Fourth-order Equation Research Articles

Oscillation of noncanonical fourth-order dynamic equations

We present two sufficient conditions for the oscillatory behaviour of a class of noncanonical fourth-order dynamic equations on arbitrary time scales. An illustrative example is included to show that new criteria improve related results reported in the literature.

Bilinear representation, bilinear Bäcklund transformation, Lax pair and analytical solutions for the fourth-order potential Ito equation describing water waves via Bell polynomials

This paper focuses on the fourth-order nonlinear potential Ito equation, which describes wave patterns in shallow waters. To reveal the integrable characteristics of the considered equation, such as bilinear BT and Lax pair, the Bell polynomials method is used. By employing this technique, the bilinear representation in the form of classical Hirota operators is derived. Moreover, with the help of bilinear form, the bilinear Bäcklund transformation and the Lax pair of the considered equation are obtained successfully. The three wave method in the form of test function is adopted to generate the analytical solutions of the considered equation. By applying this approach, ten analytical solutions are obtained successfully. For each of the obtained solutions, a 3D graph has been plotted by varying free parametric values. These graphs show the different kinds of wave behaviour, including kink-soliton, anti-kink soliton, periodic wave, dark-soliton, bright-soliton, and some complex kink and periodic-type wave solutions.

Unique continuation for a fourth-order stochastic parabolic equation

In this paper, we prove a boundary unique continuation property for a fourth-order stochastic parabolic equation evolving in a domain G⊂Rn. Our result shows that the value of the solution can be determined by the observation on an arbitrary open subset of the boundary. The quantitative version of this property can be derived by the global Carleman estimate, which is deduced from a weighted identity for a fourth-order stochastic parabolic operator. The results in this paper are also new even if the fourth-order stochastic parabolic equation reduces to the corresponding fourth-order deterministic parabolic equation.

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.

Inverse Problem for a Fourth-Order Hyperbolic Equation with a Complex-Valued Coefficient

This paper studies the existence and uniqueness of the classical solution of inverse problems for a fourth-order hyperbolic equation with a complex-valued coefficient with Dirichlet and Neumann boundary conditions. Using the method of separation of variables, formal solutions are obtained in the form of a Fourier series in terms of the eigenfunctions of a non-self-adjoint fourth-order ordinary differential operator. The proofs of the uniform convergence of the Fourier series are based on estimates of the norms of the derivatives of the eigenfunctions of a fourth-order ordinary differential operator and the uniform boundedness of the Riesz bases of the eigenfunctions.

Infinitely many high energy solutions for fourth-order elliptic equations with p-Laplacian in bounded domain

Infinitely many high energy solutions for fourth-order elliptic equations with p-Laplacian in bounded domain

On a Problem for a Fourth-Order Equation with Multiple Characteristics Containing the Third Time Derivate

On a Problem for a Fourth-Order Equation with Multiple Characteristics Containing the Third Time Derivate

Weighted fourth order equation of Kirchhoff type in dimension 4 with non-linear exponential growth

In this work, we are concerned with the existence of a ground state solution for a Kirchhoff weighted problem under boundary Dirichlet condition in the unit ball of $\mathbb{R}^{4}$. The nonlinearities have critical growth in view of Adams' inequalities. To prove the existence result, we use Pass Mountain Theorem. The main difficulty is the loss of compactness due to the critical exponential growth of the nonlinear term $f$. The associated energy function does not satisfy the condition of compactness. We provide a new condition for growth and we stress its importance to check the min-max compactness level.

Classification of single-wave and bi-wave motion through fourth-order equations generated from the Ito model

Recently, two fourth-order integrable equations were established by Wazwaz using the Boussinesq model. Tian and Zhang subsequently demonstrated that both equations are potential forms of the Ito model. This study investigates the dynamics of these equations using three effective schemes: the modified rational sine-cosine functions, Kudryashov-expansion, and the Hirota bilinear forms The study reports novel findings, including the observation that although these equations were derived from the same model, one propagates as a single-wave while the other propagates as a bi-wave. Additionally, some solutions of one equation can be obtained from the solutions of the other equation. These results are expected to be highly significant in the study of propagation solitary wave-solutions for nonlinear equations.

Coherent hybrid block method for approximating fourth-order ordinary differential equations

Conventionally, the most used method of solving fourth-order initial value problems of ordinary differential is to first reduce to a system of first-order differential equations. This approach affects the effectiveness and convergence of the numerical method due to the transformation. This paper comprises the derivation, analysis, and implementation of a new hybrid block method for direct solution of fourth-order equations. The method is derived by collocation and interpolation of an assumed basis function. The basic properties of the block method, including zero stability, error constants, consistency, order, and convergence, were analyzed. From the analysis, the block method derived was found to be zero-stable, consistent, and convergent. Errors were computed for the proposed method, and they were proven to produce approximations that agree with exact solutions and as such this shows improvement with those of existing works.

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

Super convergence analysis of fully discrete Hermite splines to simulate wave behaviour of Kuramoto–Sivashinsky equation

An improved collocation technique has been proposed to discretize the fourth-order multi-parameter non-linear Kuramoto -Sivashinsky (K-S) equation. The spatial direction has been discretized with quintic Hermite splines, whereas the temporal direction has been discretized with a weighted finite difference scheme. The fourth-order equation in space direction has been decomposed into second-order using space splitting by introducing a new variable. Space splitting has been proposed to improve the convergence of the approximate solution. The proposed equation has been analyzed on a uniform grid in both space and time directions. Error bounds for general order Hermite splines are established for fully discrete scheme. Stability analysis for the proposed scheme has also been discussed elaborately. Periodic and non periodic problems of K-S equation type have been discussed to study the technique. The error growth has been addressed by computing L2− norm and L∞− norm.

Multiple solutions to boundary-value problems for fourth-order elliptic equations

UDC 517.9 We study the existence of multiple solutions for the biharmonic problem Δ 2 u = f ( x , u ) + g ( x , u ) in Ω , u = ∂ ν u = 0 on ∂ Ω , where Ω is a bounded domain with smooth boundary in ℝ N , N > 4 , f ( x , ξ ) is odd in ξ , and g ( x , ξ ) is a perturbation term. Under certain growth conditions on f and g , we show that there are infinitely many weak solutions to the problem.

Block-Centered Finite-Difference Methods for Time-Fractional Fourth-Order Parabolic Equations

The block-centered finite-difference method has many advantages, and the time-fractional fourth-order equation is widely used in physics and engineering science. In this paper, we consider variable-coefficient fourth-order parabolic equations of fractional-order time derivatives with Neumann boundary conditions. The fractional-order time derivatives are approximated by L1 interpolation. We propose the block-centered finite-difference scheme for fourth-order parabolic equations with fractional-order time derivatives. We prove the stability of the block-centered finite-difference scheme and the second-order convergence of the discrete L2 norms of the approximate solution and its derivatives of every order. Numerical examples are provided to verify the effectiveness of the block-centered finite-difference scheme.

MIXED VIRTUAL ELEMENT APPROXIMATION OF A FOURTH ORDER OPTIMAL CONTROL PROBLEM

In this article, we study mixed virtual element methods for a distributed optimal control problem governed by a fourth order partial differential equation. By introducing an auxiliary variable, the fourth order equation can be transformed into systems of second order equations. A mixed virtual element discrete scheme for the optimal control problem is established. Moreover, a priori error estimates for auxiliary, state, adjoint state and control variable in H2 and L2 norms are derived. Finally, the theoretical finding is verified by numerical experiments.

On one problem for a fourth-order mixed-type equation that degenerates inside and on the boundary of a domain

In the article, a nonlocal boundary value problem has been investigated for a fourth-order mixed-type equation degenerating inside and on the boundary of a domain. Applying the method of separation of variables to the problem under study, the spectral problem for an ordinary differential equation is obtained. The Green function of the last problem is constructed, with the help of which it is equivalently reduced to the Fredholm integral equation of the second kind with a symmetric kernel, which implies the existence of eigenvalues and the system of eigenfunctions for the spectral problem. The theorem of expansion of a given function into a uniformly convergent series with respect to the system of eigenfunctions is proved. Using the found integral equation and Mercer's theorem, a uniform convergence of some bilinear series depending on the found eigenfunctions is proved. The order of the Fourier coefficients is established. The solution of the problem under study is written as the sum of the Fourier series with respect to the system of eigenfunctions of the spectral problem. An estimate for the problem's solution is obtained, from which its continuous dependence on the given functions follows.

Solutions of Three-Dimensional Problems of the Theory of Elasticity for Orthotropic Solids

We develop a procedure for the integration of balance equations in terms of displacements for orthotropic bodies with certain restrictions imposed on the elastic moduli of the material. The components of the vector of elastic displacements are expressed in terms of two functions, one of which satisfies the equation of the second order, while the second one satisfies a fourth-order equation. An algorithm is proposed for solving boundary-value problems for an orthotropic rectangular prism. The proposed algorithm is based on the separation of the principal and perturbed stress-strain states, the use of complete systems of eigenfunctions that are not orthogonal, and the minimization of a generalized quadratic form constructed to satisfy boundary conditions on the end face of the prism. The displacements and stresses are found for a long orthotropic rectangular prism subjected to the action of a local normal force load upon the end face.

Critical time-step size analysis and mass scaling by ghost-penalty for immersogeometric explicit dynamics

In this article, we study the effect of small-cut elements on the critical time-step size in an immersogeometric explicit dynamics context. We analyze different formulations for second-order (membrane) and fourth-order (shell-type) equations, and derive scaling relations between the critical time-step size and the cut-element size for various types of cuts. In particular, we focus on different approaches for the weak imposition of Dirichlet conditions: by penalty enforcement and with Nitsche’s method. The conventional stability requirement for Nitsche’s method necessitates either a cut-size dependent penalty parameter, or an additional ghost-penalty stabilization term. Our findings show that both techniques suffer from cut-size dependent critical time-step sizes, but the addition of a ghost-penalty term to the mass matrix serves to mitigate this issue. We confirm that this form of ‘mass-scaling’ does not adversely affect error and convergence characteristics for a transient membrane example, and has the potential to increase the critical time-step size by orders of magnitude. Finally, for a prototypical simulation of a Kirchhoff–Love shell, our stabilized Nitsche formulation reduces the solution error by well over an order of magnitude compared to a penalty formulation at equal time-step size.

Boundary-Value Problem for the Fourth-Order Equation with Multiple Characteristics in a Rectangular Domain

For a fourth-order equation with minor terms, we consider a boundary-value problem in a rectangular domain. The uniqueness of solution of the posed problem is proved by the method of energy integrals. The solution is expressed in terms of the constructed Green function. In the substantiation of uniform convergence, we establish the fact that the “small denominator” is not equal to zero.