scholarly journals Mathematical Analysis of Oxygen Uptake Rate in Continuous Process under Caputo Derivative

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 675
Author(s):  
Rubayyi T. Alqahtani ◽  
Abdullahi Yusuf ◽  
Ravi P. Agarwal
Keyword(s):  
Fractional Derivatives ◽  
Oxygen Uptake Rate ◽  
Continuous Process ◽  
Equilibrium Stability ◽  
Treatment Model ◽  
Classical Version ◽  
Fractional Differential Operator ◽  
Non Local ◽  
Fractional Differential ◽  
Steady State Solutions

In this paper, the wastewater treatment model is investigated by means of one of the most robust fractional derivatives, namely, the Caputo fractional derivative. The growth rate is assumed to obey the Contois model, which is often used to model the growth of biomass in wastewaters. The characteristics of the model under consideration are derived and evaluated, such as equilibrium, stability analysis, and steady-state solutions. Further, important characteristics of the fractional wastewater model allow us to understand the dynamics of the model in detail. To this end, we discuss several important analyses of the fractional variant of the model under consideration. To observe the efficiency of the non-local fractional differential operator of Caputo over its counter-classical version, we perform numerical simulations.

scholarly journals An Inverse Source Non-local Problem for a Mixed Type Equation with a Caputo Fractional Differential Operator

2017 ◽  
Vol 7 (2) ◽  
pp. 417-438 ◽  
Author(s):  
E. Karimov ◽  
N. Al-Salti ◽  
S. Kerbal
Keyword(s):  
Mixed Type ◽  
Type Equation ◽  
Integral Form ◽  
Rectangular Domain ◽  
Regularity Conditions ◽  
Mixed Type Equation ◽  
Fractional Differential Operator ◽  
Non Local ◽  
Fractional Differential ◽  
The Given

AbstractWe consider the unique solvability of an inverse-source problem with integral transmitting condition for a time-fractional mixed type equation in rectangular domain where the unknown source term depends only on the space variable. The solution is based on a series expansion using a bi-orthogonal basis in space, corresponding to a non-self-adjoint boundary value problem. Under certain regularity conditions on the given data, we prove the uniqueness and existence of the solution for the given problem. The influence of the transmitting condition on the solvability of the problem is also demonstrated. Two different transmitting conditions are considered — viz. a full integral form and a special case. In order to simplify the bulky expressions appearing in the proof of our main result, we establish a new property of the recently introduced Mittag-Leffler type function in two variables.

scholarly journals Mathematical model of SIR epidemic system (COVID-19) with fractional derivative: stability and numerical analysis

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Rubayyi T. Alqahtani
Keyword(s):  
Steady State ◽  
Reproduction Number ◽  
Rate Function ◽  
Backward Bifurcation ◽  
Sir Epidemic ◽  
General Incidence Rate ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Steady State Solutions ◽  
Fractional Orders

AbstractIn this paper, we study and analyze the susceptible-infectious-removed (SIR) dynamics considering the effect of health system. We consider a general incidence rate function and the recovery rate as functions of the number of hospital beds. We prove the existence, uniqueness, and boundedness of the model. We investigate all possible steady-state solutions of the model and their stability. The analysis shows that the free steady state is locally stable when the basic reproduction number $R_{0}$ R 0 is less than unity and unstable when $R_{0} > 1$ R 0 > 1 . The analysis shows that the phenomenon of backward bifurcation occurs when $R_{0}<1$ R 0 < 1 . Then we investigate the model using the concept of fractional differential operator. Finally, we perform numerical simulations to illustrate the theoretical analysis and study the effect of the parameters on the model for various fractional orders.

On the Influence of Fractional Derivative on Chaos Control of a New Fractional-Order Hyperchaotic System

2020 ◽  
pp. 115-132
Author(s):  
Ahmed Ezzat Mohamed Matouk
Keyword(s):  
Differential Operator ◽  
Fractional Order ◽  
Differential Operators ◽  
Equilibrium Points ◽  
Hyperchaotic System ◽  
Fractional Differential Operator ◽  
Quadratic Nonlinearities ◽  
Potential Applications ◽  
Non Local ◽  
Fractional Differential

The non-local fractional differential operators have potential applications in many fields of science and technology but especially in the field of dynamical systems. This chapter introduces a new hyperchaotic dynamical system involving non-local fractional differential operator with singular kernel (the Caputo type). The system involves three quadratic nonlinearities and also three equilibrium points. Existence of chaotic and hyperchaotic attractors has been illustrated. Based on Matouk's stability theory of four-dimensional fractional-order systems, the influence of the fractional differential operator on stabilizing the proposed system to its three steady states has been shown. Numerical results have been provided to verify the theoretical analysis. This kind of study is expected to add useful applications to chaos-based secure communications and text encryption.

scholarly journals New Aspects of Bloch Model Associated with Fractal Fractional Derivatives

2021 ◽  
Vol 10 (1) ◽  
pp. 323-342
Author(s):  
Ali Akgül ◽  
Ishfaq Ahmad Mallah ◽  
Subhash Alha
Keyword(s):  
Fractional Derivatives ◽  
Polynomial Interpolation ◽  
The Novel ◽  
Bloch Equations ◽  
Decay Function ◽  
Model Complex ◽  
Non Local ◽  
Fractional Differential ◽  
Novel Concept ◽  
Real World Problems

Abstract To model complex real world problems, the novel concept of non-local fractal-fractional differential and integral operators with two orders (fractional order and fractal dimension) have been used as mathematical tools in contrast to classical derivatives and integrals. In this paper, we consider Bloch equations with fractal-fractional derivatives. We find the general solutions for components of magnetization ℳ = (Mu , Mv , Mw ) by using descritization and Lagrange's two step polynomial interpolation. We analyze the model with three different kernels namely power function, exponential decay function and Mittag-Leffler type function. We provide graphical behaviour of magnetization components ℳ = (Mu , Mv , Mw ) on different orders. The examination of Bloch equations with fractal-fractional derivatives show new aspects of Bloch equations.

scholarly journals A study on multiterm hybrid multi-order fractional boundary value problem coupled with its stability analysis of Ulam–Hyers type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ahmed Nouara ◽  
Abdelkader Amara ◽  
Eva Kaslik ◽  
Sina Etemad ◽  
Shahram Rezapour ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Boundary Value Problem ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Research Work ◽  
Existence Results ◽  
Fractional Boundary Value Problem ◽  
Numerical Examples ◽  
Fractional Derivatives And Integrals ◽  
Fractional Differential

AbstractIn this research work, a newly-proposed multiterm hybrid multi-order fractional boundary value problem is studied. The existence results for the supposed hybrid fractional differential equation that involves Riemann–Liouville fractional derivatives and integrals of multi-orders type are derived using Dhage’s technique, which deals with a composition of three operators. After that, its stability analysis of Ulam–Hyers type and the relevant generalizations are checked. Some illustrative numerical examples are provided at the end to illustrate and validate our obtained results.

On the kinetics of Hadamard-type fractional differential systems

2020 ◽  
Vol 23 (2) ◽  
pp. 553-570 ◽  
Author(s):  
Li Ma
Keyword(s):  
Differential Operator ◽  
Numerical Simulations ◽  
Type Inequality ◽  
The Other ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Kinetics Of ◽  
Theoretical Results

AbstractThis paper is devoted to the investigation of the kinetics of Hadamard-type fractional differential systems (HTFDSs) in two aspects. On one hand, the nonexistence of non-trivial periodic solutions for general HTFDSs, which are considered in some functional spaces, is proved and the corresponding eigenfunction of Hadamard-type fractional differential operator is also discussed. On the other hand, by the generalized Gronwall-type inequality, we estimate the bound of the Lyapunov exponents for HTFDSs. In addition, numerical simulations are addressed to verify the obtained theoretical results.

scholarly journals On a fractional-order p-Laplacian boundary value problem at resonance on the half-line with two dimensional kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
O. F. Imaga ◽  
S. A. Iyase
Keyword(s):  
Differential Operator ◽  
Boundary Value Problem ◽  
Fractional Order ◽  
Boundary Value ◽  
Coincidence Degree ◽  
Half Line ◽  
At Resonance ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Problem At Resonance

AbstractIn this work, we consider the solvability of a fractional-order p-Laplacian boundary value problem on the half-line where the fractional differential operator is nonlinear and has a kernel dimension equal to two. Due to the nonlinearity of the fractional differential operator, the Ge and Ren extension of Mawhin’s coincidence degree theory is applied to obtain existence results for the boundary value problem at resonance. Two examples are used to validate the established results.

scholarly journals On resonant mixed Caputo fractional differential equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Assia Guezane-Lakoud ◽  
Adem Kılıçman
Keyword(s):  
Differential Equation ◽  
Fractional Derivatives ◽  
Existence Of Solutions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Caputo Fractional Derivatives ◽  
At Resonance ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
Problem At Resonance

Abstract The purpose of this study is to discuss the existence of solutions for a boundary value problem at resonance generated by a nonlinear differential equation involving both right and left Caputo fractional derivatives. The proofs of the existence of solutions are mainly based on Mawhin’s coincidence degree theory. We provide an example to illustrate the main result.

Sign in / Sign up

Export Citation Format

Share Document