On a Nonstationary Model of a Catalytic Process in a Fluidized Bed

In the half-strip 0 ≤ x ≤ h, t ≤ 0 we consider a mixed problem for an almost linear system of three first order PDEs, one of which does not involve derivatives with respect to t. We prove the existence and uniqueness of a generalized Holder continuous solution and generalized piecewise smooth and smooth solutions. For the piecewise smooth solution we prove the stabilization of some functionals as t → ∞.     

The radial basis functions method for identifying an unknown parameter in a parabolic equation with overspecified data

Parabolic partial differential equations with overspecified data play a crucial role in applied mathematics and engineering, as they appear in various engineering models. In this work, the radial basis functions method is used for finding an unknown parameter p(t) in the inverse linear parabolic partial differential equation u_t = u_xx + p(t)u + φ, in [0,1] × (0,T], where u is unknown while the initial condition and boundary conditions are given. Also an additional condition ∫₀¹k(x)u(x,t)dx = E(t), 0 ≤ t ≤ T, for known functions E(t), k(x), is given as the integral overspecification over the spatial domain. The main approach is using the radial basis functions method. In this technique the exact solution is found without any mesh generation on the domain of the problem. We also discuss on the case that the overspecified condition is in the form ∫₀^s(t) u(x,t)dx = E(t), 0 < t ≤ T, 0 < s(t) < 1, where s and E are known functions. Some illustrative examples are presented to show efficiency of the proposed method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Solvability of Mixed Problems for the Klein–Gordon–Fock Equation in the Class <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> for <Emphasis Type="Italic">p</Emphasis> ≥ 1

We prove that the mixed problem for the Klein–Gordon–Fock equation u _tt (x, t) ? u _xx (x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle Q _T = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ L _p [0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class L _p (Q _T ) for p ≥ 1. We construct the solution in explicit analytic form.     

Uniform convergence of wavelet solution to the sideways heat equation

We consider the problem uxx（x, t） = ut（x, t）, 0 ≤ x 〈 1, t ≥ 0, where the Cauchy data g（t） is given at x = 1. This is an ill-posed problem in the sense that a small disturbance on the boundary g（t） can produce a big alteration on its solution （if it exists）. We shall define a wavelet solution to obtain the well-posed approximating problem in the scaling space Vj. In the previous papers, the theoretical results concerning the error estimate are L2-norm and the solutions aren＇t stable at x = 0. However, in practice, the solution is usually required to be stable at the boundary. In this paper we shall give uniform convergence on interval x ∈ [0, 1].     

Inversion of the exponential X-ray transform. II: Numerics

The exponential X-ray transform arises in single photon emission computed tomography and is defined on functions on the plane by ??_μf(φ,x) = ∫f (x + tφ)e^μt where μ is a constant. In [MMAS(10), 561–574, 1988], we derived analytical formulae for filters K corresponding to a general point spread function E that can be used to invert the exponential X-ray transform via a filtered backprojection algorithm. Here, we use those formulae to derive expressions suitable for numerical computation of the filters corresponding to a specific family of bandlimited point spread functions and give the results of reconstructions of a mathematical phantom using these filters. Also included is an analogue of the Shepp–Logan ellipse theorem, [IEEE Trans. Nucl. Sci. (21), 21–43, 1974], for the exponential X-ray transform.     

Directional Derivative Problem for the Telegraph Equation with a Dirac Potential

In the domain Q = [0,∞)×[0,∞) of the variables (x, t), for the telegraph equation with a Dirac potential concentrated at a point (x₀, t₀) ∈ Q, we consider a mixed problem with initial (at t = 0) conditions on the solution and its derivative with respect to t and a condition on the boundary x = 0 which is a linear combination with coefficients depending on t of the solution and its first derivatives with respect to x and t (a directional derivative). We obtain formulas for the classical solution of this problem under certain conditions on the point (x₀, t₀), the coefficient of the Dirac potential, and the conditions of consistency of the initial and boundary data and the right-hand side of the equation at the point (0, 0). We study the behavior of the solution as the direction of the directional derivative in the boundary condition tends to a characteristic of the equation and obtain estimates of the difference between the corresponding solutions.     

On some properties of generalized solutions of the wave equation in the classes L p and W p 1 for p ≥ 1

For each p ≥ 1, in closed analytic form, we establish the existence of a unique generalized solution in L _p of the mixed problem for the wave equation in the rectangle [0 ≤ x ≤ 1] × [0 ≤ t ≤ T] with zero initial conditions and with boundary conditions of the first kind, one of which is homogeneous. Next, we derive necessary conditions for this solution to belong to W _p ¹ . We present examples showing that these necessary conditions are not sufficient for any p ≥ 1.     

On the one-dimensional two-phase inverse Stefan problems

New formulations of the inverse nonstationary Stefan problems are considered: (a) forx [0,1] (the inverse problem IP₁; (b) forx [0, (t)] with a degenerate initial condition (the inverse problem IP). Necessary conditions for the existence and uniqueness of a solution to these problems are formulated. On the first phase {x [0, y(t)]{, the solution of the inverse problem is found in the form of a series; on the second phase {x [y(t), 1] orx [y(t), (t)]{, it is found as a sum of heat double-layer potentials. By representing the inverse problem in the form of two connected boundary-value problems for the heat conduction equation in the domains with moving boundaries, it can be reduced to the integral Volterra equations of the second kind. An exact solution of the problem IP is found for the self similar motion of the boundariesx=y(t) andx=(t).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1058–1065, August, 1993.     

Universal fundamental systems in ordered Banach spaces

Let E be a separable Banach space ordered by a reproducing cone with empty interior. We prove the existence of operator functions A : [0, ) P (P the cone of monotone increasing linear operators) and of initial values x₀ such that the solution of x (t) = A(t) x(t), x(0) = x₀, is dense in E.Received: 27 February 2004     

Estimating the Diffusion Coefficient for Diffusions Driven by fBm

When the Hurst coefficient of a fBm B _t ^H is greater than 1/2, it is possible to define a stochastic integral with respect to B _t ^H as the pathwise limit of Riemann sums. In this article we consider diffusion equations of the type X_t = x₀ + ₀ ^T (X_s) dB_s ^H. We then construct a simple-to-use estimator of the diffusion coefficient (x), based on the number of crossings of level x of the process X _t. We then study consistency in probability of this estimator and calculate convergence rates in probability.     

Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach

The problem of determining the pair w:={F(x,t);T₀(t)} of source terms in the parabolic equation u_t=(k(x)u_x)_x+F(x,t) and Robin boundary condition −k(l)u_x(l,t)=v[u(l,t)−T₀(t)] from the measured final data μ_T(x)=u(x,T) is formulated. It is proved that both components of the Fréchet gradient of the cost functional can be found via the same solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is derived. The obtained results permit one to prove existence of a quasi-solution of the considered inverse problem, as well as to construct a monotone iteration scheme based on a gradient method.     

On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

The zero dispersion limit of the korteweg-de vries equation for initial potentials with non-trivial reflection coefficient

The inverse scattering method is used to determine the distribution limit as ? → 0 of the solution u(x, t, ?) of the initial value problem. U_t ? 6uu_x + ?²u_xxx = 0, u(x, 0) = v(x), where v(x) is a positive bump which decays sufficiently fast as x x→±α. The case v(x) ? 0 has been solved by Peter D. Lax and C. David Levermore [8], [9], [10]. The computation of the distribution limit of u(x, t, ?) as ? → 0 is reduced to a quadratic maximization problem, which is then solved.     

New approach to the numerical solution of forward-backward equations

This paper is concerned with the approximate solution of functional differential equations having the form: x′(t) = αx(t) + βx(t - 1) + γx(t + 1). We search for a solution x, defined for t ∈ [−1, k], k ∈ ℕ, which takes given values on intervals [−1, 0] and (k-1, k]. We introduce and analyse some new computational methods for the solution of this problem. Numerical results are presented and compared with the results obtained by other methods.      

Multipoint Problem for Nonisotropic Partial Differential Equations with Constant Coefficients

We investigate the well-posedness of a problem with multipoint conditions with respect to a chosen variable t and periodic conditions with respect to coordinates x ₁,..., x _p for a nonisotropic (concerning differentiation with respect to t and x ₁,..., x _p) partial differential equation with constant complex coefficients. We establish conditions for the existence and uniqueness of a solution of this problem and prove metric theorems on lower bounds for small denominators appearing in the course of the construction of its solution.     

Using fractional primal–dual to schedule split intervals with demands

We consider the problem of scheduling jobs that are given as groups of non-intersecting intervals on the real line. Each job j is associated with a t-interval (which consists of up to t segments, for some t≥1), a demand, d_j[0,1], and a weight, w(j). A feasible schedule is a collection of jobs such that, for every , the total demand of the jobs in the schedule whose t-interval contains s does not exceed 1. Our goal is to find a feasible schedule that maximizes the total weight of scheduled jobs.We present a 6t-approximation algorithm for this problem that uses a novel extension of the primal–dual schema called fractional primal–dual. The first step in a fractional primal–dual r-approximation algorithm is to compute an optimal solution, x^*, of an LP relaxation of the problem. Next, the algorithm produces an integral primal solution x, and a new LP, denoted by P′, that has the same objective function as the original problem, but contains inequalities that may not be valid with respect to the original problem. Moreover, x^* is a feasible solution of P′. The algorithm also computes a solution y to the dual of P′. The solution x is r-approximate, since its weight is bounded by the value of y divided by r.We present a fractional local ratio interpretation of our 6t-approximation algorithm. We also discuss the connection between fractional primal–dual and the fractional local ratio technique. Specifically, we show that the former is the primal–dual manifestation of the latter.     

Existence results for non-autonomous multiple-fragmentation models

We investigate an initial-value problem modelling fragmentation processes where particles split into two or more pieces at a rate, γ, that not only depends on the sizes of the particles involved but also on time. The existence of non-negative, mass-conserving solutions is established by considering a truncated version of an associated non-autonomous abstract Cauchy problem. The latter has solutions of the form u(t)=U_n(t,t₀)f, t⩾t₀, where f is the known data at some fixed time t₀⩾0 and {U_n(t,s)} is a uniformly continuous evolution system. A limit evolution system {U(t,s)} is shown to exist. Depending on the form of the known data f at time t₀, the scalar-valued function u, obtained from the limit evolution system via u(x, t)=[U(t, t₀)f](x) for a.e. x>0, t⩾t₀, is a solution of either the original initial-value problem or an integral version of this problem. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Smooth solutions to some differential-difference equations of neutral type

The paper is devoted to the scalar linear differential-difference equation of neutral type

A Strongly Semismooth Integral Function and Its Application

As shown by an example, the integral function f : ⁿ , defined by f(x) = _a ^b[B(x, t)]₊ g(t) dt, may not be a strongly semismooth function, even if g(t) 1 and B is a quadratic polynomial with respect to t and infinitely many times smooth with respect to x. We show that f is a strongly semismooth function if g is continuous and B is affine with respect to t and strongly semismooth with respect to x, i.e., B(x, t) = u(x)t + v(x), where u and v are two strongly semismooth functions in ⁿ . We also show that f is not a piecewise smooth function if u and v are two linearly independent linear functions, g is continuous and g 0 in [a, b], and n 2. We apply the first result to the edge convex minimum norm network interpolation problem, which is a two-dimensional interpolation problem.     

Connected subgraphs with small degree sums in 3‐connected planar graphs

It is well‐known that every planar graph has a vertex of degree at most five. Kotzig proved that every 3‐connected planar graph has an edge xy such that deg(x) + deg (y) ≤ 13. In this article, considering a similar problem for the case of three or more vertices that induce a connected subgraph, we show that, for a given positive integer t, every 3‐connected planar graph G with |V(G)| ≥ t has a connected subgraph H of order t such that Σ_x∈V(H) deg_G(x) ≤ 8t − 1. As a tool for proving this result, we consider decompositions of 3‐connected planar graphs into connected subgraphs of order at least t and at most 2t − 1. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 191–203, 1999