Linear and nonlinear generalized Fourier transforms

This article presents an overview of a transform method for solving linear and integrable nonlinear partial differential equations. This new transform method, proposed by Fokas, yields a generalization and unification of various fundamental mathematical techniques and, in particular, it yields an extension of the Fourier transform method.

Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test

In this article, we use the no-response test idea, introduced in Luke and Potthast (2003) and Potthast (Preprint) and the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient gamma of the equation del (.) gamma(x)del + c(x) with piecewise regular gamma and bounded function c(x). We use infinitely many Cauchy data as measurement and give a reconstructive method to localize the interface. We will base this multiwave version of the no-response test on two different proofs. The first one contains a pointwise estimate as used by the singular sources method. The second one is built on an energy (or an integral) estimate which is the basis of the probe method. As a conclusion of this, the probe and the singular sources methods are equivalent regarding their convergence and the no-response test can be seen as a unified framework for these methods. As a further contribution, we provide a formula to reconstruct the values of the jump of gamma(x), x is an element of partial derivative D at the boundary. A second consequence of this formula is that the blow-up rate of the indicator functions of the probe and singular sources methods at the interface is given by the order of the singularity of the fundamental solution.

Multi-mode approximations to wave scattering by an uneven bed

Approximations to the scattering of linear surface gravity waves on water of varying quiescent depth are Investigated by means of a variational approach. Previous authors have used wave modes associated with the constant depth case to approximate the velocity potential, leading to a system of coupled differential equations. Here it is shown that a transformation of the dependent variables results in a much simplified differential equation system which in turn leads to a new multi-mode 'mild-slope' approximation. Further, the effect of adding a bed mode is examined and clarified. A systematic analytic method is presented for evaluating inner products that arise and numerical experiments for two-dimensional scattering are used to examine the performance of the new approximations.

Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.

A mathematical model of the in vitro keratinocyte response to chromium and nickel exposure

A mathematical model describing the main mechanistic processes involved in keratinocyte response to chromium and nickel has been developed and compared to experimental in vitro data. Accounting for the interactions between the metal ions and the keratinocytes, the law of mass action was used to generate ordinary differential equations which predict the time evolution and ion concentration dependency of keratinocyte viability, the amount of metal associated with the keratinocytes and the release of cytokines by the keratinocytes. Good agreement between model predictions and existing experimental data of these endpoints was observed, supporting the use of this model to explore physiochemical parameters that influence the toxicological response of keratinocytes to these two metals.

Mathematical model for low density lipoprotein (LDL) endocytosis by hepatocytes

Individuals with elevated levels of plasma low density lipoprotein (LDL) cholesterol (LDL-C) are considered to be at risk of developing coronary heart disease. LDL particles are removed from the blood by a process known as receptor-mediated endocytosis, which occurs mainly in the liver. A series of classical experiments delineated the major steps in the endocytotic process; apolipoprotein B-100 present on LDL particles binds to a specific receptor (LDL receptor, LDL-R) in specialized areas of the cell surface called clathrin-coated pits. The pit comprising the LDL-LDL-R complex is internalized forming a cytoplasmic endosome. Fusion of the endosome with a lysosome leads to degradation of the LDL into its constituent parts (that is, cholesterol, fatty acids, and amino acids), which are released for reuse by the cell, or are excreted. In this paper, we formulate a mathematical model of LDL endocytosis, consisting of a system of ordinary differential equations. We validate our model against existing in vitro experimental data, and we use it to explore differences in system behavior when a single bolus of extracellular LDL is supplied to cells, compared to when a continuous supply of LDL particles is available. Whereas the former situation is common to in vitro experimental systems, the latter better reflects the in vivo situation. We use asymptotic analysis and numerical simulations to study the longtime behavior of model solutions. The implications of model-derived insights for experimental design are discussed.

Mathematical modelling of competitive LDL/VLDL binding and uptake by hepatocytes

Elevated levels of low-density-lipoprotein cholesterol (LDL-C) in the plasma are a well-established risk factor for the development of coronary heart disease. Plasma LDL-C levels are in part determined by the rate at which LDL particles are removed from the bloodstream by hepatic uptake. The uptake of LDL by mammalian liver cells occurs mainly via receptor-mediated endocytosis, a process which entails the binding of these particles to specific receptors in specialised areas of the cell surface, the subsequent internalization of the receptor-lipoprotein complex, and ultimately the degradation and release of the ingested lipoproteins' constituent parts. We formulate a mathematical model to study the binding and internalization (endocytosis) of LDL and VLDL particles by hepatocytes in culture. The system of ordinary differential equations, which includes a cholesterol-dependent pit production term representing feedback regulation of surface receptors in response to intracellular cholesterol levels, is analysed using numerical simulations and steady-state analysis. Our numerical results show good agreement with in vitro experimental data describing LDL uptake by cultured hepatocytes following delivery of a single bolus of lipoprotein. Our model is adapted in order to reflect the in vivo situation, in which lipoproteins are continuously delivered to the hepatocyte. In this case, our model suggests that the competition between the LDL and VLDL particles for binding to the pits on the cell surface affects the intracellular cholesterol concentration. In particular, we predict that when there is continuous delivery of low levels of lipoproteins to the cell surface, more VLDL than LDL occupies the pit, since VLDL are better competitors for receptor binding. VLDL have a cholesterol content comparable to LDL particles; however, due to the larger size of VLDL, one pit-bound VLDL particle blocks binding of several LDLs, and there is a resultant drop in the intracellular cholesterol level. When there is continuous delivery of lipoprotein at high levels to the hepatocytes, VLDL particles still out-compete LDL particles for receptor binding, and consequently more VLDL than LDL particles occupy the pit. Although the maximum intracellular cholesterol level is similar for high and low levels of lipoprotein delivery, the maximum is reached more rapidly when the lipoprotein delivery rates are high. The implications of these results for the design of in vitro experiments is discussed.

Predictive control using feedback linearization based on dynamic neural models

This paper presents a hybrid control strategy integrating dynamic neural networks and feedback linearization into a predictive control scheme. Feedback linearization is an important nonlinear control technique which transforms a nonlinear system into a linear system using nonlinear transformations and a model of the plant. In this work, empirical models based on dynamic neural networks have been employed. Dynamic neural networks are mathematical structures described by differential equations, which can be trained to approximate general nonlinear systems. A case study based on a mixing process is presented.

Planning rigid body motions using elastic curves

This paper tackles the problem of computing smooth, optimal trajectories on the Euclidean group of motions SE(3). The problem is formulated as an optimal control problem where the cost function to be minimized is equal to the integral of the classical curvature squared. This problem is analogous to the elastic problem from differential geometry and thus the resulting rigid body motions will trace elastic curves. An application of the Maximum Principle to this optimal control problem shifts the emphasis to the language of symplectic geometry and to the associated Hamiltonian formalism. This results in a system of first order differential equations that yield coordinate free necessary conditions for optimality for these curves. From these necessary conditions we identify an integrable case and these particular set of curves are solved analytically. These analytic solutions provide interpolating curves between an initial given position and orientation and a desired position and orientation that would be useful in motion planning for systems such as robotic manipulators and autonomous-oriented vehicles.

A similarity solution for a shock reflection problem in isentropic gas dynamics

A one-dimensional shock-reflection test problem in the case of slab, cylindrical, or spherical symmetry is discussed. The differential equations for a similarity solution are derived and solved numerically in conjunction with the Rankie-Hugoniot shock relations.

Similarity solutions for multicomponent flows

A one-dimensional shock-reflection test problem in the case of slab, cylindrical or spherical symmetry is discussed for multi-component flows. The differential equations for a similarity solution are derived and then solved numerically in conjunction with the Rankine-Hugoniot shock relations.