Generalized fractional order Chebyshev wavelets for solving nonlinear fractional delay-type equations

2019 ◽  
Vol 17 (03) ◽  
pp. 1950014 ◽  
Author(s):  
Amir Saeed ◽  
Umer Saeed
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Operational Matrices ◽  
Chebyshev Wavelets ◽  
Method Of Steps ◽  
Type Differential Equation ◽  
Fractional Delay ◽  
Delay Differential ◽  
Nonlinear Delay ◽  
Quasilinearization Technique

In this paper, we develop the generalized fractional order Chebyshev wavelets (GFCWs) from generalized fractional order of Chebyshev polynomials. The operational matrices for the presented wavelets are constructed and derived. We also proposed a technique by utilizing the GFCWs, the method of steps and quasilinearization technique for solving nonlinear fractional delay-type differential equations. According to the development, the method of step is used to transform the fractional nonlinear delay-type differential equation to a fractional nonlinear non-delay differential equation, and then apply the quasilinearization technique to discretize the obtained nonlinear equation. The GFCW method is utilized in each iteration of quasilinearization method for the improvement of solution. We perform the error analysis for the proposed technique. Procedure of implementation for the present method is also provided. Numerical simulation of some examples will be presented to demonstrate the benefits of computing with the present technique over existing methods in literature.

scholarly journals Hermite Wavelet Method for Fractional Delay Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Umer Saeed ◽  
Mujeeb ur Rehman
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Wavelet Method ◽  
Numerical Examples ◽  
Method Of Steps ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

We proposed a method by utilizing method of steps and Hermite wavelet method, for solving the fractional delay differential equations. This technique first converts the fractional delay differential equation to a fractional nondelay differential equation and then applies the Hermite wavelet method on the obtained fractional nondelay differential equation to find the solution. Several numerical examples are solved to show the applicability of the proposed method.

scholarly journals Sufficient and necessary conditions for oscillation of linear fractional-order delay differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Qiong Meng ◽  
Zhen Jin ◽  
Guirong Liu
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Multiple Delays ◽  
All Solutions ◽  
Sufficient And Necessary Conditions ◽  
Delay Differential ◽  
T Tau ◽  
Sufficient And Necessary Condition

AbstractThis paper studies the linear fractional-order delay differential equation $$ {}^{C}D^{\alpha }_{-}x(t)-px(t-\tau )= 0, $$ D − α C x ( t ) − p x ( t − τ ) = 0 , where $0<\alpha =\frac{\text{odd integer}}{\text{odd integer}}<1$ 0 < α = odd integer odd integer < 1 , $p, \tau >0$ p , τ > 0 , ${}^{C}D_{-}^{\alpha }x(t)=-\Gamma ^{-1}(1-\alpha )\int _{t}^{\infty }(s-t)^{- \alpha }x'(s)\,ds$ D − α C x ( t ) = − Γ − 1 ( 1 − α ) ∫ t ∞ ( s − t ) − α x ′ ( s ) d s . We obtain the conclusion that $$ p^{1/\alpha } \tau >\alpha /e $$ p 1 / α τ > α / e is a sufficient and necessary condition of the oscillations for all solutions of Eq. (*). At the same time, some sufficient conditions are obtained for the oscillations of multiple delays linear fractional differential equation. Several examples are given to illustrate our theorems.

Numerical Dynamics of Nonstandard Finite Difference Method for Nonlinear Delay Differential Equation

2018 ◽  
Vol 28 (11) ◽  
pp. 1850133 ◽  
Author(s):  
Xiaolan Zhuang ◽  
Qi Wang ◽  
Jiechang Wen
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Delay Differential Equation ◽  
Difference Method ◽  
Discrete System ◽  
Delay Differential ◽  
Nonlinear Delay Differential Equation ◽  
Nonlinear Delay ◽  
Nonstandard Finite Difference

In this paper, we study the dynamics of a nonlinear delay differential equation applied in a nonstandard finite difference method. By analyzing the numerical discrete system, we show that a sequence of Neimark–Sacker bifurcations occur at the equilibrium as the delay increases. Moreover, the existence of local Neimark–Sacker bifurcations is considered, and the direction and stability of periodic solutions bifurcating from the Neimark–Sacker bifurcation of the discrete model are determined by the Neimark–Sacker bifurcation theory of discrete system. Finally, some numerical simulations are adopted to illustrate the corresponding theoretical results.

BOUNDARY VALUE PROBLEM WITH SHIFT FOR A FRACTIONAL ORDER DELAY DIFFERENTIAL EQUATION

2019 ◽  
pp. 16-25
Author(s):  
М.Г. Мажгихова
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Fractional Order ◽  
Delay Differential Equation ◽  
Solvability Condition ◽  
Boundary Value ◽  
Explicit Representation ◽  
Existence And Uniqueness Theorem ◽  
The Green Function ◽  
Delay Differential

В работе доказана теорема существования и единственности решения краевой задачи со смещением для дифференциального уравнения дробного порядка с запаздывающим аргументом. Решение задачи выписано в терминах функции Грина. Получено условие однозначной разрешимости и показано, что оно может нарушаться только конечное число раз. In this paper we prove existence and uniqueness theorem to a boundary value problem with shift for a fractional order ordinary delay differential equation. The solution of the problem is written out in terms of the Green function. We find an explicit representation for solvability condition and show that it may only be violated a finite number of times

A Necessary and Sufficient Condition for the Oscillation of an Even Order Nonlinear Delay Differential Equation

1973 ◽  
Vol 25 (5) ◽  
pp. 1078-1089 ◽  
Author(s):  
Bhagat Singh
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Oscillatory Behavior ◽  
Necessary And Sufficient Condition ◽  
Even Order ◽  
Image Position ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Nonlinear Delay Differential Equation ◽  
Nonlinear Delay

In this paper we study the oscillatory behavior of the even order nonlinear delay differential equation(1)where(i) denotes the order of differentiation with respect to t. The delay terms τi σi are assumed to be real-valued, continuous, non-negative, non-decreasing and bounded by a common constant M on the half line (t0, + ∞ ) for some t0 ≧ 0.

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