Homotopy-based methods for fractional differential equations

The intention of this thesis is two-fold. The first aim is to describe and apply, series-based, numerical methods to fractional differential equation models. For this, it is needed to distinguish between space-fractional and time-fractional derivatives. The second goal of this thesis is to give a clear and fair numerical analysis to show that these methods, mainly based on so-called homotopy principles, may, in certain cases, provide accurate results for nonlinear models. However, it will also be stressed that, in contrast with many articles on this topic, there may be many disadvantages when applying such methods in general. Tuning of numerical parameters still plays an important role to obtain convergent series solutions. Therefore, the main focus of this work is the numerical performance of series-based methods for space- and time-fractional differential equations.
Most authors on this topic cite a particular date as the birthday of so called "Fractional Calculus". In a letter dated September 30th 1695, L’Hôpital wrote to Leibniz asking him about a particular notation he had used in his publications for the nth-derivative of a linear function. L’Hôpital posed the following question to Leibniz: "what would the result be if the order of the derivative is ½ ?” Leibniz’s response was: "An apparent paradox, from which one day useful consequences will be drawn."
Many mathematicians found, using their own notation and methodology, definitions that fit the concept of a non-integer order integral or derivative. Well-known definitions of fractional derivatives, perhaps not yet completely accepted in the calculus community, are the Riemann-Liouville, Caputo and Grünwald-Letnikov definitions.
In this thesis, we focus on a fractional order advection-diffusion-reaction model. We introduce the homotopy perturbation method to solve the model. Some theoretical results are given and are explained in terms of convergence tables and graphics. Numerical experiments illustrate the performance of this method, when applied to a test set of boundary-value problems.
Next, we deal with traveling wave solutions in time-fractional partial differential equations by using the homotopy analysis method. Primarily, we explain the importance of the convergence parameter h in the so-called h-curve which enables us, somehow, to control the convergence region of the method. Especially, for this curve, the role of an optimal h-value is emphasized. Numerical results for the time-fractional Fisher equation are given.
Finally, we study Pade ́ approximations to improve the numerical solutions which were obtained by the homotopy analysis method. A Pade ́ approximation has the potential to produce more accurate numerical solutions not only for higher time values in the differential equation, but also speeds up the computation time of the series. We introduce the rational homotopy perturbation method which makes use of a Pade ́ approximation for stationary problems. It is applied to a convective-radiative equation and to Troesch's model. We use a Pade ́ approximation with the homotopy analysis method to solve a time-fractional Fisher partial differential equation.