Numerical solution of an optimal control model of dendritic cell treatment of a growing tumour

A new optimal control model of the interactions between a growing tumour and the host immune system along with an immunotherapy treatment strategy is presented. The model is based on an ordinary differential equation model of interactions between the growing tu- mour and the natural killer, cytotoxic T lymphocyte and dendritic cells of the host immune system, extended through the addition of a control function representing the application of a dendritic cell treat- ment to the system. The numerical solution of this model, obtained from a multi species Runge–Kutta forward-backward sweep scheme, is described. We investigate the effects of varying the maximum al- lowed amount of dendritic cell vaccine administered to the system and find that control of the tumour cell population is best effected via a high initial vaccine level, followed by reduced treatment and finally cessation of treatment. We also found that increasing the strength of the dendritic cell vaccine causes an increase in the number of natural killer cells and lymphocytes, which in turn reduces the growth of the tumour.

A RBF meshless approach for modeling a fractal mobile/immobile transport model

A fractional differential equation is used to describe a fractal model of mobile/immobile transport with a power law memory function. This equation is the limiting equation that governs continuous time random walks with heavy tailed random waiting times. In this paper, we firstly propose a finite difference method to discretize the time variable and obtain a semi-discrete scheme. Then we discuss its stability and convergence. Secondly we consider a meshless method based on radial basis functions (RBFs) to discretize the space variable. In contrast to conventional FDM and FEM, the meshless method is demonstrated to have distinct advantages: calculations can be performed independent of a mesh, it is more accurate and it can be used to solve complex problems. Finally the convergence order is verified from a numerical example which is presented to describe a fractal model of mobile/immobile transport process with different problem domains. The numerical results indicate that the present meshless approach is very effective for modeling and simulating fractional differential equations, and it has good potential in the development of a robust simulation tool for problems in engineering and science that are governed by various types of fractional differential equations.

A finite volume method with linearisation in time for the solution of advection-reaction-diffusion systems

The numerical solution in one space dimension of advection--reaction--diffusion systems with nonlinear source terms may invoke a high computational cost when the presently available methods are used. Numerous examples of finite volume schemes with high order spatial discretisations together with various techniques for the approximation of the advection term can be found in the literature. Almost all such techniques result in a nonlinear system of equations as a consequence of the finite volume discretisation especially when there are nonlinear source terms in the associated partial differential equation models. This work introduces a new technique that avoids having such nonlinear systems of equations generated by the spatial discretisation process when nonlinear source terms in the model equations can be expanded in positive powers of the dependent function of interest. The basis of this method is a new linearisation technique for the temporal integration of the nonlinear source terms as a supplementation of a more typical finite volume method. The resulting linear system of equations is shown to be both accurate and significantly faster than methods that necessitate the use of solvers for nonlinear system of equations.

Hybrid frequency–time domain models for dynamic response analysis of marine structures

Time-domain models of marine structures based on frequency domain data are usually built upon the Cummins equation. This type of model is a vector integro-differential equation which involves convolution terms. These convolution terms are not convenient for analysis and design of motion control systems. In addition, these models are not efficient with respect to simulation time, and ease of implementation in standard simulation packages. For these reasons, different methods have been proposed in the literature as approximate alternative representations of the convolutions. Because the convolution is a linear operation, different approaches can be followed to obtain an approximately equivalent linear system in the form of either transfer function or state-space models. This process involves the use of system identification, and several options are available depending on how the identification problem is posed. This raises the question whether one method is better than the others. This paper therefore has three objectives. The first objective is to revisit some of the methods for replacing the convolutions, which have been reported in different areas of analysis of marine systems: hydrodynamics, wave energy conversion, and motion control systems. The second objective is to compare the different methods in terms of complexity and performance. For this purpose, a model for the response in the vertical plane of a modern containership is considered. The third objective is to describe the implementation of the resulting model in the standard simulation environment Matlab/Simulink.

A numerical method for the fractional Fitzhugh–Nagumo monodomain model

A fractional FitzHugh–Nagumo monodomain model with zero Dirichlet boundary conditions is presented, generalising the standard monodomain model that describes the propagation of the electrical potential in heterogeneous cardiac tissue. The model consists of a coupled fractional Riesz space nonlinear reaction-diffusion model and a system of ordinary differential equations, describing the ionic fluxes as a function of the membrane potential. We solve this model by decoupling the space-fractional partial differential equation and the system of ordinary differential equations at each time step. Thus, this means treating the fractional Riesz space nonlinear reaction-diffusion model as if the nonlinear source term is only locally Lipschitz. The fractional Riesz space nonlinear reaction-diffusion model is solved using an implicit numerical method with the shifted Grunwald–Letnikov approximation, and the stability and convergence are discussed in detail in the context of the local Lipschitz property. Some numerical examples are given to show the consistency of our computational approach.

A semi-alternating direction method for a 2-D fractional FitzHugh–Nagumo monodomain model on an approximate irregular domain

A FitzHugh-Nagumo monodomain model has been used to describe the propagation of the electrical potential in heterogeneous cardiac tissue. In this paper, we consider a two-dimensional fractional FitzHugh-Nagumo monodomain model on an irregular domain. The model consists of a coupled Riesz space fractional nonlinear reaction-diffusion model and an ordinary differential equation, describing the ionic fluxes as a function of the membrane potential. Secondly, we use a decoupling technique and focus on solving the Riesz space fractional nonlinear reaction-diffusion model. A novel spatially second-order accurate semi-implicit alternating direction method (SIADM) for this model on an approximate irregular domain is proposed. Thirdly, stability and convergence of the SIADM are proved. Finally, some numerical examples are given to support our theoretical analysis and these numerical techniques are employed to simulate a two-dimensional fractional Fitzhugh-Nagumo model on both an approximate circular and an approximate irregular domain.

High order unconditionally stable difference schemes for the Riesz space-fractional telegraph equation

In this paper, a class of unconditionally stable difference schemes based on the Pad´e approximation is presented for the Riesz space-fractional telegraph equation. Firstly, we introduce a new variable to transform the original dfferential equation to an equivalent differential equation system. Then, we apply a second order fractional central difference scheme to discretise the Riesz space-fractional operator. Finally, we use (1, 1), (2, 2) and (3, 3) Pad´e approximations to give a fully discrete difference scheme for the resulting linear system of ordinary differential equations. Matrix analysis is used to show the unconditional stability of the proposed algorithms. Two examples with known exact solutions are chosen to assess the proposed difference schemes. Numerical results demonstrate that these schemes provide accurate and efficient methods for solving a space-fractional hyperbolic equation.

Maximum principle and numerical method for the multi-term time–space Riesz–Caputo fractional differential equations

The maximum principle for the space and time–space fractional partial differential equations is still an open problem. In this paper, we consider a multi-term time–space Riesz–Caputo fractional differential equations over an open bounded domain. A maximum principle for the equation is proved. The uniqueness and continuous dependence of the solution are derived. Using a fractional predictor–corrector method combining the L1 and L2 discrete schemes, we present a numerical method for the specified equation. Two examples are given to illustrate the obtained results.

Quantum-Ohmic Resistance Fluctuation In Disordered Conductors - An Invariant Imbedding Approach

It is now well known that in extreme quantum limit, dominated by the elastic impurity scattering and the concomitant quantum interference, the zero-temperature d.c. resistance of a strictly one-dimensional disordered system is non-additive and non-self-averaging. While these statistical fluctuations may persist in the case of a physically thin wire, they are implicitly and questionably ignored in higher dimensions. In this work, we have re-examined this question. Following an invariant imbedding formulation, we first derive a stochastic differential equation for the complex amplitude reflection coefficient and hence obtain a Fokker-Planck equation for the full probability distribution of resistance for a one-dimensional continuum with a Gaussian white-noise random potential. We then employ the Migdal-Kadanoff type bond moving procedure and derive the d-dimensional generalization of the above probability distribution, or rather the associated cumulant function –‘the free energy’. For d=3, our analysis shows that the dispersion dominates the mobilitly edge phenomena in that (i) a one-parameter B-function depending on the mean conductance only does not exist, (ii) an approximate treatment gives a diffusion-correction involving the second cumulant. It is, however, not clear whether the fluctuations can render the transition at the mobility edge ‘first-order’. We also report some analytical results for the case of the one dimensional system in the presence of a finite electric fiekl. We find a cross-over from the exponential to the power-low length dependence of resistance as the field increases from zero. Also, the distribution of resistance saturates asymptotically to a poissonian form. Most of our analytical results are supported by the recent numerical simulation work reported by some authors.

Survival probability for a diffusive process on a growing domain

We consider the motion of a diffusive population on a growing domain, 0 < x < L(t ), which is motivated by various applications in developmental biology. Individuals in the diffusing population, which could represent molecules or cells in a developmental scenario, undergo two different kinds of motion: (i) undirected movement, characterized by a diffusion coefficient, D, and (ii) directed movement, associated with the underlying domain growth. For a general class of problems with a reflecting boundary at x = 0, and an absorbing boundary at x = L(t ), we provide an exact solution to the partial differential equation describing the evolution of the population density function, C(x,t ). Using this solution, we derive an exact expression for the survival probability, S(t ), and an accurate approximation for the long-time limit, S = limt→∞ S(t ). Unlike traditional analyses on a nongrowing domain, where S ≡ 0, we show that domain growth leads to a very different situation where S can be positive. The theoretical tools developed and validated in this study allow us to distinguish between situations where the diffusive population reaches the moving boundary at x = L(t ) from other situations where the diffusive population never reaches the moving boundary at x = L(t ). Making this distinction is relevant to certain applications in developmental biology, such as the development of the enteric nervous system (ENS). All theoretical predictions are verified by implementing a discrete stochastic model.

Distribution of the angle of rotation plane random walks

We study the probability distribution of the angle by which the tangent to the trajectory rotates in the course of a plane random walk. It is shown that the determination of this distribution function can be reduced to an integral equation, which can be rigorously transformed into a differential equation of Hill's type. We derive the asymptotic distribution for very long walks.

The non-linear response of a gyrostabilized platform: Effects of correction torque time delay and of random inputs

First, the non-linear response of a gyrostabilized platform to a small constant input torque is analyzed in respect to the effect of the time delay (inherent or deliberately introduced) in the correction torque supplied by the servomotor, which itself may be non-linear to a certain extent. The equation of motion of the platform system is a third order nonlinear non-homogeneous differential equation. An approximate analytical method of solution of this equation is utilized. The value of the delay at which the platform response becomes unstable has been calculated by using this approximate analytical method. The procedure is illustrated by means of a numerical example. Second, the non-linear response of the platform to a random input has been obtained. The effects of several types of non-linearity on reducing the level of the mean square response have been investigated, by applying the technique of equivalent linearization and solving the resulting integral equations by using laguerre or Gaussian integration techniques. The mean square responses to white noise and band limited white noise, for various values of the non-linear parameter and for different types of non-linearity function, have been obtained. For positive values of the non-linear parameter the levels of the non-linear mean square responses to both white noise and band-limited white noise are low as compared to the linear mean square response. For negative values of the non-linear parameter the level of the non-linear mean square response at first increases slowly with increasing values of the non-linear parameter and then suddenly jumps to a high level, at a certain value of the non-linearity parameter.

A simple technique for approximate solutions of the Falkner-Skan equation

A simple, sufficiently accurate and efficient method for approximate solutions of the Falkner-Skan equation is proposed here for a wide range of the pressure gradient parameter. The proposed approximate solutions are obtained utilising a known solution of another differential equation.

Circular elastic inclusions in cylindrical shells

The problem of a circular elastic inclusion in a cylindrical shell subjected to internal pressure or thermal loading is studied. The two shallow-shell equations governing the behaviour of a cylindrical shell are transformed into a single differential equation involving a curvature parameter and a complex potential function in a non-dimensional form. In the shell region, the solution is represented by Hankel functions of first kind, whereas in the inclusion region it is represented by Bessel functions of first kind. Boundary conditions at the shell-inclusion junction are expressed in a simple form involving in-plane strains and change in curvature. The effect of such inclusion parameters as extensional rigidity, bending rigidity, and thermal expansion coefficients on the stress concentrations has been determined. The results are presented in non-dimensional form for ready use.

The non-linear response of a gyrostabilized platform: Effects of correction torque time delay and of random inputs

First, the non-linear response of a gyrostabilized platform to a small constant input torque is analyzed in respect to the effect of the time delay (inherent or deliberately introduced) in the correction torque supplied by the servomotor, which itself may be non-linear to a certain extent. The equation of motion of the platform system is a third order nonlinear non-homogeneous differential equation. An approximate analytical method of solution of this equation is utilized. The value of the delay at which the platform response becomes unstable has been calculated by using this approximate analytical method. The procedure is illustrated by means of a numerical example. Second, the non-linear response of the platform to a random input has been obtained. The effects of several types of non-linearity on reducing the level of the mean square response have been investigated, by applying the technique of equivalent linearization and solving the resulting integral equations by using laguerre or Gaussian integration techniques. The mean square responses to white noise and band limited white noise, for various values of the non-linear parameter and for different types of non-linearity function, have been obtained. For positive values of the non-linear parameter the levels of the non-linear mean square responses to both white noise and band-limited white noise are low as compared to the linear mean square response. For negative values of the non-linear parameter the level of the non-linear mean square response at first increases slowly with increasing values of the non-linear parameter and then suddenly jumps to a high level, at a certain value of the non-linearity parameter.