A two-point Markov chain boundary-value problem

1993 ◽  
Vol 25 (3) ◽  
pp. 607-630 ◽  
Author(s):  
D. J. Daley ◽  
L. D. Servi
Keyword(s):  
Markov Chain ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Sample Path ◽  
Maximum Likelihood Estimates ◽  
Initial Time ◽  
Time Interval ◽  
Birth Process ◽  
Waiting Room ◽  
Time Point

The two-point Markov chain boundary-value problem discussed in this paper is a finite-time version of the quasi-stationary behaviour of Markov chains. Specifically, for a Markov chain {Xt:t = 0, 1, ·· ·}, given the time interval (0, n), the interest is in describing the chain at some intermediate time point r conditional on knowing both the behaviour of the chain at the initial time point 0 and that over the interval (0, n) it has avoided some subset B of the state space. The paper considers both ‘real time' estimates for r = n (i.e. the chain has avoided B since 0), and a posteriori estimates for r < n with at least partial knowledge of the behaviour of Xn. Algorithms to evaluate the distribution of Xr can be as small as O(n3) (and, for practical purposes, even O(n2 log n)). The estimates may be stochastically ordered, and the process (and hence, the estimates) may be spatially homogeneous in a certain sense. Maximum likelihood estimates of the sample path are furnished, but by example we note that these ML paths may differ markedly from the path consisting of the expected or average states. The scope for two-point boundary-value problems to have solutions in a Markovian setting is noted.Several examples are given, together with a discussion and examples of the analogous problem in continuous time. These examples include the basic M/G/k queue and variants that include a finite waiting room, reneging, balking, and Bernoulli feedback, a pure birth process and the Yule process. The queueing examples include Larson's (1990) ‘queue inference engine'.

A two-point Markov chain boundary-value problem

1993 ◽  
Vol 25 (03) ◽  
pp. 607-630 ◽  
Author(s):  
D. J. Daley ◽  
L. D. Servi
Keyword(s):  
Markov Chain ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Sample Path ◽  
Maximum Likelihood Estimates ◽  
Initial Time ◽  
Time Interval ◽  
Birth Process ◽  
Waiting Room ◽  
Time Point

The two-point Markov chain boundary-value problem discussed in this paper is a finite-time version of the quasi-stationary behaviour of Markov chains. Specifically, for a Markov chain {Xt :t = 0, 1, ·· ·}, given the time interval (0, n), the interest is in describing the chain at some intermediate time point r conditional on knowing both the behaviour of the chain at the initial time point 0 and that over the interval (0, n) it has avoided some subset B of the state space. The paper considers both ‘real time' estimates for r = n (i.e. the chain has avoided B since 0), and a posteriori estimates for r &lt; n with at least partial knowledge of the behaviour of Xn. Algorithms to evaluate the distribution of Xr can be as small as O(n 3) (and, for practical purposes, even O(n 2 log n)). The estimates may be stochastically ordered, and the process (and hence, the estimates) may be spatially homogeneous in a certain sense. Maximum likelihood estimates of the sample path are furnished, but by example we note that these ML paths may differ markedly from the path consisting of the expected or average states. The scope for two-point boundary-value problems to have solutions in a Markovian setting is noted. Several examples are given, together with a discussion and examples of the analogous problem in continuous time. These examples include the basic M/G/k queue and variants that include a finite waiting room, reneging, balking, and Bernoulli feedback, a pure birth process and the Yule process. The queueing examples include Larson's (1990) ‘queue inference engine'.

scholarly journals Integral presentations of the solution of a boundary value problem for impulsive fractional integro-differential equations with Riemann-Liouville derivatives

2021 ◽  
Vol 7 (2) ◽  
pp. 2973-2988
Author(s):  
Ravi Agarwal ◽  
◽  
Snezhana Hristova ◽  
Donal O'Regan ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Initial Time ◽  
Qualitative Properties ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Time Point

<abstract><p>Riemann-Liouville fractional differential equations with impulses are useful in modeling the dynamics of many real world problems. It is very important that there are good and consistent theoretical proofs and meaningful results for appropriate problems. In this paper we consider a boundary value problem for integro-differential equations with Riemann-Liouville fractional derivative of orders from $ (1, 2) $. We consider both interpretations in the literature on the presence of impulses in fractional differential equations: With fixed lower limit of the fractional derivative at the initial time point and with lower limits changeable at each impulsive time point. In both cases we set up in an appropriate way impulsive conditions which are dependent on the Riemann-Liouville fractional derivative. We establish integral presentations of the solutions in both cases and we note that these presentations are useful for furure studies of existence, stability and other qualitative properties of the solutions.</p></abstract>

The autostrada queueing problem

1984 ◽  
Vol 21 (2) ◽  
pp. 394-403 ◽  
Author(s):  
B. W. Conolly
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Waiting Room ◽  
Queue System ◽  
Room Size ◽  
Fresh Perspective ◽  
Queues In Parallel

The model considered in this note has been referred to by Haight (1958), Kingman (1961) and Flatto and McKean (1977) as two queues in parallel. Customers choose the shorter of the two queues which are otherwise independent. This system is known to be inferior to a single queue feeding the two servers, but how much? Some elementary considerations provide a fresh perspective on this awkward boundary-value problem. A procedure is proposed for the solution in the context of finite waiting-room size and some comparisons are made with the single-queue system and an independent two-queue system.

The autostrada queueing problem

1984 ◽  
Vol 21 (02) ◽  
pp. 394-403 ◽  
Author(s):  
B. W. Conolly
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Waiting Room ◽  
Queue System ◽  
Room Size ◽  
Fresh Perspective ◽  
Queues In Parallel

The model considered in this note has been referred to by Haight (1958), Kingman (1961) and Flatto and McKean (1977) as two queues in parallel. Customers choose the shorter of the two queues which are otherwise independent. This system is known to be inferior to a single queue feeding the two servers, but how much? Some elementary considerations provide a fresh perspective on this awkward boundary-value problem. A procedure is proposed for the solution in the context of finite waiting-room size and some comparisons are made with the single-queue system and an independent two-queue system.

scholarly journals QUANTUM AMPLITUDES IN BLACK-HOLE EVAPORATION: COMPLEX APPROACH AND SPIN-0 AMPLITUDE

2011 ◽  
Vol 20 (02) ◽  
pp. 133-159
Author(s):  
A. N. St. J. FARLEY ◽  
P. D. D'EATH
Keyword(s):  
Black Hole ◽  
Gravitational Field ◽  
Scalar Field ◽  
Boundary Value Problem ◽  
Boundary Data ◽  
Proper Time ◽  
Boundary Value ◽  
Constrained Systems ◽  
Time Interval ◽  
Late Time

This paper is concerned with the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat space–time and weak radiation at a very late time. We evaluate quantum amplitudes (not just probabilities) for transitions from initial to final states. This quantum description shows that no information is lost in collapse to a black hole. Boundary data for the gravitational field and (in this paper) a scalar field are posed on an initial space-like hypersurface ΣI and a final surface ΣF. These asymptotically flat three-surfaces are separated by a Lorentzian proper-time interval T (typically very large), as measured at spatial infinity. The boundary-value problem is made well-posed, both classically and quantum-mechanically, by a rotation of T into the lower-half complex plane: T → |T| exp (- iθ), with 0 < θ ≤ π/2. This corresponds to Feynman's +iϵ prescription. We consider the classical boundary-value problem and calculate the second-variation classical Lorentzian action [Formula: see text] as a functional of the boundary data. Following Feynman, the Lorentzian quantum amplitude is recovered in the limit θ → 0+ from the well-defined complex-T amplitude. Dirac's canonical approach to the quantisation of constrained systems shows that, for locally supersymmetric theories of gravity, the amplitude is exactly semi-classical, namely [Formula: see text] for weak perturbations, apart from delta functionals of the supersymmetry constraints. We treat such quantum amplitudes for weak scalar-field configurations on ΣF, taking (for simplicity) the weak final gravitational field to be spherically symmetric. The treatment involves adiabatic solutions to the scalar wave equation. This considerably extends work reported in previous papers, by giving explicit expressions for the real and imaginary parts of such quantum amplitudes.

scholarly journals Plastic Flow in a Thin Layer: the Theory, Formulations of the Boundary- Value Problems and the Applications

2021 ◽  
Vol 248 ◽  
pp. 01018
Author(s):  
Vagid Kadymov ◽  
Elena Yanovskaya
Keyword(s):  
Heat Transfer ◽  
Boundary Value Problem ◽  
Mathematical Theory ◽  
Tensile Force ◽  
Boundary Value ◽  
Initial Time ◽  
Plastic Layer ◽  
Two Dimensional ◽  
Compression Force ◽  
Thin Plastic

Two-dimensional, averaged over the thickness of the layer, mathematical theory of the spreading of a plastic layer on the plane has been studied. General and simplified mathematical formulations of boundary value problem were presented. The problem of plastic stretching of a strip by forces applied on its “clamped” ends was investigated. The analysis of various modes of the process was carried out, which are determined by both the total compression force of the ends and the total tensile force. Mathematical analogy between the process of the free spreading of a plastic layer on the plane and the process of heat transfer was studied. For known forms of a domain occupied by a thin plastic layer at the initial time and for a given law of convergence of the plates, the evolution of the boundary of a plastic layer spreading was described. The exact particular solutions of the aforementioned problem was obtained.

scholarly journals ON THE REGULARIZATION OF EQUATIONS OF THE MECHANICS OF MIXTURES OF VISCOUS COMPRESSIBLE FLUIDS

2017 ◽  
pp. 54-71
Author(s):  
Nikolay Kucher ◽  
Nikolay Kucher ◽  
Aleksandra Zhalnina ◽  
Aleksandra Zhalnina
Keyword(s):  
Boundary Value Problem ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Compressible Fluids ◽  
Time Interval ◽  
Approximation Procedure ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation

Mathematical models of multi-velocity continua, through which the motion of multicomponent mixtures are described, represent a rather extensive area of modern mechanics and mathematics. Mathematical results (statements of problems, theorems on the existence and uniqueness, properties of solutions, etc.) for such models are rather modest in comparison with the results for classical single-phase media. The present paper aims to fill this gap in some extent and is devoted to investigating the global correctness of the boundary value problem for a nonlinear system of differential equations, which is some regularity of the mathematical model of nonstationary spatial flows of a mixture of viscous compressible fluids. Construction of the solution of the problem considered in this article is a key step for the mathematical analysis of the initial model of the mixture, since it allows to obtain globally defined solutions of the latter by means of a limiting transition and, in addition, the proposed algorithm for constructing solutions to the regularized problem is practical. This algorithm is based on the finite-dimensional approximation procedure for an infinite-dimensional problem, and therefore a mathematically grounded algorithm for the numerical solution of the boundary value problem of the motion of a mixture of viscous compressible fluids in a region bounded by solid walls can be constructed on this basis. The local in time solvability of finite- dimensional problems is proved by applying the principle of contracting mappings and the local solution can be extended to an arbitrary time interval with the help of a priori estimates.

scholarly journals Ingularly perturbed equations in critical cases

2021 ◽  
Vol 84 (4) ◽  
pp. 69-75
Author(s):  
Zh.K. Daniyarova ◽  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Small Parameter ◽  
Ordinary Differential Equations ◽  
Asymptotic Expansions ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Singular Problem ◽  
Time Interval

Singularly perturbed partial differential equations with small parameters with higher derivatives deserve special attention, which often arise in a variety of applied problems and are used in describing mathematical models of diffusion processes, absorption taking into account small diffusion, filtration of liquids in porous media, chemical kinetics, chromatography, heat and mass transfer, hydrodynamics and many other fields. It is necessary to consider the creation of an asymptotic classification of solutions of singularly perturbed equations using a well-known approach to solving the boundary value problem. In this case, the singular problem is understood as the problem of constructing the asymptotics of the solution of the Cauchy problem for a system of ordinary differential equations with a small parameter with a large derivative. The asymptotics of the solution in all cases is based on the last time interval or the construction of a boundary value problem for a system with a weak clot in an asymptotically large time interval. Purpose - to construct and substantiate the asymptotics of solving a singular initial problem for a system of two nonlinear ordinary differential equations with a small parameter; To date, a number of methods have been developed for constructing asymptotic expansions of solutions to various problems. This is the method of boundary functions developed in the works of A.B. Vasilyeva, M.I. Vishik, L.A. Lusternik, V.F. Butuzov; the regularization method of S. A. Lomov, methods of averaging, VKB, splicing of asymptotic decompositions of A.M. Ilyin and others. All the above methods allow us to obtain asymptotic expansions of solutions for wide classes of equations. At the same time, such singularly perturbed problems often arise, to which ready-made methods are not applicable or do not allow to obtain an effective result. Therefore, the development of methods for solving equations remains a very urgent problem. As a result of the study, an algorithm for constructing an asymptotic classification of the initial solution of the problem with a singular perturbation is given, and approaches to estimating the residual term are also shown.

scholarly journals Application of thermal potentials to the solution of the problem of heat conduction in a region degenerates at the initial moment

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 825-836
Author(s):  
Alexey Kavokin ◽  
Adiya Kulakhmetova ◽  
Yuriy Shpadi
Keyword(s):  
Heat Transfer ◽  
Integral Equation ◽  
Heat Conduction ◽  
Liquid Metal ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Constructive Method ◽  
Initial Time ◽  
Initial Moment ◽  
Electric Contacts

In this paper, the boundary value problem for the heat equation in the region which degenerates at the initial time is considered. Such problems arise in mathematical models of the processes occurring by opening of electric contacts, in particular, at the description of the heat transfer in a liquid metal bridge and electric arcing. The boundary value problem is reduced to a Volterra integral equation of the second kind which has a singular feature. The class of solutions for the integral equation is defined and the constructive method of its solution is developed.

scholarly journals Two-point boundary value problem for a partial differential equation in spaces of periodic functions

2020 ◽  
Vol 54 (1) ◽  
pp. 79-90
Author(s):  
V.S. Ilkiv ◽  
Z.M. Nytrebych ◽  
P.Y. Pukach ◽  
M.I. Vovk
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Spatial Variable ◽  
Necessary Condition ◽  
Time Interval ◽  
Partial Differential ◽  
Finite Dimensionality ◽  
The Right

We investigate the two-point in time boundary value problem for the partial differential equations of the second-order with one spatial variable and constant coefficients. The problem is considered in in the spaces of functions which Fourier coefficients are characterized by exponential behavior on the Cartesian product of the time interval and spatial domain $\mathbb{R}/2\pi\mathbb{Z}$. The correct solvability of the problem is established, the formulas for solutions are presented, the kernel is described and the smoothness of the solution is established in the spaces of functions that are periodic in one spatial variable. We have established the conditions which are close to the necessary conditions of solvability of the problem in scale of spaces of functions with exponentially increasing (or decreasing) Fourier coefficients.We also found the asymptotic estimates demonstrating the absence of the problem of small denominators, which arises of many spatial variables and makes the boundary value problem incorrect. We have established sufficient conditions of the finite-dimensionality of the kernel of the problem and found upper bounds for its dimension. The results are obtained under the condition of minimum smoothness on the right-hand sides of two-point conditions, which is close to the necessary condition.

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