Application of a spectral method to numerical modeling of the propagation of seismic waves in porous media with energy dissipation

This paper presents an algorithm based on the spectral Laguerre method for approximation of time derivatives as applied to a problem of seismic wave propagation in porous media with energy dissipation. The initial system of equations is written as a first-order hyperbolic system in terms of velocities, stresses, and pore pressure. To numerically solve the problem, a combination of an analytical Laguerre transform and a finite-difference method is used. The method proposed in the paper is an analog of a well-known spectral method based on the Fourier transform. However, unlike the Fourier transform, the integral Laguerre transform with respect to time reduces the initial problem to a system of equations in which the expansion parameter is present only in the right-hand side of the equations as a recurrence relation. As compared to the finite-difference method, with an analytical transform in the spectral method it is possible to reduce the original problem to a system of differential equations having only derivatives with respect to the spatial coordinates. This allows using the known stable difference scheme for the recurrence solutions to similar systems. Such an approach is effective when solving dynamic problems for porous media. Because of the presence of a second longitudinal wave with low velocity, the use of difference schemes in all the coordinates to obtain stable solutions requires a small step consistent both with respect to time and space, which inevitably increases the execution time.     

Persistence of undercompressive phase boundaries for isothermal Euler equations including configurational forces and surface tension

The persistence of subsonic phase boundaries in a multidimensional Van der Waals fluid is analyzed. The phase boundary is considered as a sharp free boundary that connects liquid and vapor bulk phase dynamics given by the isothermal Euler equations. The evolution of the boundary is driven by effects of configurational forces as well as surface tension. To analyze this problem, the equations and trace conditions are linearized such that one obtains a general hyperbolic initial boundary value problem with higher‐order boundary conditions. A global existence theorem for the linearized system with constant coefficients is shown. The proof relies on the normal mode analysis and a linear form in suitable spaces that is defined using an associated adjoint problem. Especially, the associated adjoint problem satisfies the uniform backward in time Kreiss–Lopatinski? condition. A new energy‐like estimate that also includes surface energy terms leads finally to the uniqueness and regularity for the found solutions of the problem in weighted spaces. Copyright © 2016 John Wiley & Sons, Ltd.     

The time-splitting Fourier spectral method for the coupled Schrödinger-Boussinesq equations

The periodic initial boundary value problem of the coupled Schrödinger-Boussinesq equations is studied by the time-splitting Fourier spectral method. A time-splitting spectral discretization for the Schrödinger-like equation is applied, while a Crank-Nicolson/leap-frog type discretization is utilized for time derivatives in the Boussinesq-like equation. Numerical tests show that the time-splitting Fourier spectral method provides high accuracy for the coupled Schrödinger-Boussinesq equations.     

Creep buckling of glass-reinforced plastic cyclindrical shells

The buckling modes of cylindrical shells made of a polymeric composite with creep properties are considered. The initial imperfections of the shell are characterized by a Fourier series. The time dependence of a large number of harmonics of the Fourier series is investigated by means of nonlinear equations of the Timoshenko type. It is established that some are damped, while others grow at an ever-increasing rate. It is shown that in the course of time the effective shape of the deflections is transformed and at the critical moment the shell buckles in a mode that differs from the shape of the initial deflections.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 697–703, July–August, 1971.     

Blow-up of solutions for a system of nonlocal singular viscoelastic equations

The main propose of this paper is to study the blow-up of solutions of an initial boundary value problem with a nonlocal boundary condition for a system of nonlinear singular viscoelastic equations. where the blow-up of solutions in finite time with nonpositive initial energy combined with a positive initial energy are shown.     

高维广义Kuramoto-Sivashinsky型方程Fourier拟谱方法的大时间误差估计

在很多物理问题中出现如下方程:Kuramoto在研究反应扩散系统耗散结构时导出了上述方程,Sivashinsky在模拟火焰传播时也得到了它.此外,它还出现在粘性层流和Navier-Stokes方程的分枝解中.在[5-8]中,作者研究了一维情形下周期初值问题的整体吸引子和分枝解;[9]提出了广义KS型方程;[10-14]中研究了它的光滑解的存在性和t→+∞时的渐近性     

On the existence of a generalized solution of a nonlinear evolution system of equations in a domain unbounded in time

We consider a mixed problem with Dirichlet homogeneous boundary conditions and nonzero initial conditions for a nonlinear coupled evolution system of equations in a domain unbounded in time. The conditions of existence of a generalized solution are obtained. It is shown that no solution of the problem exists at a negative initial value of the energy integral.     

An analytical method for solving elastic system in inhomogeneous orthotropic media

In this paper, the three‐dimensional initial value problem for elastic system in inhomogeneous orthotropic media is considered and an analytical method is studied to solve this problem. The system is written in terms of Fourier images of displacements with respect to lateral variables. The resulting problem is reduced to integral equations of the Volterra type, whose solution is obtained by the method of successive approximations. Finally, using the real Paley‐Wiener theorem, it is shown that the solution of the initial value problem can be found by the inverse Fourier transform.     

Optimal singular and chattering modes in the problem of controlling the vibrations of a string with clamped ends

The problem of minimizing the root mean square deviation of a uniform string with clamped ends from an equilibrium position is investigated. It is assumed that the initial conditions are specified and the ends of the string are clamped. The Fourier method is used, which enables the control problem with a partial differential equation to be reduced to a control problem with a denumerable system of ordinary differential equations. For the optimal control problem in the l₂ space obtained, it is proved that the optimal synthesis contains singular trajectories and chattering trajectories. For the initial problem of the optimal control of the vibrations of a string it is also proved that there is a unique solution for which the optimal control has a denumerable number of switchings in a finite time interval.     

Global existence and blow-up of solutions for a system of Petrovsky equations

The initial-boundary value problem for a system of Petrovsky equations with memory and nonlinear source terms in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the exponential decay estimate of global solutions. Meanwhile, under suitable conditions on relaxation functions and the positive initial energy as well as non-positive initial energy, it is proved that the solutions blow up in the finite time and the lifespan estimates of solutions are also given.     

Asymptotic behavior of solutions to a class of nonlinear wave equations of sixth order with damping

We investigate the initial value problem for a class of nonlinear wave equations of sixth order with damping. The decay structure of this equation is of the regularity‐loss type, which causes difficulty in high‐frequency region. By using the Fourier splitting frequency technique and energy method in Fourier space, we establish asymptotic profiles of solutions to the linear equation that is given by the convolution of the fundamental solutions of heat and free wave equation. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear equation obtained in this paper that is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to this equation are also established. Copyright © 2016 John Wiley & Sons, Ltd.     

An analytic method for the initial value problem of the electric field system in vertical inhomogeneous anisotropic media

The time-dependent system of partial differential equations of the second order describing the electric wave propagation in vertically inhomogeneous electrically and magnetically biaxial anisotropic media is considered. A new analytical method for solving an initial value problem for this system is the main object of the paper. This method consists in the following: the initial value problem is written in terms of Fourier images with respect to lateral space variables, then the resulting problem is reduced to an operator integral equation. After that the operator integral equation is solved by the method of successive approximations. Finally, a solution of the original initial value problem is found by the inverse Fourier transform.     

Using null strain energy functions in compressible finite elasticity to generate exact solutions

In this paper we characterize those strain energy functions in unconstrained nonlinear elasticity that satisfy the equations of equilibrium identically. The idea is to construct a useful, physically reasonable strain–energy function containing one or more components which are null, in such a way that exact solutions may be obtained from the resulting equilibrium equations. We show that the dilatation is a universal null energy while there may be others that depend on the actual problem. To obtain the null energies for a given problem it is often convenient to formulate the variational problem and look at the Euler–Lagrange equations. Specific examples are used to illustrate some of the potential uses of the method in finding exact solutions for physically meaningful constitutive models.      

Using null strain energy functions in compressible finite elasticity to generate exact solutions

In this paper we characterize those strain energy functions in unconstrained nonlinear elasticity that satisfy the equations of equilibrium identically. The idea is to construct a useful, physically reasonable strain–energy function containing one or more components which are null, in such a way that exact solutions may be obtained from the resulting equilibrium equations. We show that the dilatation is a universal null energy while there may be others that depend on the actual problem. To obtain the null energies for a given problem it is often convenient to formulate the variational problem and look at the Euler–Lagrange equations. Specific examples are used to illustrate some of the potential uses of the method in finding exact solutions for physically meaningful constitutive models.     

The Kuramoto–Sivashinsky equation revisited: Low-dimensional corresponding systems

The main aim of this paper is to study the behaviors of the spatially periodic initial value problem for the Kuramoto–Sivashinsky (K–S) equation with the viscosity parameter. This is done by using spatially truncated Fourier decomposition with Fourier coefficients a system of ordinary differential equations in time variable. As a low-dimensional dynamical system we start with a system of four ordinary differential equations which has by itself interesting behaviors, specially a new behavior is found for that system. Then these results are applied to the K–S equation where some behaviors are in good agreement with some previous numerical experiments. Finally the order of truncation is increased with the resultant: chaotic behavior of the K–S equation for a value of the parameter is shown by calculation of the Lyapunov exponents.     

Splitting Integrators for Nonlinear Schrödinger Equations Over Long Times

Conservation properties of a full discretization via a spectral semi-discretization in space and a Lie–Trotter splitting in time for cubic Schrödinger equations with small initial data (or small nonlinearity) are studied. The approximate conservation of the actions of the linear Schrödinger equation, energy, and momentum over long times is shown using modulated Fourier expansions. The results are valid in arbitrary spatial dimension.     

Decay mode solution of nonlinear boundary–initial value problems for the cylindrical (spherical) Boussinesq–Burgers equations

The major target of this paper is to construct new nonlinear boundary–initial value problems for Boussinesq–Burgers Equations, and derive the solutions of these nonlinear boundary–initial value problems by the simplified homogeneous balance method. The nonlinear transformation and its inversion between the Boussinesq–Burgers Equations and the linear heat conduction equation are firstly derived; then a new nonlinear boundary–initial value problem for the Boussinesq–Burgers equations with variable damping on the half infinite straight line is put forward for the first time, and the solution of this nonlinear boundary–initial value problem is obtained, especially, the decay mode solution of nonlinear boundary–initial value problem for the cylindrical (spherical) Boussinesq–Burgers equations is obtained.     

A partition of energy in thermoelasticity of microstretch bodies

This paper is concerned with microstretch thermoelastic materials. For the mixed initial boundary value problem defined in this context, we prove that the Cesaro means of the kinetic and strain energies of a solution with finite energy become asymptotic equal as time tends to infinity.     

Interplay between Riccati,Ermakov, and Schrödinger equations to produce complex‐valued potentials with real energy spectrum

Nonlinear Riccati and Ermakov equations are combined to pair the energy spectrum of 2 different quantum systems via the Darboux method. One of the systems is assumed Hermitian, exactly solvable, with discrete energies in its spectrum. The other system is characterized by a complex‐valued potential that inherits all the energies of the former one and includes an additional real eigenvalue in its discrete spectrum. If such eigenvalue coincides with any discrete energy (or it is located between 2 discrete energies) of the initial system, its presence produces no singularities in the complex‐valued potential. Non‐Hermitian systems with spectrum that includes all the energies of either Morse or trigonometric Pöschl‐Teller potentials are introduced as concrete examples.     

On the matrix Fourier filtering problem for a class of models of nonlinear optical systems with a feedback

A novel statement of the Fourier filtering problem based on the use of matrix Fourier filters instead of conventional multiplier filters is considered. The basic properties of the matrix Fourier filtering for the filters in the Hilbert–Schmidt class are established. It is proved that the solutions with a finite energy to the periodic initial boundary value problem for the quasi-linear functional differential diffusion equation with the matrix Fourier filtering Lipschitz continuously depend on the filter. The problem of optimal matrix Fourier filtering is formulated, and its solvability for various classes of matrix Fourier filters is proved. It is proved that the objective functional is differentiable with respect to the matrix Fourier filter, and the convergence of a version of the gradient projection method is also proved.