A new method for solving variable coefficients fractional differential equations based on a hybrid of Bernoulli polynomials and block pulse functions

Abstract

Block pulse functions (BPFs) are constant functions on each subinterval and not so smooth. This property makes BPFs incapable of approximating function accurately. Therefore, the existing BPFs method is insufficiently accurate to numerically solve variable coefficients fractional differential equations (VCFDEs). To obtain highly accurate solutions with fewer computational burden, we propose an efficient numerical method to find the approximate solutions of VCFDEs. The proposed method uses a hybrid of Bernoulli polynomials and BPFs (HBPBPFs) to overcome the disadvantage that BPFs are piecewise constant and smooth enough. For this aim, a fractional integral operational matrix of HBPBPFs is derived and used to convert VCFDEs into a system of algebraic equations. The solutions of VCFDEs are obtained by solving the algebraic equations. Some simulation examples are presented to verify the effectiveness of our proposed method. Numerical results from our proposed method are compared with those from the existing BPFs numerical method. It is demonstrated that our method can obtain more accurate approximate solutions than the BPFs method.