Hybrid Fractional Problem with Periodic Boundary Conditions

The primary focus of the current study is to construct the analytic solution to hybrid fractional problem including periodic boundary conditions. Utilization of the separation of variables method (SVM) provides the solution in a Fourier series form in terms of corresponding eigenfunctions which are the solutions of corresponding fractional SturmLiouville problem in the sense of hybrid fractional derivative. The significant motivation of this study is that fractional diffusion problem with periodic boundary conditions in the constant proportional Caputo hybrid derivative (CPCHD), a combination of Riemann-Liouville integral and Caputo derivative, is considered through SVM. Special cases of CPCHD are taken into account and obtained results are compared to analyze the effect of chosen proportions. Moreover, the established solutions are given in terms of bivariate Mittag-Leffler function emerging in diverse applications. As a result, the novelty of this research is that fractional diffusion problems with periodic boundary conditions in the sense of CPCHD is considered and their solutions are obtained by means of bivariate MittagLeffler function. Examples are provided to present accuracy and efficiency of the proposed method as well as influence of the proportions in CPCHD for hybrid fractional problem.