The constant variation formulae for singular fractional differential systems with delay

AbstractThis paper considers the Caputo singular fractional differential systems with delay, and the Riemann–Liouville singular fractional differential systems with delay. A new function α−δ is defined. By the D−inverse matrix and α−δ function, two fundamental solutions are given. The constant variation formulae for singular fractional differential systems with delay are obtained

Modelling elastic structures with strong nonlinearities with application to stick-slip friction

An exact transformation method is introduced that reduces the governing equations of a continuum structure coupled to strong nonlinearities to a low dimensional equation with memory. The method is general and well suited to problems with point discontinuities such as friction and impact at point contact. It is assumed that the structure is composed of two parts: a continuum but linear structure and finitely many discrete but strong nonlinearites acting at various contact points of the elastic structure. The localised nonlinearities include discontinuities, e.g., the Coulomb friction law. Despite the discontinuities in the model, we demonstrate that contact forces are Lipschitz continuous in time at the onset of sticking for certain classes of structures. The general formalism is illustrated for a continuum elastic body coupled to a Coulomb-like friction model

Stability analysis of fractional differential equations with unknown parameters

In this paper, the stability of fractional differential equations (FDEs) with unknown parameters is studied. Using the graphical based D-decomposition method, the parametric stability analysis of FDEs is investigated without complicated mathematical analysis. To achieve this, stability boundaries are obtained firstly by a conformal mapping from s-plane to parameter space composed by unknown parameters of FDEs, and then the stability region set depending on the unknown parameters is found. The applicability of the presented method is shown considering some benchmark equations, which are often used to verify the results of a new method. Simulation examples show that the method is simple and give reliable stability results. &nbsp

General Solution and Observability of Singular Differential Systems with Delay

We study singular differential systems with delay. A general description for the solutions of singular differential systems with delay is given and a necessary and sufficient condition for exact observability of singular differential systems with delay is derived

PKS 1830-211: A Face-On Spiral Galaxy Lens

We present new Hubble Space Telescope images of the gravitational lens PKS 1830-211, which allow us to characterize the lens galaxy and update the determination of the Hubble constant from this system. The I-band image shows that the lens galaxy is a face-on spiral galaxy with clearly delineated spiral arms. The southwestern image of the background quasar passes through one of the spiral arms, explaining the previous detections of large quantities of molecular gas and dust in front of this image. The lens galaxy photometry is consistent with the Tully-Fisher relation, suggesting the lens galaxy is a typical spiral galaxy for its redshift. The lens galaxy position, which was the main source of uncertainty in previous attempts to determine H_0, is now known precisely. Given the current time delay measurement and assuming the lens galaxy has an isothermal mass distribution, we compute H_0 = 44 +/- 9 km/s/Mpc for an Omega_m = 0.3 flat cosmological model. We describe some possible systematic errors and how to reduce them. We also discuss the possibility raised by Courbin et al. (2002), that what we have identified as a single lens galaxy is actually a foreground star and two separate galaxies.Comment: 21 pp., 4 figs., accepted by ApJ, section added to discuss related work by Courbin et al. (astro-ph/0202026

Fractional model of cancer immunotherapy and its optimal control

Cancer is one of the most serious illnesses in all of the world. Although most of the cancer patients are treated with chemotherapy, radiotherapy and surgery, wide research is conducted related to experimental and theoretical immunology. In recent years, the research on cancer immunotherapy has led to major medical advances. Cancer immunotherapy refers to the stimulation of immune system to deal with cancer cells. In medical practice, it is mainly achieved by using effector cells such as activated T-cells and Interleukin-2 (IL-2), which is the main cytokine responsible for lymphocyte activation, growth and differentiation. A well-known mathematical model, named as Kirschner-Panetta (KP) model, represents richly the dynamics of the interaction between cancer cells, IL-2 and the effector cells. The dynamics of the KP model is described and the solution to which is approximated by using polynomial approximation based methods such as Adomian decomposition method and differential transform method. The rich nonlinearity of the KP model causes these approaches to become so complicated in order to deal with the representation of polynomial approximations. It is illustrated that the approximated polynomials are in good agreement with the solution obtained by common numerical approaches. In the KP model, the growth of the tumour cells can be expressed by a linear function or any limited-growth function such as logistic equation, in which the cancer population possesses an upper bound mentioned as carrying capacity. Effector cells and IL-2 construct two external sources of medical treatment to stimulate immune system to eradicate cancer cells. Since the main goal in immunotherapy is to remove the tumour cells with the least probable medication side effects, an advanced version of the model may include a time dependent external sources of medical treatment, meaning that the external sources of medical treatment could be considered as control functions of time and therefore the optimum use of medical sources can be evaluated in order to achieve the optimal measure of an objective function. With this sense of direction, two distinct strategies are explored. The first one is to only consider the external source of effector cells as the control function to formulate an optimal control problem. It is shown under which circumstances, the tumour is eliminated. The approach in the formulation of the optimal control is the Pontryagin maximum principal. Furthermore the optimal control problem will be dealt with using particle swarm optimization (PSO). It is shown that the obtained results are significantly better than those obtained by previous researchers. The second strategy is to formulate an optimal control problem by considering both the two external sources as the controls. To our knowledge, it is the first time to present a multiple therapeutic protocol for the KP model. Some MATLAB routines are develop to solve the optimal control problems based on Pontryagin maximum principal and also the PSO. As known, fractional differential equations are more appropriate to describe the persistent memory of physical phenomena. Thus, the fractional KP model is defined in the sense of Caputo differentiation operator. An effective method for numerical treatment of the model is described, namely Predictor-Corrector method of Adams-Bashforth-Moulton type. A robust MATLAB routine is coded based on the mentioned approach and the solution obtained will be compared with those of the classical KP model. The code is prepared in such a way to be able to deal with systems of fractional differential equations, in which each equation has its own fractional order (i.e. multi-order systems of fractional differential equations). The theorems for existence of solutions and the stability analysis of the fractional KP model are represented. In this regard, a frequently used method of solving fractional differential equations (FDEs) is described in details, namely multi-step generalized differential transform method (MSGDTM), then it is illustrated that the method neglects the persistent memory property and takes the incorrect approach in dealing with numerical solutions of FDEs and therefore it is unfit to be used in differential equations governed by fractional differentiation operators. The sigmoidal behavior of the solution to the logistic equation caused it to be one of the most versatile models in natural sciences and therefore the fractional logistic equation would be a relevant problem to be dealt with. Thus, a power series of Mittag-Leffer functions is introduced, the behaviour of which is in good agreement with the solution to fractional logistic equation (FLE), and then a fractional integro-differential equation is represented and proved to be satisfied with the power series of Mittag-Leffler function. The obtained fractional integro-differential equation is named as modified fractional differential equation (MFDL) and possesses a nonlinear additive term related to the solution of the logistic equation (LE). The method utilized in the thesis, may be appropriately applied to the analysis of solutions to nonlinear fractional differential equations of mathematical physics. Inverse problems to FDEs occur in many branches of science. Such problems have been investigated, for instance, in fractional diffusion equation and inverse boundary value problem for semi- linear fractional telegraph equation. The determination of the order of fractional differential equations is an issue, which has been analyzed and discussed in, for instance, fractional diffusion equations. Thus, fractional order estimation has been conducted for some classes of linear fractional differential equations, by introducing the relationship between the fractional order and the asymptotic behaviour of the solutions to linear fractional differential equations. Fractional optimal control problems, in which the system and (or) the objective function are described based on fractional derivatives, are much more complicated to be solved by using a robust and reliable numerical approach. Thus, a MATLAB routine is provided to solve the optimal control for fractional KP model and the obtained solutions are compared with those of classical KP model. It is shown that the results for fractional optimal control problems are better than classical optimal control problem in the sense of the amount of drug administration