We study the numerical solution to inverse problems in which one reconstructs the coefficients of a parabolic equation depending only on one (space or time) variable. In particular, these classes of problems arise in ...
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We study the numerical solution to inverse problems in which one reconstructs the coefficients of a parabolic equation depending only on one (space or time) variable. In particular, these classes of problems arise in the study of boundaryvalueproblems with nonlocal (integral) boundary conditions. We suggest an approach to the numerical solution to these problems with the use of the line method and the reduction of the original problem to the solution to auxiliary Cauchy problems for systems of ordinary differential equations. We present the results of numerical experiments with test problems.
We first prove two theorems on the low(2) degrees and the join property in the local structure D(<= 0'): An r.e. degree is low(2) if and only if it is bounded by an r.e. degree without the join property (in D(&...
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We first prove two theorems on the low(2) degrees and the join property in the local structure D(<= 0'): An r.e. degree is low(2) if and only if it is bounded by an r.e. degree without the join property (in D(<= 0')), and an FPF Delta(0)(2) degree is low(2) if and only if it fails to have the join property. We also study the join property in the global structure and show that for every array recursive degree, there is a degree above it which fails to satisfy the join property.
Obstacles K and L in R-d (d >= 2) are considered that are finite disjoint unions of strictly convex domains with C-3 boundaries. We show that if K and L have (almost) the same scattering length spectrum, or (almost...
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Obstacles K and L in R-d (d >= 2) are considered that are finite disjoint unions of strictly convex domains with C-3 boundaries. We show that if K and L have (almost) the same scattering length spectrum, or (almost) the same travelling times, then K = L.
We study the uniqueness of the solution of a mixed boundaryvalue problem for the biharmonic equation in the exterior of a compact set under the assumption that the generalized solution of that problem has a finite Di...
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We study the uniqueness of the solution of a mixed boundaryvalue problem for the biharmonic equation in the exterior of a compact set under the assumption that the generalized solution of that problem has a finite Dirichlet integral with weight |x| (a) . Depending on the value of the parameter a, we prove uniqueness theorems and obtain exact formulas for the dimension of the solution space of the mixed boundaryvalue problem.
We present and study a novel numerical algorithm to approximate the action of T-beta := L-beta where L is a symmetric and positive definite unbounded operator on a Hilbert space H-0. The numerical method is based on a...
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We present and study a novel numerical algorithm to approximate the action of T-beta := L-beta where L is a symmetric and positive definite unbounded operator on a Hilbert space H-0. The numerical method is based on a representation formula for T-beta in terms of Bochner integrals involving (I + t(2)L)(-1) for t is an element of (0, infinity). To develop an approximation to T-beta, we introduce a finite element approximation L-h to L and base our approximation to T-beta on T-h(beta) := L-h(-beta). The direct evaluation of T-h(beta) is extremely expensive as it involves expansion in the basis of eigenfunctions for L-h. The above mentioned representation formula holds for T-h(beta) and we propose three quadrature approximations denoted generically by Q(h)(beta). The two results of this paper bound the errors in the H-0 inner product of T-beta - T-h(beta) pi(h) ph and T-h(beta) - Q(h)(beta) where pi(h) is the H-0 orthogonal projection into the finite element space. We note that the evaluation of Q(h)(beta) involves application of (I + (t(i))(2)Lh)(-1) with t(i) being either a quadrature point or its inverse. Efficient solution algorithms for these problems are available and the problems at different quadrature points can be straightforwardly solved in parallel. numerical experiments illustrating the theoretical estimates are provided for both the quadrature error T-h(beta) - Q(h)(beta) and the finite element error T-beta - T-h(beta) pi(h).
We consider finite-difference schemes for the heat equation with variable coefficients and nonlocal boundary conditions containing real parameters alpha, beta, and gamma. We obtain a priori estimates for the solution ...
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We consider finite-difference schemes for the heat equation with variable coefficients and nonlocal boundary conditions containing real parameters alpha, beta, and gamma. We obtain a priori estimates for the solution of the difference problem, which imply the stability and convergence of the constructed difference schemes.
For any p >= 1 and any time interval [0, T], we derive a closed-form expression for a generalized solution in W (p) (1) of a mixed problem describing the radially symmetric vibrations of a three-dimensional ball wi...
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For any p >= 1 and any time interval [0, T], we derive a closed-form expression for a generalized solution in W (p) (1) of a mixed problem describing the radially symmetric vibrations of a three-dimensional ball with arbitrary initial conditions and an arbitrary boundary control on the ball surface by a boundary condition of the first kind.
We obtain necessary and sufficient conditions for the unique solvability of the periodic boundaryvalue problem for a family of nth-order linear functional differential equations with pointwise constraints on the func...
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We obtain necessary and sufficient conditions for the unique solvability of the periodic boundaryvalue problem for a family of nth-order linear functional differential equations with pointwise constraints on the functional operators.
The article discusses the solutions of boundary-valueproblems for sets of differential equations using block elements and analytical form. Topics discussed include the general boundary-value problem described by the ...
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The article discusses the solutions of boundary-valueproblems for sets of differential equations using block elements and analytical form. Topics discussed include the general boundary-value problem described by the set of partial differential equations, the exact solution of boundary-value problem based on the block-element method, and the accomplishment of quotient topology to construct the exact solution for the boundary-value problem through the implementation of the boundaries of layers.
We investigate the existence of solutions for a nonlinear fractional q-difference integral equation (q-variant of the Langevin equation) with two different fractional orders and nonlocal four-point boundary conditions...
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We investigate the existence of solutions for a nonlinear fractional q-difference integral equation (q-variant of the Langevin equation) with two different fractional orders and nonlocal four-point boundary conditions. Our results are based on some classical fixed point theorems. An illustrative example is also presented. (C) 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
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