Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders

Abstract

The differential transformation, an approach based on Taylor’s theorem, is proposed as convenient for finding an exact or approximate solution to the initial value problem with multiple Caputo fractional derivatives of generally non-commensurate orders. The multi-term differential equation is first transformed into a multi-order system and then into a system of recurrence relations for coefficients of formal fractional power series. The order of the fractional power series is discussed in relation to orders of derivatives appearing in the original equation. Application of the algorithm to an initial value problem gives a reliable and expected outcome including the phenomenon of symmetry in choice of orders of the differential transformation of the multi-order system.

Semigroup property of fractional differential operators and its applications

“…By Theorem 2.3 of Diethelm, Siegmund and Tuan [5] for any initial value (12) the equation ( 11) has unique solution. Therefore, by Theorem 4.1 for any initial value (10) the equation ( 9) has unique solution and that solution agrees with the solution of ( 11)-(12).…”

“…Let us consider Caputo fractional differential equation ( 9) on [a, b] subject to the initial conditions (10). Put…”

“…This result is a useful tool for studying multi-term fractional differential equations, since it allows us to reduce a multi-term system to a single-term system for which 2 NGUYEN DINH CONG many tools of investigation are available. There are a number of works dealing with this problem, especially the reduction to single-term fractional differential equations is used to study the asymptotic behavior of solutions to multi-order fractional differential systems [4,5,1,3,10]. However, for a proof of this weak semigroup property of Caputo fractional differential operators, Diethelm [4] requires a strong assumption of smoothness of the functions under differentiation.…”

Applications of the differential transformation to three-point singular boundary value problems for ordinary differential equations

“…It has applications in solving different types of problems for all classes of differential equations (ordinary, partial, delayed, fractional, fuzzy etc). The recent developments and applications of DTM are discussed in [15,[17][18][19] and references therein.…”

Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

“…Existence, uniqueness and stability of solution for multi-term fractional differential equations are discussed in [45][46][47][48][49]. Because of difficulty of finding the exact solutions for such equations, many new numerical techniques have been developed to investigate the numerical solutions such as Adams method [50], Haar wavelet method [51], differential transform method [52], Adams-Bashforth-Moulton method [53], collocation method based on shifted Chebyshev polynomials of the first kind [54], Boubaker polynomials method [55], matrix Mittag-Leffler functions [56], differential transform method [57] and so on.…”