Introduction to fractional calculus.

      Аннотация: The fractional derivative was introduced in 1695 by Leibnitz as a generalization of the integer order derivative and was reconsidered also by Euler, Abel, Riemann Liouville, Grunwald and Letnikov. The number of studies in this field has increased after 1930, when E.L. Post published an important article. Recent experimental investigations in physics and engineering confirm the possibility to describe a series of phenomena in terms of fractional calculus. Our book contains a series of fractional calculus problems not yet investigated and can be used as a handbook by the researchers in the field of mathematics, theoretical physics, and in experimental methods in engineering and physics. We introduced a new method, based on decomposition and Laplace transform method. We studied also the fractional differential equations with the aid of small parameters, and also using the power series method. We investigated also the fractional integral equations. We introduced the generalized the Galerkin and Ritz methods to the case of fractional differential equations. We used also a least squares method to establish solutions for the fractional differential equations.

Ключевые слова: approximate solution, fractional calculus, fractional differential equations, Integral equations, power series solutions, Galerkin, Ritz, least squares method, Laplace method, small parameter method