Second-order Hyperbolic Differential Equation Research Articles

A differential equation for Lerch's transcendent and associated symmetric operators in Hilbert space

The function , where is Lerch's transcendent, satisfies the following two-dimensional formally self- adjoint second-order hyperbolic differential equation: where . The corresponding differential expression determines a densely defined symmetric operator (the minimal operator) on the Hilbert space , where . We obtain a description of the domains of definition of some symmetric extensions of the minimal operator. We show that formal solutions of the eigenvalue problem for these symmetric extensions are represented by functional series whose structure resembles that of the Fourier series of . We discuss sufficient conditions for these formal solutions to be eigenfunctions of the resulting symmetric differential operators. We also demonstrate a close relationship between the spectral properties of these symmetric differential operators and the distribution of the zeros of some special analytic functions analogous to the Riemann zeta function. Bibliography: 15 titles.

Goursat problem for two-dimensional second-order hyperbolic operator-differential equations with variable domains

We develop a modification of the energy inequality method and use it to prove the well-posedness of the Goursat problem for linear second-order hyperbolic differential equations with operator coefficients whose domains depend on the two-dimensional time. An energy inequality for strong solutions of this Goursat problem is derived with the help of abstract smoothing operators, and we prove that the range of the problem is dense by using properties of a regularizing Cauchy problem whose inverse operator is a family of smoothing operators of a new type. We give an example of a well-posed boundary value problem for a two-dimensional complete second-order hyperbolic partial differential equation with Goursat conditions and with a boundary condition depending on the two-dimensional time.

An operator-difference scheme for abstract Cauchy problems

An abstract Cauchy problem for second-order hyperbolic differential equations containing the unbounded self-adjoint positive linear operator A ( t ) with domain in an arbitrary Hilbert space is considered. A new second-order difference scheme, generated by integer powers of A ( t ) , is developed. The stability estimates for the solution of this difference scheme and for the first- and second-order difference derivatives are established in Hilbert norms with respect to space variable. To support the theoretical statements for the solution of this difference scheme, the numerical results for the solution of one-dimensional wave equation with variable coefficients are presented.

A difference scheme for Cauchy problem for the hyperbolic equation with self-adjoint operator

A new second order absolutely stable difference scheme is presented for Cauchy problem for second-order hyperbolic differential equations containing the operator A ( t ) . This scheme makes use of this operator which is unbounded linear self-adjoint positive definite with domain in an arbitrary Hilbert space. The stability estimates for the solution of this difference scheme and for the first and second-order difference derivatives are established. The theoretical statements for the solution of this difference scheme are supported by the results of numerical experiments.

Some Remarks on Some Second-order Hyperbolic Differential Equations

We are concerned with the almost automorphic solutions to the second-order hyperbolic differential equations of type u(s) + 2B u(s) + Au(s) = f(s) (*), where A, B are densely defined closed linear operators acting in a Hilbert space H, and f : R |—> H is a vector-valued almost automorphic function. Using invariant subspaces, it will be shown that under appropriate assumptions, every solution to (*) is almost automorphic.

A class of one-step time integration schemes for second-order hyperbolic differential equations

We present a class of extended one-step time integration schemes for the integration of second-order nonlinear hyperbolic equations u tt = c 2 u xx + p( x,t,u), subject to initial conditions and boundary conditions of Dirichlet type or of Neumann type. We obtain one-step time integration schemes of orders two, three, and four; the schemes are unconditionally stable. For nonlinear problems, the second- and the third-order schemes have tridiagonal Jacobians, and the fourth-order schemes have pentadiagonal Jacobians. The accuracy and stability of the obtained schemes is illustrated computationally by considering numerical examples, including the sine-Gordon equation.

Difference scheme for hyperbolic heat conduction equation with pulsed heating boundary

In this paper, the authors have analyzed the second-order hyperbolic differential equation with pulsed heating boundary, and tried to solve this kind of equation with numerical method. After analyzing the error of the difference scheme and comparing the numerical results with the theoretical results, the authors present some examples to show how the thermal wave propagates in materials. By analyzing the calculation results, the conditions to observe the thermal wave phenomena in the laboratory are conferred. Finally the heat transfer in a complex combined structure is calculated with this method. The result is quite different from the calculated result from the parabolic heat conduction equation.

The Cauchy problem for complete second-order hyperbolic differential equations with variable domains of operator coefficients

The Cauchy problem for complete second-order hyperbolic differential equations with variable domains of operator coefficients

L∞-Estimates of Quasi-Optimal Order for Galerkin Methods to N=2, 3-Dimensional Second-Order Hyperbolic Differential Equations

Using weighted norms, L∞-error estimates of the Galerkin method for second order hyperbolic differential equations are derived.