A Review on Fractional Differential Equations and a Numerical Method to Solve Some Boundary Value Problems

Fractional differential equations can describe the dynamics of several complex and nonlocal systems with memory. They arise in many scientific and engineering areas such as physics, chemistry, biology, biophysics, economics, control theory, signal and image processing, etc. Particularly, nonlinear systems describing different phenomena can be modeled with fractional derivatives. Chaotic behavior has also been reported in some fractional models. There exist theoretical results related to existence and uniqueness of solutions to initial and boundary value problems with fractional differential equations; for the nonlinear case, there are still few of them. In this work we will present a summary of the different definitions of fractional derivatives and show models where they appear, including simple nonlinear systems with chaos. Existing results on the solvability of classical fractional differential equations and numerical approaches are summarized. Finally, we propose a numerical scheme to approximate the solution to linear fractional initial value problems and boundary value problems

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

Numerical continuation for fractional PDEs: sharp teeth and bloated snakes

Partial differential equations (PDEs) involving fractional Laplace operators have been increasingly used to model non-local diffusion processes and are actively investigated using both analytical and numerical approaches. The purpose of this work is to study the effects of the spectral fractional Laplacian on the bifurcation structure of reaction-diffusion systems on bounded domains. In order to do this we use advanced numerical continuation techniques to compute the solution branches. Since current available continuation packages only support systems involving the standard Laplacian, we first extend the pde2path software to treat fractional PDEs. The new capabilities are then applied to the study of the Allen-Cahn equation, the Swift-Hohenberg equation and the Schnakenberg system (in which the standard Laplacian is each replaced by the spectral fractional Laplacian). Our study reveals some common effects, which contributes to a better understanding of fractional diffusion in generic reaction-diffusion systems. In particular, we investigate the changes in snaking bifurcation diagrams and also study the spatial structure of non-trivial steady states upon variation of the order of the fractional Laplacian. Our results show that the fractional order can induce very significant qualitative and quantitative changes in global bifurcation structures

A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

In this paper, the fractional order of rational Bessel functions collocation method (FRBC) to solve Thomas-Fermi equation which is defined in the semi-infinite domain and has singularity at $x = 0$ and its boundary condition occurs at infinity, have been introduced. We solve the problem on semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration solves it by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results in other to show that the new method is accurate, efficient and applicable

Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart

The solution of a Caputo time fractional diffusion equation of order $0<\alpha<1$ is expressed in terms of the solution of a corresponding integer order diffusion equation. We demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem. In the context of an $N$-point finite difference time discretisation, the mapping allows for an improvement in time computational complexity from $O\left(N^2\right)$ to $O\left(N^\alpha\right)$, given a precomputation of $O\left(N^{1+\alpha}\ln N\right)$. The mapping is applied successfully to the least-squares fitting of a fractional advection diffusion model for the current in a time-of-flight experiment, resulting in a computational speed up in the range of one to three orders of magnitude for realistic problem sizes.Comment: 9 pages, 5 figures; added references for section

Fourier spectral methods for fractional-in-space reaction-diffusion equations

Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is computationally demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reactiondiffusion equations. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is show-cased by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models,together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator