Justification of a Galerkin Method for a Fractional Order Cauchy Singular Integro-Differential Equation

Now there are more than 30 different definitions of fractional order derivatives, and the number is growing. Some of them are just “mind games”, but others are introduced to solve some serious problems. In this article a new definition of a fractional order derivative is given, which generalizes the formula for differentiating Jacobi polynomials. This makes it possible to build a scale of systems of orthogonal polynomials, the closures of which are Sobolev spaces. Using these derivatives, a fractional order Cauchy singular integro-differential equation is stated. Its unique solvability is proven, and a Galerkin method for its approximate solution is justified: the convergence of the method is proven, and the error estimation is obtained.