Numerical approximation of the one-dimensional inverse Cauchy–Stefan problem using a method of fundamental solutions

We investigate an application of the method of fundamental solutions (MFS) to the one-dimensional parabolic inverse Cauchy–Stefan problem, where boundary data and the initial condition are to be determined from the Cauchy data prescribed on a given moving interface. In [B.T. Johansson, D. Lesnic, and T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan Problem, Appl. Math Model. 35 (2011), pp. 4367–4378], the inverse Stefan problem was considered, where only the boundary data is to be reconstructed on the fixed boundary. We extend the MFS proposed in Johansson et al. (2011) and show that the initial condition can also be simultaneously recovered, i.e. the MFS is appropriate for the inverse Cauchy-Stefan problem. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be efficiently obtained with small computational cost.

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.

Boundary integral equations for acoustical inverse sound-soft scattering

The shape of a plane acoustical sound-soft obstacle is detected from knowledge of the far field pattern for one time-harmonic incident field. Two methods based on solving a system of integral equations for the incoming wave and the far field pattern are investigated. Properties of the integral operators required in order to apply regularization, i.e. injectivity and denseness of the range, are proved.

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.

Quasi-lossless optical links for broad-band transmission and data processing

We present the first experimental implementation of a recently designed quasi-lossless fiber span with strongly reduced signal power excursion. The resulting fiber waveguide medium can be advantageously used both in lightwave communications and in all-optical nonlinear data processing. © 2005 IEEE.

Performance optimization of ultra-long Raman laser cavities for quasi-lossless transmission links

Performance optimization of ultra-long Raman laser links is studied theoretically and experimentally. We demonstrate that it is possible to reduce the signal power excursion by adjusting FBG reflectivity without compromising pump efficiency. Furthermore, we experimentally demonstrate an OSNR improvement of 4.3 dB in our system after 4000 km transmission by switching from conventional erbium-doped fibre amplifiers to quasi-lossless transmission. © 2007 Elsevier B.V. All rights reserved.

Signal power excursion and pump efficiency in quasi-lossless ultra-long Raman laser links

Performance optimisation of ultra-long Raman laser links is studied theoretically and experimentally. We observe that it is possible to reduce signal power excursion by adjusting FBG reflectivity without compromising pump efficiency.

42.6Gb/s RZ-ASK transmission over 2500km using quasi-lossless transmission spans

Here, for the first time, we experimentally demonstrate optical data transmission through such quasi-lossless fibre spans, which provide an entirely different nonlinear propagation regime to conventionally EDFA/Raman amplified systems. We demonstrate 42.6 Gb/s transmission over 2500 km of SMF-28 with an 82 km span length, and investigate the optimisation of the system in terms of launch power and nonlinear tolerance. Our results show the first ever data transmission experiment using an ultra-long Raman laser to provide quasi-lossless conditions in the transmission fibre. For a span length of 82 km we have shown 2500 km transmission over SMF-28 fibre at 42.6 Gb/s. Our quasi-lossless configuration had 6 dB lower optimum power and improved nonlinear tolerance compared with a conventional EDFA only system.

Optical phase conjugation for dispersion and nonlinearity compensation in a 1600km, 42 Gb/s quasi-lossless system

We experimentally investigate a long-distance, high-bit-rate transmission system which combines optical-phase-conjugation with quasi-lossless amplification. Comparison with a conventional system configuration demonstrates the possibility of obtaining both dispersion compensation and improved nonlinear tolerance using proposed scheme.

Quasi-lossless spans for broadband transmission and data processing

We present the first experimental implementation of a recently designed quasi-lossless fibre span with strongly reduced signal power excursion. The resulting fibre waveguide medium can be advantageously used both in lightwave communications and in all-optical nonlinear data processing.

Quasi-lossless data transmission with ultra-long Raman fibre laser based amplification

The project consists of an experimental and numerical modelling study of the applications of ultra-long Raman fibre laser (URFL) based amplification techniques for high-speed multi-wavelength optical communications systems. The research is focused in telecommunications C-band 40 Gb/s transmission data rates with direct and coherent detection. The optical transmission performance of URFL based systems in terms of optical noise, gain bandwidth and gain flatness for different system configurations is evaluated. Systems with different overall span lengths, transmission fibre types and data modulation formats are investigated. Performance is compared with conventional Erbium doped fibre amplifier based system to evaluate system configurations where URFL based amplification provide performance or commercial advantages.

Conversion of a chirped Gaussian pulse to a soliton or a bound multisoliton state in quasi-lossless and lossy optical fiber spans

The formation of single-soliton or bound-multisoliton states from a single linearly chirped Gaussian pulse in quasi-lossless and lossy fiber spans is examined. The conversion of an input-chirped pulse into soliton states is carried out by virtue of the so-called direct Zakharov-Shabat spectral problem, the solution of which allows one to single out the radiative (dispersive) and soliton constituents of the beam and determine the parameters of the emerging bound state(s). We describe here how the emerging pulse characteristics (the number of bound solitons, the relative soliton power) depend on the input pulse chirp and amplitude. © 2007 Optical Society of America.

Effect of solute segregation on the strength of nanocrystalline alloys:Inverse Hall-Petch relation

We have used a high-energy ball mill to prepare single-phased nanocrystalline Fe, Fe90Ni10, Fe85Al4Si11, Ni99Fe1 and Ni90Fe10 powders. We then increased their grain sizes by annealing. We found that a low-temperature anneal (T < 0.4 Tm) softens the elemental nanocrystalline Fe but hardens both the body-centered cubic iron- and face-centered cubic nickel-based solid solutions, leading in these alloys to an inverse Hall–Petch relationship. We explain this abnormal Hall–Petch effect in terms of solute segregation to the grain boundaries of the nanocrystalline alloys. Our analysis can also explain the inverse Hall–Petch relationship found in previous studies during the thermal anneal of ball-milled nanocrystalline Fe (containing ∼1.5 at.% impurities) and electrodeposited nanocrystalline Ni (containing ∼1.0 at.% impurities).

Optical regeneration using self-phase modulation and quasi-continuous filtering

We propose a scheme for 211 optical regeneration based on self-phase modulation in fiber and quasi-continuous filtering. Numerical simulations demonstrate the possibility of increasing the transmission reach from 3500 to more than 6000 km at 10 Gb/s using 100-km spans. Spectral broadening is shown to be small using this technique, indicating its suitability for wavelength-division-multiplexing regeneration.