Determining the temperature from Cauchy data in corner domains

We consider the problem of reconstruction of the temperature from knowledge of the temperature and heat flux on a part of the boundary of a bounded planar domain containing corner points. An iterative method is proposed involving the solution of mixed boundary value problems for the heat equation (with time-dependent conductivity). These mixed problems are shown to be well-posed in a weighted Sobolev space.

A procedure for the reconstruction of a stochastic stationary temperature field

An iterative procedure is proposed for the reconstruction of a temperature field from a linear stationary heat equation with stochastic coefficients, and stochastic Cauchy data given on a part of the boundary of a bounded domain. In each step, a series of mixed well-posed boundary-value problems are solved for the stochastic heat operator and its adjoint. Well-posedness of these problems is shown to hold and convergence in the mean of the procedure is proved. A discretized version of this procedure, based on a Monte Carlo Galerkin finite-element method, suitable for numerical implementation is discussed. It is demonstrated that the solution to the discretized problem converges to the continuous as the mesh size tends to zero.

Quasi-separation of the biharmonic partial differential equation

In this paper, we consider analytical and numerical solutions to the Dirichlet boundary-value problem for the biharmonic partial differential equation on a disc of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well known that the solution method of separation in polar coordinates is not possible, in general. However, in this paper, for circular domains in the plane, it is shown that a method, here called quasi-separation of variables, does lead to solutions of the partial differential equation. These solutions are products of solutions of two ordinary linear differential equations: a fourth-order radial equation and a second-order angular differential equation. To be expected, without complete separation of the polar variables, there is some restriction on the range of these solutions in comparison with the corresponding separated solutions of the second-order harmonic differential equation in the plane. Notwithstanding these restrictions, the quasi-separation method leads to solutions of the Dirichlet boundary-value problem on a disc with centre at the origin, with boundary conditions determined by the solution and its inward drawn normal taking the value 0 on the edge of the disc. One significant feature for these biharmonic boundary-value problems, in general, follows from the form of the biharmonic differential expression when represented in polar coordinates. In this form, the differential expression has a singularity at the origin, in the radial variable. This singularity translates to a singularity at the origin of the fourth-order radial separated equation; this singularity necessitates the application of a third boundary condition in order to determine a self-adjoint solution to the Dirichlet boundary-value problem. The penultimate section of the paper reports on numerical solutions to the Dirichlet boundary-value problem; these results are also presented graphically. Two specific cases are studied in detail and numerical values of the eigenvalues are compared with the results obtained in earlier studies.

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions

We consider a Cauchy problem for the Laplace equation in a two-dimensional semi-infinite region with a bounded inclusion, i.e. the region is the intersection between a half-plane and the exterior of a bounded closed curve contained in the half-plane. The Cauchy data are given on the unbounded part of the boundary of the region and the aim is to construct the solution on the boundary of the inclusion. In 1989, Kozlov and Maz'ya [10] proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems in bounded domains. We extend their approach to our setting and in each iteration step mixed boundary value problems for the Laplace equation in the semi-infinite region are solved. Well-posedness of these mixed problems are investigated and convergence of the alternating procedure is examined. For the numerical implementation an efficient boundary integral equation method is proposed, based on the indirect variant of the boundary integral equation approach. The mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing the feasibility of the proposed method.

Stereoacuity as an indicator of prism adaptation

Purpose: The purpose of this study was to determine whether stereoacuity can be used as an indicator of prism adaptation. In particular, we wanted to know whether the time required for stereoacuity to return to the initial level after viewing through a prism can be used to determine the degree of adaptation. Materials and Methods: Eighteen subjects participated in this study. Stereoacuity and dissociated phoria were determined using the TNO stereotest and the Maddox rod, respectively. Prism vergences were measured using a prism bar. For each participant, prism power equivalent to the blur point of base-in (BI) and base-out (BO) fusional vergence at 40 cm was divided and placed in front of both eyes. At 0, 3, 6, 9 and 12 min after prism introduction, the stereoacuity was measured, and at 0 and 12 min, the heterophoria was measured. Results: The repeated measures ANOVA showed a significant difference between the mean stereoacuity for BI and BO prisms at the different measurement times (p < 0.05). For BO prism, the initial value was different between 0 and 3 min after the prism introduction, whereas for BI prism, a difference in stereoacuity was found between the pre-prism value and the value at 0, 3 and 6 min. The size of the heterophoria with BO and BI prisms was different from 0 to 12 min (p < 0.05). Conclusion: The time required for stereoacuity to return to baseline level was more than 3 min for BO, and more than 6 min for BI prism. In addition, the time required to return to baseline values was not similar for the stereoacuity and heterophoria. The recovery of stereoacuity is slower when adapting to divergence, as when looking from near to far. This implies that stereopsis responds faster to near targets than to distant one, and may precede complete phoria adaptation. © 2014 Informa Healthcare USA, Inc. All rights reserved: reproduction in whole or part not permitted.

An iterative procedure for solving a Cauchy problem for second order elliptic equations

An iterative method for reconstruction of solutions to second order elliptic equations by Cauchy data given on a part of the boundary, is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the elliptic operator and its adjoint. The convergence proof of this method in a weighted L2 space is included. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A procedure for the temperature reconstruction in corner domains from Cauchy data

An iterative procedure is proposed for the reconstruction of a stationary temperature field from Cauchy data given on a part of the boundary of a bounded plane domain where the boundary is smooth except for a finite number of corner points. In each step, a series of mixed well-posed boundary value problems are solved for the heat operator and its adjoint. Convergence is proved in a weighted L2-space. Numerical results are included which show that the procedure gives accurate and stable approximations in relatively few iterations.

A variational conjugate gradient method for determining the fluid velocity of a slow viscous flow

The problem considered is that of determining the fluid velocity for linear hydrostatics Stokes flow of slow viscous fluids from measured velocity and fluid stress force on a part of the boundary of a bounded domain. A variational conjugate gradient iterative procedure is proposed based on solving a series of mixed well-posed boundary value problems for the Stokes operator and its adjoint. In order to stabilize the Cauchy problem, the iterations are ceased according to an optimal order discrepancy principle stopping criterion. Numerical results obtained using the boundary element method confirm that the procedure produces a convergent and stable numerical solution.

An iterative method for reconstruction of temperature

An iterative method for the reconstruction of a stationary three-dimensional temperature field, from Cauchy data given on a part of the boundary, is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the heat operator and its adjoint. A convergence proof of this method in a weighted L 2-space is include

New insights in the deactivation of sulfonic modified SBA-15 catalysts for biodiesel production from low-grade oleaginous feedstock

Arenesulfonic-acid functionalized SBA-15 materials have been used in the production of biodiesel from low grade oleaginous feedstock. These materials display an outstanding catalytic activity, being able to promote the transformation of crude palm oil with methanol into fatty acid methyl esters with high yield (85%) under mild reaction conditions. However, high sensitivity of the catalyst against poisoning by different substances has also been detected. Thus, alkaline metal cations, such as sodium or potassium exert a negative influence on the catalytic activity of these materials, being necessary amounts around 500 ppm of sodium in the reaction media to decrease the catalytic activity of these materials to a half of its initial value in just two reaction runs. The deactivation of arenesulfonic acid functionalized SBA-15 materials seems to occur in this case by ion exchange of the acid protons at the sulfonic groups. Organic unsaponifiable compounds like lecithin or retinol also induce a negative influence in the catalytic activity of these sulfonic acid-based materials, though not so intense as in the case of alkaline metals. The deactivating mechanism associated to the influence of the organic compounds seems to be linked to the adsorption of such substances onto the catalytic acid sites as well as on the silica surface. The accumulation of lecithin in the surface of catalyst, observed by means of thermogravimetric analysis, suggest the creation of a strong interaction, probably by ion pair, between this compound and the sulfonic acid group.

On the Stabilization of the Wave Equation by the Boundary

* Partially supported by CNPq (Brazil)

Inhomogeneous Fractional Diffusion Equations

2000 Mathematics Subject Classification: Primary 26A33; Secondary 35S10, 86A05

Operational Calculi for the Euler Operator

2000 Mathematics Subject Classification: 44A40, 44A35

Some Fractional Extensions of the Temperature Field Problem in Oil Strata

This survey is devoted to some fractional extensions of the incomplete lumped formulation, the lumped formulation and the formulation of Lauwerier of the temperature field problem in oil strata. The method of integral transforms is used to solve the corresponding boundary value problems for the fractional heat equation. By using Caputo’s differintegration operator and the Laplace transform, new integral forms of the solutions are obtained. In each of the different cases the integrands are expressed in terms of a convolution of two special functions of Wright’s type.

Impulsive Fractional Differential Inclusions Involving the Caputo Fractional Derivative

Mathematics Subject Classification: 26A33, 34A37.