Abstract
In this paper, a new set of functions called fractional-order Euler functions (FEFs) is constructed to obtain the solution of fractional integro-differential equations. The properties of the fractional-order Euler functions are utilized to construct the operational matrix of fractional integration. By using the matrix and the functions approximation, the fractional integro-differential equations are reduced to systems of algebraic equations. The convergence analysis of fractional-order Euler functions approximation is given. Illustrative examples are included to demonstrate the high precision and good performance of the new scheme.
Highlights
Fractional calculus is a branch of mathematical analysis
Some interesting applications of fractional calculus can be found in viscoelasticity [1], electromagnetic waves [2], chaotic systems [3], physical systems [4], optimization [5], nonlinear dynamical systems [6], heat transfer modeling [7], and dynamics of interfaces between nanoparticles and substrates [8]
We are planning to generalize new functions based on Euler polynomials and to acquire numerical solution of the fractional integro-differential equations without discretizing
Summary
Fractional calculus is a branch of mathematical analysis. Fractional integral and differential models have attracted great interest due to their applications in many fields of science and engineering. Some fractional-order functions (polynomial or wavelet) have been proposed to solve fractional differential equations. Bhrawy et al [31] proposed the fractional-order generalized Laguerre functions to find numerical solution of systems of fractional differential equations. Yuzbasi [32] presented a collocation method based on the fractional-order Bernstein polynomials for the fractional Riccati type differential equations. We are planning to generalize new functions based on Euler polynomials and to acquire numerical solution of the fractional integro-differential equations without discretizing. This method is accurate, advantageous, and easy to implement in solving the fractional integro-differential equations. Euler polynomials form a complete basis over the interval [0, 1]
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