Abstract
In this paper, a numerical method for solving the fractional differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting ofblock-pulse functions and Taylor polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the initial value problems for fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.
Highlights
Fractional differential equations (FDEs) are generalized from integer order ones, which are obtained by replacing integer order derivatives by fractional ones
The main advantage of using orthogonal functions is that they reduce the dynamical system problems to those of solving a system of algebraic equations by using the operational matrices of differentiation or integration
The hybrid functions consisting of the combination of block-pulse functions with Legendre polynomials, Chebyshev polynomials, Taylor series, or Bernoulli polynomials [17,18,19,20] have been successfully used for solving selected smooth and non-smooth problems arising in diverse areas of science and engineering
Summary
Fractional differential equations (FDEs) are generalized from integer order ones, which are obtained by replacing integer order derivatives by fractional ones. The main advantage of using orthogonal functions is that they reduce the dynamical system problems to those of solving a system of algebraic equations by using the operational matrices of differentiation or integration. The hybrid functions consisting of the combination of block-pulse functions with Legendre polynomials, Chebyshev polynomials, Taylor series, or Bernoulli polynomials [17,18,19,20] have been successfully used for solving selected smooth and non-smooth problems arising in diverse areas of science and engineering. An exact Riemann-Liouville fractional integral operator for the hybrid of block-pulse functions and Taylor polynomials is given. This operator is utilized to reduce the solution of the fractional order differential equations to the solution of algebraic equations
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