Existence of solutions of infinite system of nonlinear sequential fractional differential equations

Abstract In a recent paper (Filomat 32:4577–4586, 2018) the authors have investigated the existence and uniqueness of a solution for a nonlinear sequential fractional differential equation. To present an analytical improvement for Fazli–Nieto's results with some conditions removed based on a new technique is the main objective of this paper. In addition, we introduce an infinite system of nonlinear sequential fractional differential equations and discuss the existence of a solution for them in the classical Banach sequence spaces c 0 $c_{0}$ and ℓ p $\ell_{p}$ by applying the Darbo fixed point theorem. Moreover, the proposed method is applied to several examples to show the clarity and effectiveness

On the spectral distribution of kernel matrices related to\ud radial basis functions

This paper focuses on the spectral distribution of kernel matrices related to radial basis functions. The asymptotic behaviour of eigenvalues of kernel matrices related to radial basis functions with different smoothness are studied. These results are obtained by estimated the coefficients of an orthogonal expansion of the underlying kernel function. Beside many other results, we prove that there are exactly (k+d−1/d-1) eigenvalues in the same order for analytic separable kernel functions like the Gaussian in Rd. This gives theoretical support for how to choose the diagonal scaling matrix in the RBF-QR method (Fornberg et al, SIAM J. Sci. Comput. (33), 2011) which can stably compute Gaussian radial basis function interpolants

A dynamic gradient approach to Pareto optimization with nonsmooth convex objective functions

In a general Hilbert framework, we consider continuous gradient-like dynamical systems for constrained multiobjective optimization involving non-smooth convex objective functions. Our approach is in the line of a previous work where was considered the case of convex di erentiable objective functions. Based on the Yosida regularization of the subdi erential operators involved in the system, we obtain the existence of strong global trajectories. We prove a descent property for each objective function, and the convergence of trajectories to weak Pareto minima. This approach provides a dynamical endogenous weighting of the objective functions. Applications are given to cooperative games, inverse problems, and numerical multiobjective optimization

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

A particle method for the homogeneous Landau equation

We propose a novel deterministic particle method to numerically approximate the Landau equation for plasmas. Based on a new variational formulation in terms of gradient flows of the Landau equation, we regularize the collision operator to make sense of the particle solutions. These particle solutions solve a large coupled ODE system that retains all the important properties of the Landau operator, namely the conservation of mass, momentum and energy, and the decay of entropy. We illustrate our new method by showing its performance in several test cases including the physically relevant case of the Coulomb interaction. The comparison to the exact solution and the spectral method is strikingly good maintaining 2nd order accuracy. Moreover, an efficient implementation of the method via the treecode is explored. This gives a proof of concept for the practical use of our method when coupled with the classical PIC method for the Vlasov equation.Comment: 27 pages, 14 figures, debloated some figures, improved explanations in sections 2, 3, and