| Abstract [eng] |
This final degree project aims to combine two highly applicable scientific fields: solitary wave theory and fractional calculus. Solitons are waves with exceptional stability and particle properties, widely used in many fields of science (optics, hydrodynamics, biophysics, medicine, etc.) that study nonlinear dynamic models. Fractional calculus is a concept that generalizes the order of classical differentiation and integration operators and allows to model real-world phenomena with more accuracy than using traditional calculus. Since both of these scientific fields are relatively new, the question of the existence of solitary solutions to fractional differential equations is almost unexplored and requires more detailed analysis and new research techniques. Thus, the aim of this final degree project was to develop an analytical framework for the analysis of solutions to Caputo fractional differential equations, applicable in the case of solitary solutions. Proposed technique is based on the construction of the characteristic ordinary differential equation corresponding to the original fractional differential equation (when the characteristic equation exists). Analytic solutions to fractional differential equation are then expressed in the form of infinite function series. This thesis presents an algorithm for the analytical continuation of such series as well as its extension to the negative half-line which results in solution branching out into several different complex solutions. In order to illustrate the efficacy of the proposed scheme, fractional Riccati differential equation was investigated and it was shown that the solitary solutions of this equation approach the kink solution of the ordinary Riccati differential equation as the initial condition corresponding to the fractional derivative value approaches zero. |
