Abstract
In this paper, we propose a new method called the inverse fractional natural transform method (IFNTM). We present theoretical results and apply them to obtain approximate solutions of linear fractional ordinary differential equations (LFODEs) and partial differential equations (LFPDEs). The fractional derivatives are described in the Caputo sense. The algorithm described in this study is expected to be further employed to solve similar linear problems in fractional calculus.
Highlights
In recent years, interest in the fractional differential equations has been stimulated due to their wide applications [1–4] in various fields of engineering and science [7, 8, 12, 17, 20]
We present some background about the nature of the natural transform method (NTM)
5 Conclusion In this paper, we proved three theorems related to the fractional natural transform method (FNTM), and we successfully applied the new method to obtain solutions to two lin
Summary
Interest in the fractional differential equations has been stimulated due to their wide applications [1–4] in various fields of engineering and science [7, 8, 12, 17, 20]. We give approximate analytical and exact solutions to two linear fractional differential equations: First, consider the linear fractional initial value problem of the form [19]. Theorem 2.5 If R(s, u) is the natural transform of f (t), the natural transform of the Riemann–Liouville fractional integral for the function f (t) of order α denoted by Jα[f (t)] is given by. Theorem 2.7 If n is any positive integer, n – 1 ≤ α < n, and R(s, u) is the natural transform of a function f (t), the natural transform Rcα(s, u) of the Caputo fractional derivative of the function f (t) of order α denoted by cDαf (t) is given by cDαf (t).
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