Hyers–Ulam stability of a first order partial differential equation

Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives

“…In recent years, many researchers have focused on the study of Hyers-Ulam stability of differential equations, and gained a series of results (see [1]- [7] and [9]- [14] and the references therein).…”

“…András and Mészáros presented Hyers-Ulam stability of dynamic equations on time scales via Picard operators (see [1]). Regarding partial differential equations, in [9], Lungu and Popa discussed Hyers-Ulam stability of a first order partial differential equation, and in [2], Gordji, Cho, Ghaemi, and Alizadeh investigated stability of second order partial differential equations. Hegyi and Jung discussed the stability of Laplace's equation (see [3]).…”

The necessary and sufficient conditions of Hyers-Ulam stability for a class of parabolic equation

“…Since then, there have been many papers dealing with the Hyers-Ulam stability for ordinary differential equations [1-3, 8, 13-15, 17-19] and partial differential equations [4,9]. Lungu and Popa [9] investigated the sufficient conditions of Hyers-Ulam stability for the following first order linear partial differential equation p(x, y) ∂u ∂x + q(x, y) ∂u ∂y = p(x, y)r(x)u + f(x, y).…”

“…However, to the best knowledge of the authors, there exists no work in the literature which discusses the Hyers-Ulam stability of parabolic equations. Motivated by the above work [4,9], in this paper, we will consider the Hyers-Ulam stability for the following parabolic equation…”

The Fixed Point Approach to the Stability of Fractional Differential Equations with Causal Operators

“…In 1978, Rassias [13] provided a remarkable generalization of the Hyers-Ulam stability of mappings by considering variables, which is named Hyers-Ulam-Rassias stability. In the past few years, the two kinds of stabilities for ordinary differential equations, functional equations and partial differential equations have been studied extensively by Rezaei et al [14], Lungu et al [15], Rus [16][17][18], Jung [19], Forti [20], Miura et al [21], Takahasi et al [22]. Moreover, many generalized results are obtained, particularly, the results in Wang et al [23,24] and Li et al [25] devoted to the Mittag-Leffler-Ulam stabilities and the generalized Mittag-Leffler stability.…”