Existence of solution for a fractional‐order Lotka‐Volterra reaction‐diffusion model with Mittag‐Leffler kernel

Khan, Hasib; Li, Yongjin; Khan, Aftab; Khan, Aziz

doi:10.1002/mma.5590

Cited by 78 publications

(29 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Among the vital areas in the concept of noninteger-order differential equations is the concept of existence and the uniqueness of solutions in a dynamical system. Recently, the theory has attracted many researchers ' attention [36]. By means of fixed point theorem, we report the existence and uniqueness of (5).…”

Section: Existence and Uniqueness Resultsmentioning

confidence: 99%

Caputo SIR Model for COVID-19 under Optimized Fractional Order

Ullah

Băleanu

2020

Preprint

View full text Add to dashboard Cite

Everyone is talking about coronavirus from last couple of months due to its exponential spread throughout the globe. Lives have become paralyzed and as many as 180 countries are so far affected with 928,287 (14 September, 2020) deaths within couple of months. Ironically, 29,185,779 are still active cases. Having seen such drastic situation, a relatively simple epidemiological SIR model with Caputo derivative is suggested unlike more sophisticated models being proposed nowadays in the current literature. Major aim of the present research study is to look for possibilities and extents to which the SIR model fits the real data for the cases chosen from 01 April to 15 March, 2020, Pakistan. To further analyze qualitative behavior of the Caputo SIR model, uniqueness conditions under the Banach contraction principle are discussed and stability analysis with basic reproduction number is investigated using Ulam-Hyers and its generalized version. Best parameters have been obtained via nonlinear least-squares curve fitting technique. The infectious compartment of the Caputo SIR model fits the real data better than classical version of the SIR model [4]. Average absolute relative error under the Caputo operator is about 48% smaller than the one obtained in the classical case (ν = 1). Time series and 3D contour plots offer social distancing to be the most effective measure to control the epidemic.

show abstract

Section: Existence and Uniqueness Resultsmentioning

confidence: 99%

Caputo SIR Model for COVID-19 under Optimized Fractional Order

Ullah

Băleanu

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…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 …”

Section: Introductionmentioning

confidence: 99%

Nonlinear impulsive Langevin equation with mixed derivatives

Rizwan

Zada

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

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...

show abstract

“…Integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermoelasticity, underground water flow, and population dynamics. For a detailed description of some recent work on the integral boundary conditions, we refer the reader to some recent papers [33][34][35] and the references therein [36][37][38][39].…”

Section: Introductionmentioning

confidence: 99%

On the Generalization of a Solution for a Class of Integro-Differential Equations with Nonseparated Integral Boundary Conditions

Xing

Jiao

Liu

2020

Mathematical Problems in Engineering

View full text Add to dashboard Cite

In this paper, the existence and uniqueness results of the generalization nonlinear fractional integro-differential equations with nonseparated type integral boundary conditions are investigated. A natural formula of solutions is derived and some new existence and uniqueness results are obtained under some conditions for this class of problems by using standard fixed point theorems and Leray-Schauder degree theory, which extend and supplement some known results. Some examples are discussed for the illustration of the main work. ⎧ ⎨ ⎩(1)

show abstract

Existence of solution for a fractional‐order Lotka‐Volterra reaction‐diffusion model with Mittag‐Leffler kernel

Cited by 78 publications

References 37 publications

Caputo SIR Model for COVID-19 under Optimized Fractional Order

Caputo SIR Model for COVID-19 under Optimized Fractional Order

Nonlinear impulsive Langevin equation with mixed derivatives

On the Generalization of a Solution for a Class of Integro-Differential Equations with Nonseparated Integral Boundary Conditions

Contact Info

Product

Resources

About