Pseudo-hyperbolic Equations Research Articles

Existence of the global solutions of the Cauchy problem for a system of semilinear pseudohyperbolic equations with structural dissipation

Existence of the global solutions of the Cauchy problem for a system of semilinear pseudohyperbolic equations with structural dissipation

Hyperbolic and pseudo-hyperbolic equations in the theory of vibration

Longitudinal vibration of bars is usually considered in mathematical physics in terms of a classical model described by the wave equation under the assumption that the bar is thin and relatively long. More general theories have been formulated taking into consideration the effect of the lateral motion of a relatively thick bar (beam). The mathematical formulation of these models includes higher-order derivatives in the equation of motion. Rayleigh derived the simplest generalization of the classical model in 1894, by including the effects of lateral motion and neglecting the shear stress. Bishop obtained the next generalization of the theory in 1952. The Rayleigh–Bishop model is described by a fourth-order partial differential equation not containing the fourth-order time derivative. He took into account the effects of shear stress. Both Rayleigh’s and Bishop’s theories consider lateral displacement being proportional to the longitudinal strain. The Bishop model was generalized by Mindlin and Herrmann. They considered the lateral displacement proportional to an independent function of time and longitudinal coordinate. This result is formulated as a system of two differential equations of second order, which could be replaced by a single equation of fourth order resolved with respect to the highest order time derivative. To obtain a more general class of equations, the longitudinal and lateral displacements are expressed in the form of a power series expansion in the lateral coordinate. In this paper, all of the above-mentioned equations are considered in the framework of a general theory of hyperbolic equations, with the aim of classifying the equations into general groups. The solvability of the corresponding problems is also discussed.

Solvability conditions of the Cauchy problem for pseudohyperbolic equations

We consider a class of strictly pseudohyperbolic equations with lower order terms. Energy estimates are obtained and solvability conditions of the Cauchy problem in Sobolev spaces with an exponential weight are established.

Cauchy problem for a class of fourth-order semilinear pseudohyperbolic equations with structural damping

A sufficient condition for the existence of global solutions is established, and the rate of decay of solutions to the Cauchy problem for a semilinear pseudohyperbolic equation with structural damping is found. The nonexistence of global solutions is also investigated. An analogue of the critical Fujita exponent is obtained for the considered problem.

On the solvability of spatial nonlocal boundary value problemsfor one-dimensional pseudoparabolic and pseudohyperbolic equations

In the present work we study the solvability of spatial nonlocal boundary value problems for linear one-dimensional pseudoparabolic and pseudohyperbolic equations with constant coefficients, but with general nonlocal boundary conditions by A.A. Samarsky and integral conditions with variables coefficients. The proof of the theorems of existence and uniqueness of regular solutions is carried out by the method of Fourier. The study of solvability in the classes of regular solutions leads to the study of a system of integral equations of Volterra of the second kind. In particular cases nongeneracy conditions of the obtained systems of integral equations in explicit form are given.

On an Initial Boundary Value Problem for a Class of Odd Higher Order Pseudohyperbolic Integrodifferential Equations

This paper is devoted to the study of the well-posedness of an initial boundary value problem for an odd higher order nonlinear pseudohyperbolic integrodifferential partial differential equation. We associate to the equationnnonlocal conditions andn+1classical conditions. Upon some a priori estimates and density arguments, we first establish the existence and uniqueness of the strongly generalized solution in a class of a certain type of Sobolev spaces for the associated linear mixed problem. On the basis of the obtained results for the linear problem, we apply an iterative process in order to establish the well-posedness of the nonlinear problem.

Superconvergence analysis of splitting positive definite nonconforming mixed finite element method for pseudo-hyperbolic equations

In this paper, a new splitting positive definite nonconforming mixed finite element method is proposed for pseudo-hyperbolic equations, in which a quasi-Wilson quadrilateral element is used for the flux p, and the bilinear element is used for u. Superconvergence results in ‖ · ‖div,h norm for p and optimal error estimates in L2 norm for u are derived for both semi-discrete and fully discrete schemes under almost uniform meshes.

Nonconforming <i>H</i><sup>1</sup>-Galerkin Mixed Finite Element Method for Pseudo-Hyperbolic Equations

Based on H1-Galerkin mixed finite element method with nonconforming quasi-Wilson element, a numerical approximate scheme is established for pseudo-hyperbolic equations under arbitrary quadrilateral meshes. The corresponding optimal order error estimate is derived by the interpolation technique instead of the generalized elliptic projection which is necessary for classical error estimates of finite element analysis.

A splitting positive definite mixed finite element method for two classes of integro-differential equations

In this article, we establish a new mixed finite element procedure to solve the second-order hyperbolic and pseudo-hyperbolic integro-differential equations, in which the mixed element system is symmetric positive definite without requiring the LBB consistency condition. Convergence analysis shows that the method yields the approximate solutions with optimal accuracy in L2(Ω) norm for u and in H( div;Ω) norm for the flux σ. Numerical experiments are given to verify the theoretical results.

H<sup>1</sup> -Galerkin Expanded Mixed Finite Element Methods for Pseudo-Hyperbolicequations

H1-Galerkin mixed finite element method combining with expanded mixed element method are discussed for a class of second-order pseudo-hyperbolic equations. The methods possesses the advantage of mixed finite element while avoiding directly inverting the permeability tensor, which is important especially in a low permeability zone. Depended on the physical quantities of interest, the methods are discussed. The existence and uniqueness of numerical solutions of the scheme are derived and an optimal order error estimate for the methods is obtained.

Analysis of split weighted least-squares procedures for pseudo-hyperbolic equations

In this paper, we introduce two novel split weighted least-squares finite element procedures for pseudo-hyperbolic equations arising in the modelling of nerve conduction process. By selecting the weighted least-squares functional properly, each procedure can be split into two independent symmetric positive definite sub-procedures. One of sub-procedures is for the primitive unknown variable, which is the same as the standard Galerkin finite element procedure and the other is for the introduced flux variable. Optimal order error estimates are developed and the numerical example is given to show the efficiency of the introduced schemes.

Existence and non-existence of global solutions of the Cauchy problem for higher order semilinear pseudo-hyperbolic equations

We consider the Cauchy problem for a higher order pseudo-hyperbolic equation. Using the L p → L q type estimation for the corresponding linear problem, the existence and non-existence criteria of global solutions are found. The existence and uniqueness of smooth global solutions are also investigated. We also establish the behavior of solutions and their derivatives as t → + ∞ .

Formula omitted]-Galerkin mixed finite element methods for pseudo-hyperbolic equations

H 1 -Galerkin mixed finite element methods are discussed for a class of second-order pseudo-hyperbolic equations. Depending on the physical quantities of interest, two methods are discussed. Optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension. An extension to problems in two and three space variables is also discussed, the existence and uniqueness are derived and it is showed that the H 1 -Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.

A remark on least-squares Galerkin procedures for pseudohyperbolic equations

In this paper, we introduce two split least-squares Galerkin finite element procedures for pseudohyperbolic equations arising in the modelling of nerve conduction process. By selecting the least-squares functional properly, the procedures can be split into two sub-procedures, one of which is for the primitive unknown variable and the other is for the flux. The convergence analysis shows that both the two methods yield the approximate solutions with optimal accuracy in L 2 ( Ω ) norm for u and u t and ( L 2 ( Ω ) ) 2 norm for the flux σ . Moreover, the two methods get approximate solutions with first-order and second-order accuracy in time increment, respectively. A numerical example is given to show the efficiency of the introduced schemes.

Least-squares Galerkin procedures for pseudohyperbolic equations

Two least-squares Galerkin finite element schemes are formulated to solve pseudohyperbolic equations. The advantage of this method is that it is not subject to the LBB condition. The convergence analysis shows that the methods both yield the approximate solutions with optimal accuracy in ( L 2 ( Ω ) ) 2 × L 2 ( Ω ) . Moreover, the two methods get the approximate solutions with first-order and second-order accuracy in time increment, respectively. Numerical example shows that the two schemes are effective.

Unique solvability of pseudohyperbolic equations with singular right-hand sides

In this paper, we consider the solvability of the Cauchy problem for pseudohyperbolic equations (partial differential equations of third order). For the case in which the right-hand side is a generalized function (distribution) of finite order, we establish a theorem on the unique solvability for a sufficiently general pseudohyperbolic operator. The method of proof is based on a specially constructed “scale” of a priori inequalities for the direct and adjoint operators.

Pseudo-stable and Pseudo-unstable Fiber Bundles for Dynamic Equations on Measure Chains

Invariant fiber bundles are the generalization of invariant manifolds from classical discrete or continuous dynamical systems to non-autonomous dynamic equations on measure chains. In this paper, we present a self-contained proof of their existence and smoothness. Our main result generalizes the so-called "Hadamard-Perron-Theorem" for hyperbolic finite-dimensional diffeomorphisms to pseudo-hyperbolic time-dependent non-regressive dynamic equations in Banach spaces. The proof of their smoothness uses a fixed point theorem of Vanderbauwhede-Van Gils.

Controllability and Optimization of Pseudohyperbolic Systems

The problems of singular optimization and controllability of systems described by the boundary-value problems for pseudohyperbolic equations are studied. The existence and uniqueness theorems for solutions to basic boundary-value problems for pseudohyperolic equations in the nonclassic Sobolev-like spaces under wide assumptions as to right-hand sides are proved. Based on these results, final impulse-point controllability of the systems and existence of optimal control are studied.

Identification problems for pseudohyperbolic integrodifferential operator equations

Identification problems for pseudohyperbolic integrodifferential operator equations

Blow-up and die-out of solutions of nonlinear pseudo-hyperbolic equations of generalized nerve conduction type

This paper considers the initial boundary value problems with three types of the boundary conditions for nonlinear pseudo-hyperbolic equations of generalized nerve conduction type, using the eigenfunction method, the conditions for which the solutions blow-up and die-out in the finite time are got.