A two-station queue with dependent preparation and service times

We discuss a single-server multi-station alternating queue where the preparation times and the service times are auto- and cross-correlated. We examine two cases. In the first case, preparation and service times depend on a common discrete time Markov chain. In the second case, we assume that the service times depend on the previous preparation time through their joint Laplace transform. The waiting time process is directly analysed by solving a Lindley-type equation via transform methods. Numerical examples are included to demonstrate the effect of the auto-correlation of and the cross-correlation between the preparation and service times.     

On a Certain Class of Integral Equations Associated with Hankel Transforms

This paper deals with a new class of Fredholm integral equations of the first kind associated with Hankel transforms of integer order. Analysis of the equations is based on operators transforming Bessel functions of the first kind into kernels of Weber–Orr integral transforms. Their inverse operators are established by means of new inversion theorems for the Hankel and Weber–Orr integral transforms of functions belonging to L ¹ and L ². These operators together with the proven Paley–Wiener’s theorem for the Weber–Orr transform enable to regularize the equations and, in special cases, to derive explicit solutions. The integral equations analyzed in this paper can be employed instead of dual integral equations usually treated with the Cooke–Lebedev method. An example manifests that it may be preferable because of the possibility to control norms of operators in the regularized equations.      

On Grushin’s equation

The paper deals with nonlinear problems for equations of Grushin type. We prove some nonexistence results via Pokhozhaev’s identity. In the rest of the paper we prove some results on smoothness near the boundary of eigenfunctions by using an explicit formula for fundamental solutions and the Kelvin transform for the operator. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 95–105, January, 1998. The author wishes to express his thanks to the International Center for Theoretical Physics of Trieste for support and hospitality.     

A highly accurate collocation Trefftz method for solving the Laplace equation in the doubly connected domains

A highly accurate new solver is developed to deal with the Dirichlet problems for the 2D Laplace equation in the doubly connected domains. We introduce two circular artificial boundaries determined uniquely by the physical problem domain, and derive a Dirichlet to Dirichlet mapping on these two circles, which are exact boundary conditions described by the first kind Fredholm integral equations. As a direct result, we obtain a modified Trefftz method equipped with two characteristic length factors, ensuring that the new solver is stable because the condition number can be greatly reduced. Then, the collocation method is used to derive a linear equations system to determine the unknown coefficients. The new method possesses several advantages: mesh‐free, singularity‐free, non‐illposedness, semi‐analyticity of solution, efficiency, accuracy, and stability. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Existence,uniqueness and a priori estimates for a nonlinear integro-differential equation

The paper deals with the explicit calculus and the properties of the fundamental solution K of a parabolic operator related to a semilinear equation that models reaction diffusion systems with excitable kinetics. The initial value problem in all of the space is analyzed together with continuous dependence and a priori estimates of the solution. These estimates show that the asymptotic behavior is determined by the reaction mechanism. Moreover it’s possible a rigorous singular perturbation analysis for discussing travelling waves with their characteristic times. This work has been performed under the auspices of the G.N.F.M. of I.N.D.A.M. and M.I.U.R. (P.R.I.N. 2007) “Waves and stability in continuous media”.     

On solvability of one nonlinear integral equation in the class of analytic functions

We consider a nonlinear integral equation of a special type that appears in the inverse spectral theory of integro-differential operators and whose unique solvability in the class of square-integrable functions is known. However, for some applied issues in order to construct effective algorithms for solving equations of this type, it is required to establish their solvability in the class of analytic functions. Assuming the free term of the nonlinear equation under consideration to be an entire function of exponential type, we prove that so is its solution. Leaning on this result we provide a constructive procedure for solving this equation in the class of square-integrable functions, which can be easily implemented numerically.     

A limiting absorption principle for scattering problems with unbounded obstacles

A generalized mode matching method that applies to a wide class of scattering problems is developed in the time harmonic two‐dimensional Helmholtz case. This method leads by variational means to an integro‐differential formulation whose unknown is the trace of the field on an unbounded one‐dimensional interface. The well‐posedness is proved after a careful study of the rather original functional framework. Owing to a fundamental density result—based upon some properties of a singular integral operator similar to the Hilbert transform—the limiting absorption principle related to this original formulation is established. Finally, two other situations are emphasized. Copyright © 2001 John Wiley & Sons, Ltd.     

A construction of multiscale bases for Petrov–Galerkin methods for integral equations

In this paper, a construction of multiscale bases for Petrov–Galerkin methods for Fredholm integral equations of the second kind is proposed. The properties of multiscale bases are presented including additional order of vanishing moments, compact supports and stability. Communicated by A. Zhou Dedicated to Professor Charles A. Micchelli on the occasion of his sixtieth birthday with friendship and esteem Mathematics subject classifications (2000) 41A10, 65R20, 65D15. Min Huang: Supported in part by Professor Yuesheng Xu's support under the program of “One Hundred Distinguished Young Scientists” of the Chinese Academy of Sciences and by the Graduate Innovation Foundation of the Chinese Academy of Sciences.     

Boundedness of solutions for Duffing-type equation

The boundedness of all solutions is shown for Duffing-type equations wherep ₁,p ₂,...,p _2n are of period 1 and of Lipschitzian continuity andp _n+1,...,p _2n are of Zygmundian continuity. This conclusion implies that the boundedness phenomenon for the Duffing-type equations does not require the smoothness in the time-variable, thus answering the question posed by Dieckerhoff and Zehnder.     

Absorbing boundary conditions and optimized Schwarz waveform relaxation

The numerical evaluation of the transforms in the title, and their inverses, is considered, using a variety of decomposition, truncation, and quadrature methods. Extensive numerical testing is provided and an application given to the numerical evaluation of the kernel of a Fredholm integral equation of interest in mixed boundary value problems on wedge-shaped domains. AMS subject classification (2000) 44A15, 65D30, 65R10     

A new fractional derivative for solving time fractional diffusion wave equation

The time fractional diffusion wave equation, which can be used to describe wave diffusion process in this article, was studied. First of all, the diffusion wave equation can be extended to a generalized form in the sense of the regularized version of the $k$-Hilfer–Prabhakar ( $k$-H-P) fractional operator involving the $k$-Mittag- function. Then, the analytical solution can be obtained for this considered equation by using the Laplace transform method and the Fourier transform method. As a result, a novel and general solution have been found. The unconventional solution may show new result and phenomenon to wave diffusion process. Thereby, this research provides a window for discovering new diffusion mechanisms.     

A new method of solving Bessel's differential equation using the -transform

In the present paper, a new method of solving Bessel's differential equation is given using the -transform.     

Exact soliton solutions for the general fifth Korteweg-de Vries equation

With the aid of computer symbolic computation system such as Maple, the extended hyperbolic function method and the Hirota’s bilinear formalism combined with the simplified Hereman form are applied to determine the soliton solutions for the general fifth-order KdV equation. Several new soliton solutions can be obtained if we taking parameters properly in these solutions. The employed methods are straightforward and concise, and they can also be applied to other nonlinear evolution equations in mathematical physics. The article is published in the original.     

Method of boundary integral equations as applied to the numerical solution of the three-dimensional Dirichlet problem for the laplace equation in a piecewise homogeneous medium

A Dirichlet problem is considered in a three-dimensional domain filled with a piecewise homogeneous medium. The uniqueness of its solution is proved. A system of Fredholm boundary integral equations of the second kind is constructed using the method of surface potentials, and a system of boundary integral equations of the first kind is derived directly from Green’s identity. A technique for the numerical solution of integral equations is proposed, and results of numerical experiments are presented.     

A penalized Fischer-Burmeister NCP-function

We introduce a new NCP-function in order to reformulate the nonlinear complementarity problem as a nonsmooth system of equations. This new NCP-function turns out to have stronger theoretical properties than the widely used Fischer-Burmeister function and other NCP-functions suggested previously. Moreover, numerical experience indicates that a semismooth Newton method based on this new NCP-function performs considerably better than the corresponding method based on the Fischer-Burmeister function. Received: March 10, 1997 / Accepted: February 15, 2000?Published online May 12, 2000     

A diffusion equation in transparent media

In this paper we study the homogeneous relativistic heat equation (HRHE) obtained as asymptotic limit of the so-called relativistic heat equation (RHE) when the kinematic viscosity ν → ∞. These equations were introduced in the theory of radiation hydrodynamics to guarantee a bounded speed of propagation of radiating energy. We shall prove that this is indeed true, and we shall construct some explicit solutions of the HRHE exhibiting fronts propagating at light speed.     

Cauchy problem for Mathieu’s equation at parametric resonance

Mathieu’s equation is solved by an asymptotic averaging method in the fourth approximation for the first to fourth resonance domains and in the third approximation for the zero resonance domain. The general periodic and aperiodic solutions on characteristic curves are found, and the general solution is obtained in instability domains and stability-domain areas adjacent to the characteristic curves. All the solutions are explicitly found in the form of functions of an argument without using the auxiliary parameter employed in Whittaker’s method. Simple formulas depending on two parameters of the equation are derived for the characteristic exponent in instability domains and for the frequency of slow oscillations in stability domains near the characteristic curves. The theory is developed by analyzing the resonances exhibited by Mathieu’s equation.