This dissertation presents new numerical methods for the solution of fractional differential equations of single and distributed order that find application in the different fields of physics and engineering. We start by presenting the relationship between fractional derivatives and processes like anomalous diffusion, and, we then develop new numerical methods for the solution of the time-fractional diffusion equations. The first numerical method is developed for the solution of the fractional diffusion equations with Neumann boundary conditions and the diffusivity parameter depending on the space variable. The method is based on finite differences, and, we prove its convergence (convergence order of O(Δx² + Δt²^-α), 0 < α < 1) and stability. We also present a brief description of the application of such boundary conditions and fractional model to real world problems (heat flux in human skin). A discussion on the common substitution of the classical derivative by a fractional derivative is also performed, using as an example the temperature equation. Numerical methods for the solution of fractional differential equations are more difficult to develop when compared to the classical integer-order case, and, this is due to potential singularities of the solution and to the nonlocal properties of the fractional differential operators that lead to numerical methods that are computationally demanding. We then study a more complex type of equations: distributed order fractional differential equations where we intend to overcome the second problem on the numerical approximation of fractional differential equations mentioned above. These equations allow the modelling of more complex anomalous diffusion processes, and can be viewed as a continuous sum of weighted fractional derivatives. Since the numerical solution of distributed order fractional differential equations based on finite differences is very time consuming, we develop a new numerical method for the solution of the distributed order fractional differential equations based on Chebyshev polynomials and present for the first time a detailed study on the convergence of the method. The third numerical method proposed in this thesis aims to overcome both problems on the numerical approximation of fractional differential equations. We start by solving the problem of potential singularities in the solution by presenting a method based on a non-polynomial approximation of the solution. We use the method of lines for the numerical approximation of the fractional diffusion equation, by proceeding in two separate steps: first, spatial derivatives are approximated using finite differences; second, the resulting system of semi-discrete ordinary differential equations in the initial value variable is integrated in time with a non-polynomial collocation method. This numerical method is further improved by considering graded meshes and an hybrid approximation of the solution by considering a non-polynomial approximation in the first sub-interval which contains the origin in time (the point where the solution may be singular) and a polynomial approximation in the remaining intervals. This way we obtain a method that allows a faster numerical solution of fractional differential equations (than the method obtained with non-polynomial approximation) and also takes into account the potential singularity of the solution. The thesis ends with the main conclusions and a discussion on the main topics presented along the text, together with a proposal of future work.

In recent decades, significant interest, based on physics and engineering applications, has developed on so-called anomalous diffusion processes that possess different spread functions with classical ones. The resulting differential equation whose fundamental solution matches this decay process is best modeled by an equation containing a fractional order derivative. This dissertation mainly focuses on some inverse problems for fractional diffusion equations. After some background introductions and preliminaries in Section 1 and 2, in the third section we consider our first inverse boundary problem. This is where an unknown boundary condition is to be determined from overposed data in a time- fractional diffusion equation. Based upon the fundamental solution in free space, we derive a representation for the unknown parameters as the solution of a nonlinear Volterra integral equation of second kind with a weakly singular kernel. We are able to make physically reasonable assumptions on our constraining functions (initial and given boundary values) to be able to prove a uniqueness and reconstruction result. This is achieved by an iterative process and is an immediate result of applying a certain fixed point theorem. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method. In the fourth section a reaction-diffusion problem with an unknown nonlinear source function, which has to be determined from overposed data, is considered. A uniqueness result is proved and a numerical algorithm including convergence analysis under some physically reasonable assumptions is presented in the one-dimensional case. To show effectiveness of the proposed method, some results of numerical simulations are presented. In Section 5, we also attempted to reconstruct a nonlinear source in a heat equation from a number of known input sources. This represents a new research even for the case of classical diffusion and would be the first step in a solution method for the fractional diffusion case. While analytic work is still in progress on this problem, Newton and Quasi-Newton method are applied to show the feasibility of numerical reconstructions. In conclusion, the fractional diffusion equations have some different properties with the classical ones but there are some similarities between them. The classical tools like integral equations and fixed point theory still hold under slightly different assumptions. Inverse problems for fractional diffusion equations have applications in many engineering and physics areas such as material design, porous media. They are trickier than classical ones but there are also some advantages due to the mildly ill-conditioned singularity caused by the new kernel functions.