Higher-order Equations Research Articles

Existence of solutions to nonlinear parabolic equations via majorant integral kernel

We establish the existence of solutions to the Cauchy problem for a large class of nonlinear parabolic equations including fractional semilinear parabolic equations, higher-order semilinear parabolic equations, and viscous Hamilton–Jacobi equations by using the majorant kernel introduced in Ishige et al. (2020).

Nonlinear analysis of rectangular magnetoelectroelastic moderately thick laminated plates under multi-field coupling loads

Based on the first-order shear deformation theory (FSDT) and combined with von Karman’s nonlinear strain–displacement relationship, the nonlinear dynamic model of the rectangular magnetoelectroelastic (MEE) laminated plate is established. Then the nonlinear motion control equations of the structure are derived by using Hamilton principle. Through introducing dimensionless parameters, these equations are processed by converted into the dimensionless form. Given simply supported boundary conditions, the nonlinear higher-order equations in the governing equations are transformed into algebraic expression by the Galerkin method. In the numerical examples, the influences of size factors, temperature variation, stacking sequence and external loads on the deflection of the MEE laminated plate are studied. In addition, the distribution rule of electric and magnetic potential along the thickness direction of the MEE laminated plate with two different stacking sequences is given.

Existence of Solutions for Coupled Higher-Order Fractional Integro-Differential Equations with Nonlocal Integral and Multi-Point Boundary Conditions Depending on Lower-Order Fractional Derivatives and Integrals

In this article, we investigate the existence and uniqueness of solutions for a nonlinear coupled system of Liouville–Caputo type fractional integro-differential equations supplemented with non-local discrete and integral boundary conditions. The nonlinearity relies both on the unknown functions and their fractional derivatives and integrals in the lower order. The consequence of existence is obtained utilizing the alternative of Leray–Schauder, while the result of uniqueness is based on the concept of Banach contraction mapping. We introduced the concept of unification in the present work with varying parameters of the multi-point and classical integral boundary conditions. With the help of examples, the main results are well demonstrated.

Symmetry reductions and exact solutions of two new generalized negative KdV type equations

We give a simple geometric interpretation of the mapping of the negative KdV equation (−ψxxψ)t=(ψ2)x as proposed by Qiao and Li (2011) [26] and the Fuchssteiner equation (utxxux)x+4(uutux)x+2ut=0 using geometry of projective connection on S1 or stabilizer set of the Virasoro orbit. We propose a similar connection between (−ψxxψ)t=(ψ3)x and (−ψxxψ)t=(ψ4)x with the higher-order negative KdV equations of Fuchssteiner type described as (utxxx3u2+uxx)x+10((uut)x3u2+uxx)x+3ut=0 and (utxxxxuxxx+16uux)x+20(uutxxuxxx+16uux)x+30(uxutxuxxx+16uux)x+18(uxxut+64u2utuxxx+16uux)x+4ut=0 respectively. Then we perform the symmetry analysis of these newly found equations and study one-dimensional optimal classifications of their inequivalent subalgebras. We then obtain several exact solutions for these two equations by using those subalgebras. Further we establish some traveling wave solutions for these two equations. Moreover we construct some new exact solutions of the negative KdV equation which have not been reported in earlier studies. We illustrate the physical significance of some of the obtained solutions by performing numerical simulations and they appeared to be several kinds of soliton solutions.

Analysis of characteristics of rogue waves for higher-order equations

Analysis of characteristics of rogue waves for higher-order equations

Complex solutions to the higher-order nonlinear boussinesq type wave equation transform

Complex solutions to the higher-order nonlinear boussinesq type wave equation transform

Compactness of Solutions to Higher-Order Elliptic Equations

Abstract We use blow up analysis for local integral equations to prove compactness of solutions to higher-order critical elliptic equations provided the potentials only have non-degenerate zeros. Secondly, corresponding to Schoen’s Weyl tensor vanishing conjecture for the Yamabe equation on manifolds, we establish a Laplacian vanishing rate of the potentials at blow up points of solutions.

Rogue waves of the sixth-order nonlinear Schrödinger equation on a periodic background

In this paper, we construct the rogue wave solutions of the sixth-order nonlinear Schrödinger equation on a background of Jacobian elliptic functions dn and cn by means of the nonlinearization of a spectral problem and Darboux transformation approach. The solutions we find present the dynamic phenomena of higher-order nonlinear wave equations.

Fast spatial behavior in higher order in time equations and systems

In this work, we consider the spatial decay for high-order parabolic (and combined with a hyperbolic) equation in a semi-infinite cylinder. We prove a Phragmén-Lindelöf alternative function and, by means of some appropriate inequalities, we show that the decay is of the type of the square of the distance to the bounded end face of the cylinder. The thermoelastic case is also considered when the heat conduction is modeled using a high-order parabolic equation. Though the arguments are similar to others usually applied, we obtain new relevant results by selecting appropriate functions never considered before.

Influence of ions nonextensivity on the dynamics of dust acoustic double layers in a magnetized self-gravitating dusty plasma

The existence and polarity of dust acoustic double layers (DADLs) in a magnetized self-gravitating opposite polarity dusty plasma (OPDP) have been investigated. The constituents of the OPDP model are two species of positively and negatively charged dust grains, Maxwellian electrons, and nonextensive distributed ions. Employing the reduction perturbation method to solve basic fluid equations for dust grains, the extended Korteweg-de Vries (E-KdV) equation was derived, and its solution was extracted and analyzed. Results show that DADLs behavior is influenced by several new relevant configurational parameters, including the nonextensivity of ions, strength, and direction of the static magnetic field, as well as the self-gravitational field. In the proposed OPDP model positive and negative double-layer polarities may be formed. The polarity of OPDP is severely affected by the nonextensivity of plasma ions and the mass ratio of positive dusts to negatives. Results of higher-order nonlinear equation for solitary structure deny the presence of solitons in the physical condition of modeled plasma. In this work, simultaneous effects of an external magnetic field, self-gravitational force, and plasma ions’ nonextensivity have been studied extensively, which can be applied in various material processing plasmas, fusion plasmas, to space plasmas such as cometary tails, and Earth’s mesosphere.

Global existence and estimates for Blackstock's model of thermoviscous flow with second sound phenomena

According to the Lighthill scheme of approximation procedures, we establish the Blackstock-Cattaneo model in this paper to describe nonlinear acoustics' propagations with second sound phenomena in perfect gases under the irrotational flow, which is an approximated higher-order equation for Navier-Stokes-Cattaneo equations. This paper is devoted to the mathematical analysis of the linear and nonlinear higher-order evolution equations in Rn. Concerning the linearized Cauchy problem for the Blackstock-Cattaneo equation, by means of the Fourier analysis and the WKB analysis, we will derive sharp L2 estimates of solutions with additional L1 regularity for initial data. For another, we demonstrate global (in time) well-posedness for the nonlinear Blackstock-Cattaneo equation with small initial data and suitable regularity of Sobolev solutions.

Growth properties of meromorphic solutions of some higher-order linear differential-difference equations

Abstract This paper is devoted to the study of the growth of meromorphic solutions of homogeneous and non-homogeneous linear differential-difference equations ∑ i = 0 n ∑ j = 0 m A i ⁢ j ⁢ f ( j ) ⁢ ( z + c i ) = 0 , \displaystyle\sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}f^{(j)}(z+c_{i})=0, ∑ i = 0 n ∑ j = 0 m A i ⁢ j ⁢ f ( j ) ⁢ ( z + c i ) = F , \displaystyle\sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}f^{(j)}(z+c_{i})=F, where A i ⁢ j {A_{ij}} ( i = 0 , … , n {i=0,\ldots,n} , j = 0 , … , m {j=0,\ldots,m} ), F are meromorphic functions and c i {c_{i}} ( 0 , … , n {0,\ldots,n} ) are non-zero distinct complex numbers. Under some conditions on the coefficients, we extend early results due to Zhou and Zheng.

Strichartz estimates for higher-order Schrödinger equations and their applications

We consider the higher-order linear Schrödinger equations which are formal finite Taylor expansions of the linear pseudo-relativistic Schrödinger equation. In this paper, we establish global-in-time Strichartz estimates for these higher-order equations which hold uniformly in the speed of light. As nonlinear applications, we show that the higher-order Hartree(-Fock) equations approximate the corresponding pseudo-relativistic equation on an arbitrarily long time interval, with higher accuracy than the non-relativistic model. We also prove small data scattering for the higher-order nonlinear Schrödinger equations.

An inverse problem for investigating the time-dependent coefficient in a higher-order equation

An inverse problem for investigating the time-dependent coefficient in a higher-order equation

The asymptotic behavior for a binary alloy in energy and material science: The unified method and its applications

The Cahn–Hilliard system was proposed to the first time by Chan and Hilliard in 1958. This model (or system of equations) has intrinsic participation energy and materials sciences and depicts significant characteristics of two phase systems relating to the procedures of phase separation when the temperature is constant. For instance, it can be noticed when a binary alloy (“Aluminum + Zinc” or “Iron + Chromium”) is cooled down adequately. In this case, partially or totally nucleation (nucleation means the appearance of nuclides in the material) is observed: the homogeneous material in the initial state gradually turns into inhomogeneous, giving rise to a very accurate dispersive microstructure. Next, when the time scale is slower the microstructure becomes coarse. In this work, to the first time, the unified method is presented to investigate some physical interpretations for the solutions of the Cahn–Hilliard system when its coefficients varying with time, and to show how phase separation of one or two components and their concentrations occurs dynamically in the system. Finally, 2D and 3D plots are introduced to add more comprehensive study which help to understand the physical phenomena of this model. The technique applied in this analysis is powerful and efficient, as evidenced by the computational work and results. This technique can also solve a large number of higher-order evolution equations.

Non-uniqueness of the solution to the problem Cauchy for one equation of high order with fractional Caputo derivative

In the article, by the method of A.N. Tikhonov, a nontrivial solution of the homogeneous Cauchy problem for a homogeneous high-order equation with the Caputo derivative of order 1/2 is constructed.

The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients

Abstract In this paper, the stochastic asymptotic behavior of the nonautonomous stochastic higher-order Kirchhoff equation with variable coefficients is studied. By using the Galerkin method, the solution of this kind of equation is obtained, and stochastic dynamical system under this kind of equation is obtained; by using the uniform estimation, the existence of the family of D k {{\mathcal{D}}}_{k} -absorbing sets of the stochastic dynamical system Φ k {\Phi }_{k} is obtained, and the asymptotic compactness of Φ k {\Phi }_{k} is proved by the decomposition method. Finally, the D k {{\mathcal{D}}}_{k} -stochastic attractor family of the stochastic dynamical system Φ k {\Phi }_{k} in V m + k ( Ω ) × V k ( Ω ) {V}_{m+k}\left(\Omega )\times {V}_{k}\left(\Omega ) is obtained.

On the stability of thin-walled circular cylindrical shells under static and periodic radial loading

In this paper, we study the dynamic behavior of the cylindrical shells subjected to static and time-varying (periodic) radial loading. Flügge-Luré-Byrne shell theory has been used to derive dynamic equations in the form of nonlinear partial differential equations (PDEs). Linearization of PDEs about the steady-state periodic solutions, followed by Galerkin projection, yields a system of ordinary differential equations (ODEs) with time-varying periodic coefficients. Those are similar to the higher-order Mathieu-Hill equations and their stability analyses have been ascertained using Floquet theory. Dynamic instabilities of the cylindrical shells have been presented through stability charts in the plane of forcing parameters, namely, the static component of the pressure, the amplitude, and frequency of the fluctuating component of the pressure. Also, possible sub and super harmonic resonances have been explored through the stability charts. Four different boundary conditions, viz., simply supported, clamped–free, pinned–free (axial constraint at the pinned end) and pinned–free (axial constraint at the free end) have been considered in the present analysis. Presented charts can be very useful during the design of the shell structures subjected to static/dynamic loading.

Mathematical modeling and numerical analysis for the higher order Boussinesq system

This study deals with higher-order asymptotic equations for the water-waves problem. We considered the higher-order/extended Boussinesq equations over a flat bottom topography in the well-known long wave regime. Providing an existence and uniqueness of solution on a relevant time scale of order 1/√ε and showing that the solution’s behavior is close to the solution of the water waves equations with a better precision corresponding to initial data, the asymptotic model is well-posed in the sense of Hadamard. Then we compared several water waves solitary solutions with respect to the numerical solution of our model. At last, we solve explicitly this model and validate the results numerically.

Explicit Solution to Large Deformation of Cantilever Beam by Improved Homotopy Analysis Method II: Vertical and Horizontal Displacements

Explicit solutions to vertical and horizontal displacements are derived for large deformation of a cantilever beam under point load at the free end by an improved homotopy analysis method (IHAM). Quadratic and cubic nonlinear differential equations are adopted to construct more proficient nonlinear equations for vertical and horizontal displacements respectively combined with their currently available nonlinear displacement equations. Higher-order nonlinear iterative homotopy equations are established to solve the vertical and horizontal displacements by combining simultaneous equations of the constructed nonlinear equations and the auxiliary linear equations. The convergence range of vertical displacement is extended by the homotopy-Páde approximation. The explicit solutions to the vertical and horizontal displacements are in favorable agreements with the respective exact solutions. The convergence ranges for a relative error of 1% by the improved homotopy analysis method for vertical and horizontal displacements increases by 60% and 7%, respectively. These explicit formulas are helpful in practical engineering design for very slender structures, such as high-rise buildings and long bridges.