NEW OPERATIONAL MATRIX OF RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE OF ORTHONORMAL BERNOULLI POLYNOMIALS FOR THE NUMERICAL SOLUTION OF SOME DISTRIBUTED-ORDER TIME-FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

In this article, the orthonormal Bernoulli polynomials (OBPs) and their properties are applied for concluding a general technique for forming a new operational matrix of the distributed-order (DO) fractional derivative. Then, we apply tau approach and obtained operational matrix to solve some DO time-fractional partial differential equations including distributed-order Rayleigh-Stokes problem (DRSP) for a generalized second-grade fluid and DO anomalous sub-diffusion equation. Our methodology reduces the solution of these problems to a set of algebraic equations. By analysis the error of approximation by the obtained matrix and comparing between the numerical solutions and exact result, we can conclude that this operational matrix is valid to solve the mentioned equations. Also, to confirm the accuracy and the validity of our technique three examples are provided. Finally, we compare obtained results from this approach with the achieved results from relevant studies.