scholarly journals Fractional differential equation with uncertainty

2021 ◽  
Vol 23 (08) ◽  
pp. 181-185
Author(s):  
Karanveer Singh ◽  
◽  
R N Prajapati ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Fractional Order Differential Equation ◽  
First Order ◽  
Fractional Differential ◽  
First Order Differential Equations

We consider a fractional order differential equation with uncertainty and introduce the concept of solution. It goes beyond ordinary first-order differential equations and differential equations with uncertainty.

Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

scholarly journals Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 718 ◽  
Author(s):  
Emad R. Attia ◽  
Hassan A. El-Morshedy ◽  
Ioannis P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Oscillation Criteria ◽  
Lower Upper ◽  
First Order ◽  
Upper Limit ◽  
First Order Differential Equations

New sufficient criteria are obtained for the oscillation of a non-autonomous first order differential equation with non-monotone delays. Both recursive and lower-upper limit types criteria are given. The obtained results improve most recent published results. An example is given to illustrate the applicability and strength of our results.

scholarly journals A Refinement of Quasilinearization Method for Caputo's Sense Fractional-Order Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Coşkun Yakar ◽  
Ali Yakar
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Unique Solution ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Quasilinearization Method ◽  
Fractional Order Differential Equation ◽  
Fractional Order Differential Equations ◽  
Quasilinearization Technique

The method of the quasilinearization technique in Caputo's sense fractional-order differential equation is applied to obtain lower and upper sequences in terms of the solutions of linear Caputo's sense fractional-order differential equations. It is also shown that these sequences converge to the unique solution of the nonlinear Caputo's sense fractional-order differential equation uniformly and semiquadratically with less restrictive assumptions.

scholarly journals Fractional Differential Equations in Terms of Comparison Results and Lyapunov Stability with Initial Time Difference

2010 ◽  
Vol 2010 ◽  
pp. 1-16 ◽  
Author(s):  
Coşkun Yakar
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Time Difference ◽  
Initial Time ◽  
Qualitative Behavior ◽  
Fractional Order Differential Equation ◽  
Comparison Results ◽  
Caputo’S Derivative

The qualitative behavior of a perturbed fractional-order differential equation with Caputo's derivative that differs in initial position and initial time with respect to the unperturbed fractional-order differential equation with Caputo's derivative has been investigated. We compare the classical notion of stability to the notion of initial time difference stability for fractional-order differential equations in Caputo's sense. We present a comparison result which again gives the null solution a central role in the comparison fractional-order differential equation when establishing initial time difference stability of the perturbed fractional-order differential equation with respect to the unperturbed fractional-order differential equation.

scholarly journals Boundary value problems for a class of stochastic nonlinear fractional order differential equations

2020 ◽  
Vol 4 (2) ◽  
pp. 152-159
Author(s):  
McSylvester Ejighikeme Omaba ◽  
◽  
Louis O. Omenyi ◽  
Keyword(s):  
Differential Equation ◽  
Boundary Value Problems ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Gaussian White Noise ◽  
Type Equation ◽  
Fractional Order Differential Equation ◽  
Fractional Differential Operator ◽  
Fractional Differential

Consider a class of two-point Boundary Value Problems (BVP) for a stochastic nonlinear fractional order differential equation \(D^\alpha u(t)=\lambda\sqrt{I^\beta[\sigma^2(t,u(t))]}\dot{w}(t)\  ,0< t< 1\) with boundary conditions \(u(0)=0,\,\,u'(0)=u'(1)=0,\) where \(\lambda>0\) is a level of the noise term, \(\sigma:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}\) is continuous, \(\dot{w}(t)\) is a generalized derivative of Wiener process (Gaussian white noise), \(D^\alpha\) is the Riemann-Liouville fractional differential operator of order \(\alpha\in (3,4)\) and \(I^\beta,\,\,\beta>0\) is a fractional integral operator. We formulate the solution of the equation via a stochastic Volterra-type equation and investigate its existence and uniqueness under some precise linearity conditions using contraction fixed point theorem. A case of the above BVP for a stochastic nonlinear second order differential equation for \(\alpha=2\) and \(\beta=0\) with \(u(0)=u(1)=0\) is also studied.

A Series of New Chaotic Attractors via Switched Linear Integer Order and Fractional Order Differential Equations

2015 ◽  
Vol 25 (01) ◽  
pp. 1550008 ◽  
Author(s):  
Fei Xu ◽  
Ross Cressman ◽  
Xiao-Bao Shu ◽  
Xinzhi Liu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Chaotic Attractors ◽  
Fractional Order Systems ◽  
Fractional Order Differential Equation ◽  
Integer Order ◽  
Fractional Order Differential Equations ◽  
Dynamical Behaviors

This article investigates the design of a series of new chaotic attractors. A switching control with different switching surfaces is designed to link two systems of linear integer order differential equations. Under such control, the linked systems have rich dynamical behaviors such as chaos. We also investigate the dynamical behaviors of the corresponding linear fractional order differential equation systems with switching controls. It is shown that such fractional order systems have chaotic behaviors as well.

scholarly journals Solution of Extraordinary Differential Equations with Physical Reasoning by Obtaining Modal Reaction Series

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
Shantanu Das
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Decomposition Method ◽  
Physical Reality ◽  
Physical Principle ◽  
Time Frequency ◽  
Fractional Differential ◽  
Initial States

Mathematical modeling of many engineering and physics problem leads to extraordinary differential equations like Nonlinear, Delayed, and Fractional Order. An effective method is required to analyze the mathematical model which provides solutions conforming to physical reality. A Fractional Differential Equation (FDE), where the leading differential operator is Riemann-Liouvelli (RL) type requires fractional order initial states which are sometimes hard to physically relate. Therefore, we must be able to solve these extraordinary systems, in space, time, frequency, area, volume, with physical reality conserved. Extra Ordinary Differential equation Systems and its solution, with Physical Principle, of action-reaction and equivalent mathematical decomposition method, are obtained as an aid for Physicists and Engineers to tackle the process dynamics with ease. This reactions-chain generates internal modes from zeroth mode reaction to first mode second mode and to infinite modes; instantaneously in parallel time or space-scales; and the sum of all these modes gives entire system reaction. This modal reaction as explained by physics theory exactly matches the principle of Adomian Decomposition Method (ADM). Fractional Differential Equation (FDE) with Riemann-Liouvelli formulation linear and non-linear is solved as per ADM. In this formulation of FDE by RL method it is found that there is no need to worry about the fractional initial states; instead one can use integer order initial states (the conventional ones) to arrive at solution of FDE. This new finding too is highlighted in this paper-along with several other problems to give physical insight to the solution of extraordinary differential equation systems. This way one gets insight to Physics of General Differential Equation Systems-and its solution-by Physical Principle and equivalent mathematical decomposition method. This facilitates ease in modeling.

scholarly journals Mathematical modelling of the mass-spring-damper system - A fractional calculus approach

2012 ◽  
Vol 22 (5) ◽  
pp. 5-11 ◽  
Author(s):  
José Francisco Gómez Aguilar ◽  
Juan Rosales García ◽  
Jesus Bernal Alvarado ◽  
Manuel Guía
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Mathematical Modelling ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Time Derivative ◽  
System A ◽  
Fractional Differential ◽  
Mass Spring ◽  
Derivatives Of

In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter char­acterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

scholarly journals Solution of Fractional Differential Equations Utilizing Symmetric Contraction

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Aftab Hussain
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fractional Differential Equations ◽  
Metric Space ◽  
Metric Spaces ◽  
Order Differential Equation ◽  
Fractional Order Differential Equation ◽  
Continuous Mappings ◽  
Fractional Differential ◽  
Fixed Point Technique

The aim of this paper is to present another family of fractional symmetric α - η -contractions and build up some new results for such contraction in the context of ℱ -metric space. The author derives some results for Suzuki-type contractions and orbitally T -complete and orbitally continuous mappings in ℱ -metric spaces. The inspiration of this paper is to observe the solution of fractional-order differential equation with one of the boundary conditions using fixed-point technique in ℱ -metric space.

DIFFERENTIAL EQUATIONS IN A HYPERCOMPLEX SYSTEM

2020 ◽  
Vol 69 (1) ◽  
pp. 7-11
Author(s):  
A.K. Abirov ◽  
◽  
N.K. Shazhdekeeva ◽  
T.N. Akhmurzina ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Inhomogeneous System ◽  
Homogeneous Solutions ◽  
First Order ◽  
Zero Divisors ◽  
Hypercomplex System ◽  
Independent Variable ◽  
The Right

The article considers the problem of solving an inhomogeneous first-order differential equation with a variable with a constant coefficient in a hypercomplex system. The structure of the solution in different cases of the right-hand side of the differential equation is determined. The structure of solving the equation in the case of the appearance of zero divisors is shown. It turns out that when the component of a hypercomplex function is a polynomial of an independent variable, the differential equation turns into an inhomogeneous system of real variables from n equations and its solution is determined by certain methods of the theory of differential equations. Thus, obtaining analytically homogeneous solutions of inhomogeneous differential equations in a hypercomplex system leads to an increase in the efficiency of modeling processes in various fields of science and technology.

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