. The traveling wave reductions to the equation are used to obtain the real and imaginary parts of nonlinear differential equation. Using the Painlevé analysis to the system of equations it is shown that this system does not have the general solution with four arbitrary constants. However the equation can have exact solution with the smaller number of arbitrary constants. Conditions for some parameters of the mathematical model are found for solution of the system of equations. Exact solutions for the system of equations are found by the means of the simplest equation method. Exact solutions are given using the Jacobi elliptic functions.

Asymptotic Expansion of the Solution of the Equation of a Slow Axisymmetric Electrovortex Flow Between Two Planes

Electrovortex flows are of great interest both from the viewpoints of theoretical magnetohydrodynamics and applications. They arise when the electric current of variable density passes through a conducting medium (such as a liquid metal). The interaction between the current and self magnetic field of the current induces a nonpotential electromagnetic force that causes a vortex flow of the liquid. As a model problem, we consider a stationary flow between two parallel planes. The flow is described by the stream function for which we obtain a nonlinear fourth-order partial differential equation. For moderate values of the electrovortex flow parameter, we investigate the problem by the asymptotic series expansion in the powers of this parameter. We describe the process of successive approximations of various degrees and present the flow pattern that is given by two first terms. This asymptotic solution is shown to be rather close to the numerical solution of the problem.

A convergent finite element scheme for a fourth-order liquid crystal model

Abstract In this manuscript we propose and analyse a fully discrete, unconditionally stable finite element scheme for a recently developed director model for liquid crystalline flows (Metzger, S. (2020) On a novel approach for modeling liquid crystalline flows. Commun. Math. Sci., 18, 359–378). The model consists of nonlinear fourth-order partial differential equations describing the evolution of the director field and Navier–Stokes equations governing the velocity field. We employ a stable splitting approach to reduce the computational complexity by decoupling the update of the director field from the update of the velocity field. We also perform a rigorous passage to the limit as the spatial and temporal discretization parameters simultaneously tend to zero, and show that a subsequence of finite element approximations converges towards a weak solution of the original model. Passing to the limit in the nonlinear terms requires us to derive the strong convergence of the gradient of the director field from uniform bounds for its discrete Laplacian. Furthermore, we present simulations underlining the practicability of the proposed scheme, investigate its convergence properties and discuss the differences between the underlying model and already established Ericksen–Leslie-type models.

Optical solitons of model with integrable equation for wave packet envelope

Abstract We consider the nonlinear fourth-order partial differential equation that can be used for describing solitary waves in nonlinear optics. The Cauchy problem for this equation is not solved by the inverse scattering transform. However we demonstrate that nonlinear ordinary differential equation for description of the wave packet envelope possesses the Painleve property and is integrable. The Lax pair to this nonlinear ordinary differential equation is presented. Using the determinant for the Lax pair matrix, we find the first integrals of a nonlinear ordinary differential equation. The general solution of the fourth-order nonlinear differential equation is given via the ultraelliptic integrals. Special cases of exact solutions for the fourth-order equation are expressed in terms of the Jacobi elliptic sine. Optical solitons of the original partial differential equation are found.

Porous Three-Dimensional Scaffold Generation for 3D Printing

In this paper, we present an efficient numerical method for arbitrary shaped porous structure generation for 3D printing. A phase-field model is employed for modeling phase separation phenomena of diblock copolymers based on the three-dimensional nonlocal Cahn–Hilliard (CH) equation. The nonlocal CH equation is a fourth-order nonlinear partial differential equation. To efficiently solve the governing equation, an unconditionally gradient stable convex splitting method for temporal discretization with a Fourier spectral method for the spatial discretization is adopted. The standard fast Fourier transform is used to speed up the computation. A new local average concentration function is introduced to the original nonlocal CH equation so that we can locally control the morphology of the structure. The proposed algorithm is simple to implement and complex shaped structures can also be implemented with corresponding signed distance fields. Various numerical tests are performed on simple and complex structures. The computational results demonstrate that the proposed method is efficient to generate irregular porous structures for 3D printing.

Lump Solutions of a Nonlinear PDE Combining with a New Fourth-Order TermDx2Dt2∗

A nonlinear PDE combining with a new fourth-order termDx2Dt2is studied. Adding three new fourth-order derivative terms and some second-order derivative terms, we formulate a combined fourth-order nonlinear partial differential equation, which possesses a Hirota’s bilinear form. The class of lump solutions is constructed explicitly through Hirota’s bilinear method. Their dynamical behaviors are analyzed through plots.

Solution for 4th-order nonlinear axisymmetric surface diffusion by inverse method

We present a method for constructing similarity solutions of a fourth-order nonlinear partial differential equation for axisymmetric surface diffusion by extending an inverse method previously used for the second-order one-dimensional nonlinear diffusion equation. After imposing a solution profile, both a feasible surface tension, and a compatible mobility function are deduced simultaneously. Although the profile is not one-to-one, an optimization algorithm is implemented to construct a mobility function that is a function of surface orientation, with no practical difference in mobility between different arms of the many-to-one profile. It is shown that the solution of the linear model well approximates the solution of the nonlinear model, in which the surface tension and mobility are close to constant for a wide range of surface angles, even when nonlinear geometric terms are included.

Nonlinear dynamics of dewetting thin films

Fluid films spreading on hydrophobic solid surfaces exhibit complicated dynamics that describe transitions leading the films to break up into droplets. For viscous fluids coating hydrophobic solids this process is called "dewetting". These dynamics can be represented by a lubrication model consisting of a fourth-order nonlinear degenerate parabolic partial differential equation (PDE) for the evolution of the film height. Analysis of the PDE model and its regimes of dynamics have yielded rich and interesting research bringing together a wide array of different mathematical approaches. The early stages of dewetting involve stability analysis and pattern formation from small perturbations and self-similar dynamics for finite-time rupture from larger amplitude perturbations. The intermediate dynamics describes further instabilities yielding topological transitions in the solutions producing sets of slowly-evolving near-equilibrium droplets. The long-time behavior can be reduced to a finite-dimensional dynamical system for the evolution of the droplets as interacting quasi-steady localized structures. This system yields <i>coarsening</i>, the successive re-arrangement and merging of smaller drops into fewer larger drops. To describe macro-scale applications, mean-field models can be constructed for the evolution of the number of droplets and the distribution of droplet sizes. We present an overview of the mathematical challenges and open questions that arise from the stages of dewetting and how they relate to issues in multi-scale modeling and singularity formation that could be applied to other problems in PDEs and materials science.

Analysis and application of the interpolating element-free Galerkin method for extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling

In this paper, the interpolating element-free Galerkin method is applied for solving the nonlinear biharmonic extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling. At first, a finite difference formula is utilized for obtaining a time-discrete scheme. The unconditional stability and convergence of the time-discrete method are proved by the energy method. Then, we use the interpolating element-free Galerkin method to approximate the spatial derivatives. An error analysis of the interpolating element-free Galerkin method is proposed for this nonlinear equation. Moreover, this method is compared with some other meshless local weak-form techniques. The main aim of this paper is to show that the interpolating element-free Galerkin is a suitable technique for solving the nonlinear fourth-order partial differential equations especially extended Fisher–Kolmogorov equation. The numerical experiments confirm the analytical results and show the good efficiency of the interpolating element-free Galerkin method for solving this nonlinear biharmonic equation.

Steady states and dynamics of a thin-film-type equation with non-conserved mass

We study the steady states and dynamics of a thin-film-type equation with non-conserved mass in one dimension. The evolution equation is a non-linear fourth-order degenerate parabolic partial differential equation (PDE) motivated by a model of volatile viscous fluid films allowing for condensation or evaporation. We show that by changing the sign of the non-conserved flux and breaking from a gradient flow structure, the problem can exhibit novel behaviours including having two distinct classes of co-existing steady-state solutions. Detailed analysis of the bifurcation structure for these steady states and their stability reveals several possibilities for the dynamics. For some parameter regimes, solutions can lead to finite-time rupture singularities. Interestingly, we also show that a finite-amplitude limit cycle can occur as a singular perturbation in the nearly conserved limit.

Local existence of a fourth-order dispersive curve flow on locally Hermitian symmetric spaces and its application

A fourth-order nonlinear dispersive partial differential equation arises in the field of mathematical physics, the solution of which is a curve flow on the two-dimensional unit sphere. In recent ten years, a geometric generalization of the sphere-valued physical model has been considered and the solvability of the initial value problem has been investigated. In particular, in the author's previous work, time-local existence and uniqueness result of the solution was established under the assumption that the solution is a closed curve flow on a compact Riemann surface with constant curvature. In the present paper, we propose a new geometric generalization of the sphere-valued physical model. As a main result, we show time-local existence of a solution to the initial value problem under the assumption that the solution is a closed curve flow on a compact locally Hermitian symmetric space. The proof is based on the geometric energy method combined with a gauge transformation to overcome the difficulty of the so-called loss of derivatives. Interestingly, the results can be applied to construct a generalized bi-Schrödinger flow proposed by Ding and Wang. The assumption on the manifold plays a crucial role both to enjoy a good solvable structure of the initial value problem and to reduce the generalized bi-Schrödinger flow equation to the one considered in the present paper.