Toward strictly singular fractional operator restricted by Fredholm-Volterra in Sobolev space

Abstract

In this paper, a reliable numerical technique is proposed for solving a class of singular fractional differential equations involving Fredholm and Volterra operators subjected to suitable three-point boundary conditions.  The solution methodology is presented based on reproducing-kernel method (RKM), which is used directly without employing linearization and perturbation.  However, a favorable Hilbert spaces are construcred, and then the orthonormal function system is generated by using Gram-Schmidt orthogonalization process.  Error analysis is given in Sobolev space.  Numerical example is tested to multipoint singular fractional differential problems with Fredholm and Volterra operators to show the theoretical statements of the RKHS method.  The results obtained indicate that the RKHS method is easy to implement, reliability and capability with a great potential of such singular problems.