Abstract
In this paper, by introducing the concepts of Ulam type stability for ODEs into the equations involving conformable fractional derivative, we utilize the technique of conformable fractional Laplace transform to investigate the Ulam–Hyers and Ulam–Hyers–Rassias stability for several kinds of linear differential equations in the frame of conformable fractional derivative.
Highlights
1 Introduction Over the past few decades, the problem of analysis on Ulam type stability has already been proved to be an important subject in the area of stability theory and mathematical modeling
One can refer to [2,3,4,5,6,7,8] and the references therein, where the Ulam type stability of different type equations was discussed via different approaches, like the Gronwall lemma and the method of Picard operators [2, 3], the technique of integral factors [4, 5], the Laplace transform approach [6,7,8]
In order to overcome these or other difficulties, in [20], the authors introduced a new simple and well-behaved local derivative called conformable fractional derivative, which is just defined on a basic limit definition and satisfies almost all the properties that the classical integer-order derivative owns
Summary
Over the past few decades, the problem of analysis on Ulam type stability (which was first posed by Ulam in [1]) has already been proved to be an important subject in the area of stability theory and mathematical modeling. According to [37], we can recall that a function x ∈ C1([t0, T], R) is the solution of (1) if x satisfies Ttα0 x(t) + βx(t) = f (t), t ∈ (t0, T] and x(t0) = x0 This method is applied to study the Ulam type stability of linear nonhomogenous conformable fractional differential equations, linear Langevin equations described by two same conformable fractional derivatives, and linear conformable integrodifferential equations. Remark 3.2 The existence and uniqueness results of the solutions for initial value problem (1) have been studied in considerable detail (see [23, 37]) It can be derived from [21, 23, 25] that the general solution of (1) is given by t x(t) = Eα(–β, t – t0)x0 + Eα(–β, t – t0)Eα(β, τ – t0)(τ – t0)α–1f (τ ) dτ.
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