Properties of a method of fundamental solutions for the parabolic heat equation

We show that a set of fundamental solutions to the parabolic heat equation, with each element in the set corresponding to a point source located on a given surface with the number of source points being dense on this surface, constitute a linearly independent and dense set with respect to the standard inner product of square integrable functions, both on lateral- and time-boundaries. This result leads naturally to a method of numerically approximating solutions to the parabolic heat equation denoted a method of fundamental solutions (MFS). A discussion around convergence of such an approximation is included.

An iterative method for a Cauchy problem for the heat equation

An iterative method for reconstruction of the solution to a parabolic initial boundary value problem of second order from Cauchy data is presented. The data are given on a part of the boundary. At each iteration step, a series of well-posed mixed boundary value problems are solved for the parabolic operator and its adjoint. The convergence proof of this method in a weighted L2-space is included.

Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

A numerical method for the Dirichlet initial boundary value problem for the heat equation in the exterior and unbounded region of a smooth closed simply connected 3-dimensional domain is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and an integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the parabolic problem to a sequence of stationary elliptic problems which are solved by a boundary layer approach giving a sequence of boundary integral equations of the first kind to solve. Under the assumption that the boundary surface of the solution domain has a one-to-one mapping onto the unit sphere, these integral equations are transformed and rewritten over this sphere. The numerical discretisation and solution are obtained by a discrete projection method involving spherical harmonic functions. Numerical results are included.

Time trials on second-order and variable-learning-rate algorithms

The performance of seven minimization algorithms are compared on five neural network problems. These include a variable-step-size algorithm, conjugate gradient, and several methods with explicit analytic or numerical approximations to the Hessian.

Discrimination between strain and temperature effects using first and second-order diffraction from a long period grating

We study the effects of temperature and strain on the spectra of the first and second-order diffraction attenuation bands of a single long-period grating (LPG) in step-index fibre. The primary and second-order attenuation bands had comparable strength with the second-order bands appearing in the visible and near-infra red parts of the spectrum. Using first and second-order diffraction to the eighth cladding mode a sensitivity matrix was obtained with limiting accuracy given by cross-sensitivity of ~1.19% of the measurement. The sensing scheme presented as a limiting temperature and strain resolution of ±0.7 °C and ~±25 µ.

What is second-order vision for? Discriminating illumination versus material changes

The human visual system is sensitive to second-order modulations of the local contrast (CM) or amplitude (AM) of a carrier signal. Second-order cues are detected independently of first-order luminance signals; however, it is not clear why vision should benet from second-order sensitivity. Analysis of the first-and second-order contents of natural images suggests that these cues tend to occur together, but their phase relationship varies. We have shown that in-phase combinations of LM and AM are perceived as a shaded corrugated surface whereas the anti-phase combination can be seen as corrugated when presented alone or as a flat material change when presented in a plaid containing the in-phase cue. We now extend these findings using new stimulus types and a novel haptic matching task. We also introduce a computational model based on initially separate first-and second-order channels that are combined within orientation and subsequently across orientation to produce a shading signal. Contrast gain control allows the LM + AM cue to suppress responses to the LM-AM when presented in a plaid. Thus, the model sees LM -AM as flat in these circumstances. We conclude that second-order vision plays a key role in disambiguating the origin of luminance changes within an image. © ARVO.

Simultaneous measurement of strain and temperature using the first- and second-order diffraction wavelengths of Bragg gratings

A method of discriminating between temperature and strain effects in fibre sensing using a conventionally written, in-fibre Bragg grating is presented. The technique uses wavelength information from the first and second diffraction orders of the grating element to determine the wavelength dependent strain and temperature coefficients, from which independent temperature and strain measurements can be made. The authors present results that validate this matrix inversion technique and quantify the strain and temperature errors which can arise for a given uncertainty in the measurement of the reflected wavelength.

A procedure for determining a spacewise dependent heat source and the initial temperature

The inverse problem of determining a spacewise dependent heat source, together with the initial temperature for the parabolic heat equation, using the usual conditions of the direct problem and information from two supplementary temperature measurements at different instants of time is studied. These spacewise dependent temperature measurements ensure that this inverse problem has a unique solution, despite the solution being unstable, hence the problem is ill-posed. We propose an iterative algorithm for the stable reconstruction of both the initial data and the source based on a sequence of well-posed direct problems for the parabolic heat equation, which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for a typical benchmark test example, which has the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure gives accurate numerical approximations in relatively few iterations.

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.

Solvability and asymptotics of the heat equation with mixed variable lateral conditions and applications in the opening of the exocytotic fusion pore in cells

We investigate a mixed problem with variable lateral conditions for the heat equation that arises in modelling exocytosis, i.e. the opening of a cell boundary in specific biological species for the release of certain molecules to the exterior of the cell. The Dirichlet condition is imposed on a surface patch of the boundary and this patch is occupying a larger part of the boundary as time increases modelling where the cell is opening (the fusion pore), and on the remaining part, a zero Neumann condition is imposed (no molecules can cross this boundary). Uniform concentration is assumed at the initial time. We introduce a weak formulation of this problem and show that there is a unique weak solution. Moreover, we give an asymptotic expansion for the behaviour of the solution near the opening point and for small values in time. We also give an integral equation for the numerical construction of the leading term in this expansion.

Quasi-lossless transmission using second-order Raman amplification and fibre Bragg gratings

A novel distributed amplification scheme for quasi-lossless transmission is presented. The system is studied numerically and shown to be able to strongly reduce signal power variations in comparison with currently employed schemes of similar complexity. As an example, variations of less than 3.1 dB for 100 km distance between pumps and below 0.42 dB for 60 km are obtained when using standard single-mode fibre as the transmission medium with an input signal average power of 0 dBm, and a total pump power of about 1.7 W. © 2004 Optical Society of America.

Simultaneous measurement of strain and temperature using the first- and second-order diffraction wavelengths of Bragg gratings

A method of discriminating between temperature and strain effects in fibre sensing using a conventionally written, in-fibre Bragg grating is presented. The technique uses wavelength information from the first and second diffraction orders of the grating element to determine the wavelength dependent strain and temperature coefficients, from which independent temperature and strain measurements can be made. The authors present results that validate this matrix inversion technique and quantify the strain and temperature errors which can arise for a given uncertainty in the measurement of the reflected wavelength.

Discrimination between strain and temperature effects using first and second-order diffraction from a long-period grating

We study the effects of temperature and strain on the spectra of the first and second-order diffraction attenuation bands of a single long-period grating (LPG) in step-index fibre. The primary and second-order attenuation bands had comparable strength with the second-order bands appearing in the visible and near-infra red parts of the spectrum. Using first and second-order diffraction to the eighth cladding mode a sensitivity matrix was obtained with limiting accuracy given by cross-sensitivity of ∼ 1.19% of the measurement. The sensing scheme presented as a limiting temperature and strain resolution of ± 0.7 °C and ∼ ± 25 με. © 2002 Elsevier Science B.V. All rights reserved.

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 variational method for identifying a spacewise-dependent heat source

The inverse problem of determining a spacewise-dependent heat source for the parabolic heat equation using the usual conditions of the direct problem and information from one supplementary temperature measurement at a given instant of time is studied. This spacewise-dependent temperature measurement ensures that this inverse problem has a unique solution, but the solution is unstable and hence the problem is ill-posed. We propose a variational conjugate gradient-type iterative algorithm for the stable reconstruction of the heat source based on a sequence of well-posed direct problems for the parabolic heat equation which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterative procedure at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented which have the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure yields stable and accurate numerical approximations after only a few iterations.