On the coefficient inverse problem of heat conduction in a degenerating domain

ABSTRACT

In the paper, we consider a coefficient inverse problem for the heat equation in a degenerating angular domain. It has been shown that the inverse problem for the homogeneous heat equation with homogeneous boundary conditions has a nontrivial solution up to a constant factor consistent with the integral condition. Moreover, the solution of the considered inverse problem is found in explicit form. In conclusion, statements of possible generalizations and the results of numerical calculations are given.     

Two inverse problems of finding a coefficient in a parabolic equation

For a parabolic equation, we consider inverse problems of reconstructing a coefficient that depends on the space variables alone. The first problem is to find a lower-order coefficient c(x) multiplying u(x, t), and the second problem is to find the coefficient a(x) multiplying Δu. As additional information, the integral of the solution with respect to time with some weight function is given. The coefficients of the equation depend both on time and on the space variables. We obtain sufficient conditions for the existence of generalized solutions of our problems; moreover, for the first problem, we also prove uniqueness and construct an iterative sequence that converges to the desired coefficient almost everywhere in the domain. We present examples of input data of these problems for which the assumptions of our theorems are necessarily true.     

Numerical Solutions to the Darboux Problem with Functional Dependence

The paper deals with the Darboux problem for the equation D _{xy z} (x,y) = f(x,y,z( _x,y ) where z( _x,y ) is a function defined by . We construct a general class of difference methods for this problem. We prove the existence and uniqueness of solutions to implicit functional difference equations by means of a comparison method; moreover we give an error estimate. The convergence of explicit difference schemes is proved under a general assumption that given functions satisfy nonlinear estimates of the Perron type. Our results are illustrated by a numerical example.     

An elegant 3-basis for inverse semigroups

It is well known that in every inverse semigroup the binary operation and the unary operation of inversion satisfy the following three identities:

An Asymptotic Expansion of the Double Gamma Function

The Barnes double gamma function G(z) is considered for large argument z. A new integral representation is obtained for log G(z). An asymptotic expansion in decreasing powers of z and uniformly valid for |Arg z|<π is derived from this integral. The expansion is accompanied by an error bound at any order of the approximation. Numerical experiments show that this bound is very accurate for real z. The accuracy of the error bound decreases for increasing Arg z.     

Some remarks on forced kdv

For the KdV equation with forcing q(0,t) = Q(t) and initial data q(x,0) = q₀: (x) (0≤x <∞) we show how to solve the problem by inverse scattering. The time evolution of spectral data is obtained via a differential integral equation using intrinsic half line spectral data     

Unique solvability of the inverse problem of determination of the leading coefficient in a parabolic equation

We study existence and uniqueness of the solution for the inverse problem of determination of the unknown coefficient ϱ(t) multiplying u _t in a nondivergence parabolic equation. As additional information, the integral of the solution over the domain of space variables with some given weight function is specified. The coefficients of the equation depend both on time and on the space variables.     

A reconstruction method for a two-dimensional inverse eigenvalue problem

In this paper we describe a method for constructing approximate solutions of a two-dimensional inverse eigenvalue problem. Here we consider the problem of recovering a functionq(x, y) from the eigenvalues of — +q(x, y) on a rectangle with Dirichlet boundary conditions. The potentialq(x, y) is assumed to be symmetric with respect to the midlines of the rectangle. Our method is a generalization of an algorithm Hald presented for the construction of symmetric potentials in the one-dimensional inverse Sturm-Liouville problem. Using a projection method, the inverse spectral problem is reduced to an inverse eigenvalue problem for a matrix. We show that if the given eigenvalues are small perturbations of simple eigenvalues ofq=0, then the matrix problem has a solution. This solution is used to construct a functionq which has the same lowest eigenvalues as the unknownq, and several numerical examples are given to illustrate the methods.     

Inverse scattering in one-dimensional nonconservative media

The inverse scattering problem arising in wave propagation in one-dimensional non-conservative media is analyzed. This is done in the frequency domain by considering the Schrödinger equation with the potentialikP(x)+Q(x), wherek ² is the energy andP(x) andQ(x) are real integrable functions. Using a pair of uncoupled Marchenko integral equations,P(x) andQ(x) are recovered from an appropriate set of scattering data including bound-state information. Some illustrative examples are provided.Dedicated to M.G. Kreîn, one of the founding fathers of inverse scattering theory.     

Inverse problem of scattering theory for a class one-dimensional Schrödinger equation

Abstract

In this work, direct and inverse scattering problem on the real axis for the Schrödinger equation with piecewise-constant coefficient are studied. Using the new integral representations for solutions, the scattering data is defined, the main integral equations of the inverse scattering problem are obtained, the spectral characteristics of the scattering data are investigated and uniqueness theorem for the solution of inverse problem is proved.     

Cancellation in Skew Lattices

Distributive lattices are well known to be precisely those lattices that possess cancellation: x úy = x úzx \lor y = x \lor z and x ùy = x ùzx \land y = x \land z imply y = z. Cancellation, in turn, occurs whenever a lattice has neither of the five-element lattices M ₃ or N ₅ as sublattices. In this paper we examine cancellation in skew lattices, where the involved objects are in many ways lattice-like, but the operations ù\land and ú\lor no longer need be commutative. In particular, we find necessary and sufficient conditions involving the nonoccurrence of potential sub-objects similar to M ₃ or N ₅ that ensure that a skew lattice is left cancellative (satisfying the above implication) right cancellative (x úz = y úzx \lor z = y \lor z and x ùz = y ùzx \land z = y \land z imply x = y) or just cancellative (satisfying both implications). We also present systems of identities showing that left [right or fully] cancellative skew lattices form varieties. Finally, we give some positive characterizations of cancellation.     

Stability and instability criteria for Kaplan–Yorke solutions

We derive sufficient conditions for the stability and instability of periodic solutions p:\mathbbR ? \mathbbRp:\mathbb{R} \to \mathbb{R} of Kaplan–Yorke type to the equation $\ifmmode\expandafter\dot\else\expandafter\.\fi{x}(t) = \alpha f(x(t),x(t - 1)),$\ifmmode\expandafter\dot\else\expandafter\.\fi{x}(t) = \alpha f(x(t),x(t - 1)), where f is even in the first and odd in the second argument. The criteria are based on the monotonicity of the coefficient in a transformed version of the variational equation. For the special case of cubic f, we show that this monotonicity property is satisfied if and only if the set { (p(t),p(t - 1))|t ? \mathbbR} ì \mathbbR² \{ (p(t),p(t - 1))|t \in \mathbb{R}\} \subset \mathbb{R}^{2} is contained in a region E defined by a quadratic form (bounded by an an ellipse or a hyperbola). The coefficients of this quadratic form are expressible in terms of the Taylor coefficients of f. Further, the parameter α in the equation and the amplitude z of the periodic solution are related by an elliptic integral. Using the relation between this integral and the arithmeticgeometric mean, we obtain upper and lower estimates on this relation, and on the inverse function. Combining these estimates with the inequality that defines the region E, we obtain stability criteria explicit in terms of the Taylor coefficients of f. These criteria go well beyond local stability analysis, as examples show.     

On the Nevanlinna characteristic of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">z</Emphasis>+<Emphasis Type="Italic">η</Emphasis>) and difference equations in the complex plane

We investigate the growth of the Nevanlinna characteristic of f(z+η) for a fixed η∈C in this paper. In particular, we obtain a precise asymptotic relation between T(r,f(z+η)) and T(r,f), which is only true for finite order meromorphic functions. We have also obtained the proximity function and pointwise estimates of f(z+η)/f(z) which is a discrete version of the classical logarithmic derivative estimates of f(z). We apply these results to give new growth estimates of meromorphic solutions to higher order linear difference equations. This also allows us to solve an old problem of Whittaker (Interpolatory Function Theory, Cambridge University Press, Cambridge, 1935) concerning a first order difference equation. We show by giving a number of examples that all of our results are best possible in certain senses. Finally, we give a direct proof of a result in Ablowitz, Halburd and Herbst (Nonlinearity 13:889–905, 2000) concerning integrable difference equations. This research was supported in part by the Research Grants Council of the Hong Kong Special Administrative Region, China (HKUST6135/01P). The second author was also partially supported by the National Natural Science Foundation of China (Grant No. 10501044) and the HKUST PDF Matching Fund.     

Confidence Intervals for Discrete Approximations to Ill-Posed Problems

We consider the linear model Y = Xβ + ε that is obtained by discretizing a system of first-kind integral equations describing a set of physical measurements. The n vector β represents the desired quantities, the m x n matrix X represents the instrument response functions, and the m vector Y contains the measurements actually obtained. These measurements are corrupted by random measuring errors ε drawn from a distribution with zero mean vector and known variance matrix. Solution of first-kind integral equations is an ill-posed problem, so the least squares solution for the above model is a highly unstable function of the measurements, and the classical confidence intervals for the solution are too wide to be useful. The solution can often be stabilized by imposing physically motivated nonnegativity constraints. In a previous article (O'Leary and Rust 1986) we developed a method for computing sets of nonnegatively constrained simultaneous confidence intervals. In this article we briefly review the simultaneous intervals and then show how to compute nonnegativity constrained one-at-a-time confidence intervals. The technique gives valid confidence intervals even for problems with m < n. We demonstrate the methods using both an overdetermined and an underdetermined problem obtained by discretizing an equation of Phillips (Phillips 1962).     

Estimates on the weak solution of semilinear hyperbolic systems

The Cauchy problem for a semilinear hyperbolic system of the type

On the unsteady Poiseuille flow in a pipe

We show that in an unsteady Poiseuille flow of a Navier–Stokes fluid in an infinite straight pipe of constant cross-section, σ, the flow rate, F(t), and the axial pressure drop, q(t), are related, at each time t, by a linear Volterra integral equation of the second type, where the kernel depends only upon t and σ. One significant consequence of this result is that it allows us to prove that the inverse parabolic problem of finding a Poiseuille flow corresponding to a given F(t) is equivalent to the resolution of the classical initial-boundary value problem for the heat equation.     

The Operator Factorization Method in Inverse Obstacle Scattering

The standard factorization method from inverse scattering theory allows to reconstruct an obstacle pointwise from the normal far field operator F. The kernel of this method is the study of the first kind Fredholm integral equation (F^* F)^1/4 f = Φ_z with the right-hand part In this paper we extend the factorization method to cover some kinds of boundary conditions which leads to non-normal far field operators. We visualize the scatterer explicitly in terms of the singular system of the selfadjoint positive operator F_# = [(ReF)^* (ReF)]^1/2 + ImF. The following characterization criterium holds: a given point z is inside the obstacle if and only if the function Φ_z belongs to the range of F_#^1/2. Our operator approach provides the tool for treatment of a wide class of inverse elliptic problems.     

Explicit solution of the inverse kinematic problem in the non-Herglotz case

The inverse kinematic problem is solved in the half space R ₊ ^ν+1 ={(x,z)|z?0,x∈R^ν, ν?1 under the assumption that the index of refraction can be represented in the form $$n^2 (x,z) = K^2 (z) + \sum\limits_{j = 1}^\nu {\Phi _j^2 (x_j ),} n_z< 0.$$ . The solution obtained is a generalization of the Herglotz-Wiechert formula. A formula is presented for the solution of the inverse kinematic problem in the general case of separation of variables in the eikonal equation.     

Semiseparable Integral Operators and Explicit Solution of an Inverse Problem for a Skew-Self-Adjoint Dirac-Type System

Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl matrix function is treated in the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl function and a procedure to solve the inverse problem is given. The case of the generalized Weyl functions of the form f(l) exp{-2ilD}{\phi(\lambda)\,{\rm exp}\{-2i{\lambda}D\}}, where f{\phi} is a strictly proper rational matrix function and D = D* ≥ 0 is a diagonal matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable integral operators and recovery of the Dirac-type system are obtained for this case.     

On the Inverse Scattering of the Time-Dependent Schrödinger Equation and the Associated Kadomtsev-Petviashvili (I) Equation

The Kadomtsev-Petviashvili equation, a two-spatial-dimensional analogue of the Korteweg-deVries equation, arises in physical situations in two different forms depending on a certain sign appearing in the evolution equation. Here we investigate one of the two cases. The initial-value problem, associated with initial data decaying sufficiently rapidly at infinity, is linearized by a suitable extension of the inverse scattering transform. Essential is the formulation of a nonlocal Riemann-Hilbert problem in terms of scattering data expressible in closed form in terms of given initial data. The lump solutions, algebraically decaying solitons, are given a definite spectral characterization. Pure lump solutions are obtained by solving a linear algebraic system whose coefficients depend linearly on x, y, t. Many of the above results are also relevant to the problem of inverse scattering for the so-called time-dependent Schrödinger equation.