Stability Analysis of Multi-point Boundary Value Problem for Sequential Fractional Differential Equations with Non-instantaneous Impulses

Zada, Akbar; Ali, Sartaj

doi:10.1515/ijnsns-2018-0040

Cited by 59 publications

(32 citation statements)

References 33 publications

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“…Furthermore, Hyers result was generalized by Rassias, so this notion is given a name known as Ulam‐Hyers‐Rassias stability. Ulam's stability has been well investigated for different differential equations with different approaches, we recommend the reader to previous studies …”

Section: Introductionmentioning

confidence: 99%

Stability analysis of first‐order impulsive nonautonomous system on timescales

Zada

Pervaiz

Shah

et al. 2020

Math Methods in App Sciences

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In this manuscript, we present the existence, uniqueness, -Ulam-Hyers stability, and -Ulam-Hyers-Rassias stability of semilinear nonautonomous impulsive dynamic systems on timescales, with the help of fixed point approach. We use Grönwall inequality on timescale, abstract Gröwall lemma, and Picard operator as basic tools to develop our main results. At the end, an example is given to demonstrate the validity of our main theoretical result. KEYWORDSBanach fixed point theorem, dynamic system, impulses, timescale, semilinear nonautonomous system, -Ulam-Hyers stability MSC CLASSIFICATION 34N05; 34G20; 35B35; 45J05 Math Meth Appl Sci. 2020;43:5097-5113. wileyonlinelibrary.com/journal/mma

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Section: Introductionmentioning

confidence: 99%

Stability analysis of first‐order impulsive nonautonomous system on timescales

Zada

Pervaiz

Shah

et al. 2020

Math Methods in App Sciences

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“…This stability analysis is very useful in many applications, such as numerical analysis, optimization, etc., where finding the exact solution is quite difficult. For a detailed study of Ulam type stability with different approaches, we recommend [51][52][53] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Analysis of coupled systems of implicit impulsive fractional differential equations involving Hadamard derivatives

et al. 2019

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We present some results on the existence, uniqueness and Hyers-Ulam stability to the solution of an implicit coupled system of impulsive fractional differential equations having Hadamard type fractional derivative. Using a fixed point theorem of Kransnoselskii's type, the existence and uniqueness results are obtained. Along these lines, different kinds of Hyers-Ulam stability are discussed. An example is given to illustrate the main theorems.

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“…In last few decades, the theory of impulsive fractional differential equations is well utilized in medicine, mechanical engineering, ecology, biology, and astronomy. There are some remarkable monographs, considering fractional differential equations with impulses.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear impulsive Langevin equation with mixed derivatives

Rizwan

Zada

2019

Math Methods in App Sciences

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In this paper, we consider a nonlocal boundary value problem of nonlinear impulsive Langevin equation with mixed derivatives. Some sufficient conditions are constructed to observe the existence, uniqueness, and generalized Ulam-Hyers-Rassias stability of our proposed model, with the help of Diaz-Margolis' fixed-point approach over generalized complete metric space. We give an example that supports our main result. KEYWORDS Caputo derivative, Langevin equation, noninstantaneous impulses, Ulam-Hyers-Rassias stability MSC CLASSIFICATION26A33; 34A08; 34B27 INTRODUCTIONAt Wisconsin University, Ulam raised a question about the stability of functional equations in the year 1940. The question of Ulam was under what conditions does there exist an additive mapping near an approximately additive mapping? 1 In 1941, Hyers was the first mathematician who gave partial answer to Ulam's question, 2 over Banach space. Afterwards, stability of such form is known as Ulam-Hyers stability. In 1978, Rassias 3 provided a remarkable generalization of the Ulam-Hyers stability of mappings by considering variables. For more information and different approaches about the topic, we refer the reader to the previous studies. [4][5][6][7][8][9][10][11][12][13][14][15][16] An equation of the form m d 2 x dt 2 = dx dt + (t) is called Langevin equation, introduced by Paul Langevin in 1908. Langevin equations are broadly used to describe stochastic problems in image processing, physics, astronomy, chemistry, defence system, electrical, and mechanical engineering. Brownian motion is well described by the Langevin equations when the random oscillation force is supposed to be Gaussian noise. For the removal of noise, mathematicians used fractional order differential equations, also it performs well in reducing the staircase effects as compared with ordinary differential equations. Thus, it is very important to study fractional Langevin equations, for more details, see previous studies. [17][18][19][20] Fractional order differential equations are the generalizations of the classical integer order differential equations. Fractional calculus has become a speedily developing area, and its applications can be found in diverse fields ranging from physical sciences, porous media, electrochemistry, economics, electromagnetics, medicine, and engineering to biological sciences. Progressively, fractional differential equations play a very important role in fields such as thermodynamics, statistical physics viscoelasticity, nonlinear oscillation of earthquakes, defence, optics, control, electrical circuits, signal processing, and astronomy. There are some outstanding articles that provide the main theoretical tools for the qualitative analysis of this research field and, at the same time, shows the interconnection as well as the distinction between integral models of classical and fractional differential equations, see previous studies. [21][22][23][24][25][26][27][28][29][30][31][32][33] Impulsive fractional differential equations are used to describe both physi...

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Stability Analysis of Multi-point Boundary Value Problem for Sequential Fractional Differential Equations with Non-instantaneous Impulses

Cited by 59 publications

References 33 publications

Stability analysis of first‐order impulsive nonautonomous system on timescales

Stability analysis of first‐order impulsive nonautonomous system on timescales

Analysis of coupled systems of implicit impulsive fractional differential equations involving Hadamard derivatives

Nonlinear impulsive Langevin equation with mixed derivatives

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