Ulam stability and data dependence for fractional differential equations with Caputo derivative

By Jinrong Wang, Linli Lv, Yong Zhou. 2011

307 citations recorded by the source corpus.

In this paper, Ulam stability and data dependence for fractional differential equations with Caputo fractional derivative of orderare studied. We present four types of Ulam stability results for the fractional differential equation in the case of 0 < � < 1 and b = +1 by virtue of the Henry-Gronwall inequality. Meanwhile, we give an interesting data dependence results for the fractional differential equation in the case of 1 < � < 2 and b < +1 by virtue of a generalized Henry-Gronwall inequality with mixed integral term. Finally, examples are given to illustrate our theory results.

Subfields: Applied Mathematics, Modeling and Simulation, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f0461aab1c052f6856e6224dbe338cd862ff5297

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Modeling and Simulation

In Statistics and Probability

Citations in the corpus

On the Ulam–Hyers–Rassias stability for nonlinear fractional differential equations using the $$\psi $$ψ-Hilfer operator (2017, background): “Studies about UlamHyers and Ulam-Hyers-Rassias stability for fractional differential equations can be found in [22, 23, 24, 25].”
Existence and Stability Results for Nonlinear Boundary Value Problem for Implicit Differential Equations of Fractional Order (2015, methodology): “For more detailed definitions of the Hyers-Ulam stability and the generalized Hyers-Ulam stability, we refer the reader to the papers [1, 7, 8, 14, 16, 19, 20, 22, 26, 29, 34, 35, 36] and the books [13, 27, 28].”
Stability of the fractional Volterra integro‐differential equation by means of ψ‐Hilfer operator (2018, background): “[30] Benchohra and Lazreg [31], Wang and Li [32] and more recently Sousa and Capelas de Oliveira [33, 34, 35] have investigated stabilities of UlamHyers, Ulam-Hyers-Rassias, semi-Ulam-Hyers-Rassias, among others, aiming new results involving different types of stability.”
Hyers–Ulam Stability and Existence of Solutions for Differential Equations with Caputo–Fabrizio Fractional Derivative (2019, unknown): “3, [37]), we introduce the following definition.”
Existence and Ulam stability for nonlinear implicit fractional differential equations with Hadamard derivative (2017, result): “[7, 12, 14, 15, 16, 33] and books [11, 25, 26, 27] on this subject have been published in order to generalize the results of Hyers in many directions.”
HYERS-ULAM-RASSIAS STABILITY OF FRACTIONAL DIFFERENTIAL EQUATION (2015, background): “[19], Cadariu and Radu [20] and Jung [21].”
Some Existence and Stability Results for Hilfer-fractional Implicit Differential Equations with Nonlocal Conditions (2018, unknown): “For detailed study of Ulam-type stability with different approaches, we recommend papers such as [17,29,30].”
Continuous grey model with conformable fractional derivative (2020, background): “Wu first proposed a grey prediction model based on the Caputo fractional derivative, and the time response sequence of the model was directly derived from the fractional derivative of Caputo, which achieved good results [34].”
Ulam–Hyers–Mittag-Leffler stability of fractional-order delay differential equations (2014, methodology): “Motivated by [10,11,18,21], we will investigate existence, uniqueness and Ulam–Hyers– Mittag-Leffler stability of the following fractional-order differential equation with modification of the argument c D t x(t) = f (t, x(t), x(g(t))), t ∈ I ⊂ R, α ∈ (0, 1).”