Applications of new smart algorithm based on kernel method for variable fractional functional boundary value problems

This paper studies the variable fractional functional boundary value problems (VFF–BVPs) by considering Caputo fractional derivative. We use the reproducing kernel method (RKM) without the orthogonalization process as a smart scheme. For this purpose, we construct a reproducing kernel that does not satisfy the boundary condition of VFF–BVP. With this kernel, we can better approximate the solutions for VFF–BVP. Using this method increases the accuracy of the approximate solution so that a significant error analysis can be produced. Finally, two numerical examples are solved to illustrate the efficiency of the present method.