A multiple-scales solution of the undular hydraulic jump problem

A non-linear third-order ODE governing the free surface of an undular hydraulic jump is solved by means of a multiple-scales analysis. The resulting solutions are compared with direct numerical solutions of the ODE, and good agreement is found. Furthermore, new flow features, which are not evident in the numerical solutions, are revealed by the multiple-scales solutions. The results of the analysis are also compared with numerical solutions of the full equations of motion and with experimental data, and satisfactory agreement is obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A series solution of the fin problem with a temperature-dependent thermal conductivity

In this work, we apply the homotopy analysis method to solve fin problems with temperature-dependent thermal conductivity. The results are compared with the solutions obtained by the Adomian decomposition method (ADM), homotopy perturbation method (HPM) and with the Taylor series expansion method as well. The results of the comparisons have shown that the ADM and HPM are just an approximation of the present study and the accuracy of the solution obtained by the present study is much better than the latter algorithms.     

Asymptotic solution of the problem of nonstationary thermal conductivity of laminated anisotropic inhomogeneous shells

External asymptotic expansions of the solutions of the problem of nonstationary thermal conductivity of laminated anisotropic inhomogeneous shells under different boundary conditions on faces are constructed. We analyze the obtained two-dimensional resolving equations and investigate the asymptotic properties of the solutions of the problem of thermal conductivity. A physical justification of some features of the asymptotic expansion of temperature is presented.     

A heat balance integral solution of the thermistor problem with a modified electrical conductivity

A key driver of thermistor response is a thermally-dependent electrical conductivity. Several variants have been proposed in the literature and it is the purpose of this paper to explore the applicability of a bulk model by way of using the heat balance integral approach to predict the response of a positive temperature coefficient thermistor. Conclusions are presented regarding the sensitivity of the response to electrical conductivity.     

Stability estimates for the solution to the problem of determining the kernel of a viscoelastic equation

For an integrodifferential equation corresponding to a two-dimensional viscoelastic problem, we study the problem of defining the spatial part of the kernel involved in the integral term of the equation. The support of the sought function is assumed to belong to a compact domain Ω. As information for solving this inverse problem, the traces of the solution to the direct Cauchy problem and its normal derivative are given for some finite time interval on the boundary of Ω. An important feature in the statement of the problem is the fact that the solution of the direct problem corresponds to the zero initial data and a force impulse in time localized on a fixed straight line disjoint with Ω. The main result of the article consists in obtaining a Lipschitz estimate for the conditional stability of the solution to the inverse problem under consideration.     

Interpolation solution to a boundary value problem of harmonic field

This article is a improvement on author's early work (Acta Mathematica Scientia, Vol.30 No.2 Ser.A 2010). In this article, there are two new contributions: 1) The restrictive conditions on approximation domain boundary is improved essentially. 2) The Fejér points is extended by perturbed Fejér points with stable order of approximation.     

On the hydraulic lock-exchange problem

Summary This paper discusses the fundamental aspects of the hydraulic lock-exchange problem. In a first part the limit of small density ratios of the two fluids involved is discussed by considering the problem of emptying a horizontal water-filled channel by means of pressurized air.To study the problem for arbitrary density ratios, the general case of two fluids of different densities in a closed horizontal channel that are initially separated by a vertical splitter plate is considered. At timet=0 (say) the separating plate is suddenly removed and thus initiates the lock-exchange flow. Generally, a complete lock-exchange flow involves an essentially loss-free gravity current of low-density fluid penetrating into a part of the channel that was originally filled with high-density fluid. An expansion wave that runs in the same direction follows this gravity current at a somewhat lower speed and connects it to a counter-running gravity current of high-density fluid penetrating into a part of the channel that was originally filled with low-density fluid. At density ratiosr0.281 this gravity current of high-density fluid exhibits a dissipative character, i.e. in a frame of reference moving with the gravity current the flow behind it is subcritical. As the density ratio exceeds the value 0.281 the gravity current may essentially become loss-free. In this case the flow behind it becomes supercritical and the loss-free gravity current is followed by an internal hydraulic jump.

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Zusammenfassung Diese Arbeit befasst sich mit den grundlegenden Aspekten der hydraulischen Ausgleichsströmungen. In einem ersten Teil wird der Grenzfall eines kleinen Dichteverhältnisses zweier Fluida anhand der Betrachtung des Entleerungsproblems eines horizontalen wassergefüllten Kanals mittels Pressluft diskutiert.Zur Untersuchung des Problems für beliebige Dichteverhältnisse wird der allgemeine Fall zweier Fluida verschiedener Dichte in einem geschlossenen horizontalen Kanal betrachtet, die anfänglich durch eine vertikale Trennwand voneinander getrennt sind. Zu einer Zeitt=0 wird die Trennwand plötzlich entfernt, und die Ausgleichsströmung setzt ein. Im allgemeinen besteht eine vollständige Ausgleichsströmung aus einem Schwerestrom eines leichten Fluids, der in einen Kanalbereich eindringt, welcher ursprünglich mit einem schweren Fluid gefüllt war. Eine Expansionswelle, die mit etwas kleinerer Geschwindigkeit in die gleiche Richtung läuft, folgt diesem Schwerestrom and verbindet ihn mit einem in Gegenrichtung laufenden Schwerestrom hoher Dichte, der in einem Kanalbereich eindringt, welcher ursprünglich mit leichtem Fluid gefüllt war. Für Dichteverhältnisser0.281 weist dieser Schwerestrom einen dissipativen Charakter auf-oder anders ausgedrückt-in einem Bezugssystem, das sich mit dem Schwerestrom bewegt, ist die nachfolgende Strömung unterkritisch. Wenn das Dichteverhältnis den Wert 0.281 übersteigt, kann der Schwerestrom grundsätzlich verlustfrei sein. In diesem Fall wird die nachfolgende Strömung überkritisch und der verlustfreie Schwerestrom wird von einem inneren hydraulischen Sprung gefolgt.

     

Uniqueness and nonuniqueness of the solution to the problem of determining the source in the heat equation

An initial–boundary value problem for the two-dimensional heat equation with a source is considered. The source is the sum of two unknown functions of spatial variables multiplied by exponentially decaying functions of time. The inverse problem is stated of determining two unknown functions of spatial variables from additional information on the solution of the initial–boundary value problem, which is a function of time and one of the spatial variables. It is shown that, in the general case, this inverse problem has an infinite set of solutions. It is proved that the solution of the inverse problem is unique in the class of sufficiently smooth compactly supported functions such that the supports of the unknown functions do not intersect. This result is extended to the case of a source involving an arbitrary finite number of unknown functions of spatial variables multiplied by exponentially decaying functions of time.     

General exact solution of the fin problem with the power law temperature‐dependent thermal conductivity

In this paper, the general exact implicit solution of the second‐order nonlinear ordinary differential equation governing heat transfer in rectangular fin is obtained using Lie point symmetry method. General relationship among the fin efficiency, the rate of heat transfer from the entire fin, the fin effectiveness, and the thermo‐geometric fin parameter is obtained for any value of the mode of heat transfer n and the constant β. Copyright © 2015 John Wiley & Sons, Ltd.     

Inverse problem of determining parameters of inhomogeneity of a body from acoustic field measurements

This work is devoted to development of methods for solving inverse problems in acoustics. Propagation of an acoustic field in a body located in the free space is considered. In the inverse problem, an iterative method for reconstructing the parameters of inhomogeneity of a body from a known acoustic field is applied. The theorem on convergence of the method is proven. Numerical results for inhomogeneous bodies of complex form are presented.     

Self-similar solution of the plane problem of the evolution of a hydraulic fracture crack in an elastic medium

The plane problem of the evolution of a hydraulic fracture crack in an elastic medium is considered. It is established that a self-similar solution is only possible at a constant rate of fluid injection. The solution for the value of the crack opening is presented in the form of a series expansion in Chebyshev polynomials of the second kind, and expansion coefficients are obtained as a solution of the algebraic set of equations which arise when projecting the balance equation for injected fluid mass on Chebyshev polynomials. When there is no part of the region unfilled with fluid (a fluid lag), the gradient of the crack opening at the crack tip turns out to be singular when the finiteness of the medium stress intensity factor is taken into account. According to the estimate made, the rate of convergence of the series expansion for the solution in Chebyshev polynomials is fairly rapid for a small injection intensity.     

Numerical solution of an inverse electrocardiography problem for a medium with piecewise constant electrical conductivity

A numerical method is proposed for solving an inverse electrocardiography problem for a medium with a piecewise constant electrical conductivity. The method is based on the method of boundary integral equations and Tikhonov regularization.     

Exterior problem of magnetic field in Darwin model and its numerical solution

In this paper, we consider the solutions of magnetic field in the Darwin model to the Maxwell's equations in 2D unbounded domain. We first deduce the variational formulation and prove the well‐posedness of the weak solution, and then prove the existence and uniqueness of the infinite element solution. Error estimate and the numerical examples are provided. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

On uniqueness in the inverse conductivity problem

We derive a uniqueness proof of inclusions of different (analytic) conductivities in the equation div(a grad u) = 0 in Ω under the minimal assumptions: (i) the boundaries of inclusions are only Lipschitz and (ii) we have no topological assumptions. For any Dirichlet data g, we are given the Neumann data h; in other words, results of all possible boundary measurements are known. For this purpose, we use and modify the construction of singular solution of elliptic equations due to Alessandrini [1]. Copyright © 1999 John Wiley & Sons, Ltd.     

Series solution of the differential equation determining the nth-shell one-electron density of a bare Coulomb problem in quantum physics

The differential equation determining the nth-shell one-electron density of a bare Coulomb problem is analytically investigated by using the differential function method and extended homogeneous balance idea. New series solution of the differential equation has been obtained in quantum physics for the first time. The explicit series analytical expression for the mathematical characteristic of the bare Coulomb problem in the density-functional theory can further describe the dynamical behaviors of the nth-shell one-electron density.