Numerical Analysis of Fractional Neutral Functional Differential Equations Based on Generalized Volterra-Integral Operators

2017 ◽  
Vol 12 (3) ◽  
Author(s):  
Xiao-Li Ding ◽  
Juan J. Nieto
Keyword(s):  
Differential Equation ◽  
Integral Operator ◽  
Differential Equations ◽  
Numerical Method ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Waveform Relaxation ◽  
Neutral Functional Differential Equation ◽  
Neutral Functional Differential Equations ◽  
Functional Differential

We use waveform relaxation (WR) method to solve numerically fractional neutral functional differential equations and mainly consider the convergence of the numerical method with the help of a generalized Volterra-integral operator associated with the Mittag–Leffler function. We first give some properties of the integral operator. Using the proposed properties, we establish the convergence condition of the numerical method. Finally, we provide a new way to prove the convergence of waveform relaxation method for integer-order neutral functional differential equation, which is a special case of fractional neutral functional differential equation. Compared to the existing proof in the literature, our proof is concise and original.

scholarly journals Asymptotic behaviour of solutions to neutral functional differential equations

1989 ◽  
Vol 40 (3) ◽  
pp. 345-355
Author(s):  
Shaozhu Chen ◽  
Qingguang Huang
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Functional Differential Equation ◽  
Necessary Conditions ◽  
Functional Differential Equations ◽  
Neutral Functional Differential Equation ◽  
Neutral Functional Differential Equations ◽  
Functional Differential

Sufficient or necessary conditions are established so that the neutral functional differential equation [x(t) − G(t, xt)]″ + F(t, xt) = 0 has a solution which is asymptotic to a given solution of the related difference equation x(t) = G(t, xt) + a + bt, where a and b are constants.

scholarly journals Oscillation of Second-Order Neutral Functional Differential Equations with Mixed Nonlinearities

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Shurong Sun ◽  
Tongxing Li ◽  
Zhenlai Han ◽  
Yibing Sun
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Second Order ◽  
Neutral Functional Differential Equation ◽  
Neutral Functional Differential Equations ◽  
Mixed Nonlinearities ◽  
Functional Differential

We study the following second-order neutral functional differential equation with mixed nonlinearities(r(t)|(u(t)+p(t)u(t-σ))'|α-1(u(t)+p(t)u(t-σ))′)′+q0(t)|u(τ0(t))|α-1u(τ0(t))+q1(t)|u(τ1(t))|β-1u(τ1(t))+q2(t)|u(τ2(t))|γ-1u(τ2(t))=0, whereγ>α>β>0,∫t0∞(1/r1/α(t))dt<∞. Oscillation results for the equation are established which improve the results obtained by Sun and Meng (2006), Xu and Meng (2006), Sun and Meng (2009), and Han et al. (2010).

Positive periodic solutions for singular high-order neutral functional differential equations

2018 ◽  
Vol 68 (2) ◽  
pp. 379-396 ◽  
Author(s):  
Fanchao Kong ◽  
Zhiguo Luo ◽  
Shiping Lu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
High Order ◽  
Positive Periodic Solutions ◽  
Neutral Functional Differential Equation ◽  
Neutral Functional Differential Equations ◽  
Small Force ◽  
Functional Differential

Abstract In this paper, we establish new results on the existence of positive periodic solutions for the following high-order neutral functional differential equation (NFDE) $$\begin{array}{} (x(t)-cx(t-\sigma)) ^{(2m)}+f(x(t)) x'(t)+g(t,x(t-\delta))=e(t). \end{array}$$ The interesting thing is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the corresponding ones known in the literature. Two examples are given to illustrate the effectiveness of our results.

Asymptotic Behavior for Nonoscillatory Solutions of Second Order Nonlinear Functional Differential Equation

2012 ◽  
Vol 616-618 ◽  
pp. 2137-2141
Author(s):  
Zhi Min Luo ◽  
Bei Fei Chen
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Nonoscillatory Solutions ◽  
Nonlinear Functional ◽  
Nonlinear Functional Differential Equation ◽  
Functional Differential

This paper studied the asymptotic behavior of a class of nonlinear functional differential equations by using the Bellman-Bihari inequality. We obtain results which extend and complement those in references. The results indicate that all non-oscillatory continuable solutions of equation are asymptotic to at+b as under some sufficient conditions, where a,b are real constants. An example is provided to illustrate the application of the results.

scholarly journals Mild Solutions of Neutral Stochastic Partial Functional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
T. E. Govindan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Local Lipschitz Condition ◽  
Functional Differential ◽  
Special Case ◽  
Partial Functional Differential Equation

This paper studies the existence and uniqueness of a mild solution for a neutral stochastic partial functional differential equation using a local Lipschitz condition. When the neutral term is zero and even in the deterministic special case, the result obtained here appears to be new. An example is included to illustrate the theory.

scholarly journals Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jun Zhou ◽  
Jun Shen
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Positive Solutions ◽  
Functional Differential Equation ◽  
Continuous Dependence ◽  
Mixed Type ◽  
Functional Differential Equations ◽  
State Dependence ◽  
Functional Differential ◽  
Iterative Functional Differential Equation

<p style='text-indent:20px;'>In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for the general iterative functional differential equation <inline-formula><tex-math id="M1">\begin{document}$ \dot{x}(t) = f(t,x(t),x^{[2]}(t),...,x^{[n]}(t)). $\end{document}</tex-math></inline-formula> As <inline-formula><tex-math id="M2">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula>, this equation can be regarded as a mixed-type functional differential equation with state-dependence <inline-formula><tex-math id="M3">\begin{document}$ \dot{x}(t) = f(t,x(t),x(T(t,x(t)))) $\end{document}</tex-math></inline-formula> of a special form but, being a nonlinear operator, <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula>-th order iteration makes more difficulties in estimation than usual state-dependence. Then we apply our results to the existence, uniqueness, boundedness, asymptotics and continuous dependence of solutions for the mixed-type functional differential equation. Finally, we present two concrete examples to show the boundedness and asymptotics of solutions to these two types of equations respectively.</p>

scholarly journals Oscillation properties of nonlinear neutral differential equations ofnth order

2004 ◽  
Vol 2004 (71) ◽  
pp. 3893-3900
Author(s):  
T. Candan ◽  
R. S. Dahiya
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Asymptotic Properties ◽  
Neutral Functional Differential Equation ◽  
Neutral Differential Equations ◽  
Functional Differential ◽  
Oscillation Properties

We consider the nonlinear neutral functional differential equation[r(t)[x(t)+∫abp(t,μ)x(τ(t,μ))dμ](n−1)]′+δ∫cdq(t,ξ)f(x(σ(t,ξ)))dξ=0with continuous arguments. We will develop oscillatory and asymptotic properties of the solutions.

scholarly journals Study of periodic and nonnegative periodic solutions of nonlinear neutral functional differential equations via fixed points

2016 ◽  
Vol 8 (2) ◽  
pp. 255-270
Author(s):  
Mouataz Billah Mesmouli ◽  
Abdelouaheb Ardjouni ◽  
Ahcene Djoudi
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Neutral Functional Differential Equations ◽  
Completely Continuous ◽  
Continuous Map ◽  
Functional Differential ◽  
Large Contraction

Abstract In this paper, we study the existence of periodic and non-negative periodic solutions of the nonlinear neutral differential equation $${{\rm{d}} \over {{\rm{dt}}}}{\rm{x}}({\rm{t}}) = - {\rm{a}}\;({\rm{t}})\;{\rm{h}}\;({\rm{x}}\;({\rm{t}})) + {{\rm{d}} \over {{\rm{dt}}}}{\rm{Q}}\;({\rm{t}},\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))) + {\rm{G}}\;({\rm{t}},\;{\rm{x}}({\rm{t}}),\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))).$$ We invert this equation to construct a sum of a completely continuous map and a large contraction which is suitable for applying the modificatition of Krasnoselskii’s theorem. The Caratheodory condition is used for the functions Q and G.

scholarly journals Oscillation of a functional differential equation arising from an industrial problem

1978 ◽  
Vol 26 (3) ◽  
pp. 323-329 ◽  
Author(s):  
Hiroshi Onose
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
First Order ◽  
Functional Differential ◽  
Image Position ◽  
Industrial Problem

AbstractIn the last few years, the oscillatory behavior of functional differential equations has been investigated by many authors. But much less is known about the first-order functional differential equations. Recently, Tomaras (1975b) considered the functional differential equation and gave very interesting results on this problem, namely the sufficient conditions for its solutions to oscillate. The purpose of this paper is to extend and improve them, by examining the more general functional differential equation

Non-linear functional differential equations and abstract integral equations

1979 ◽  
Vol 84 (1-2) ◽  
pp. 71-91 ◽  
Author(s):  
F. Kappel ◽  
W. Schappacher
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Space Setting ◽  
Non Linear ◽  
Functional Differential ◽  
Order Spline ◽  
The Given

SynopsisThe equivalence between solutions of functional differential equations and an abstract integral equation is investigated. Using this result we derive a general approximation result in the state space C and consider as an example approximation by first order spline functions. During the last twenty years C1-semigroups of linear transformations have played an important role in the theory of linear autonomous functional differential equations (cf. for instance the discussion in [9, Section 7.7]). Applications of non-linear semigroup theory to functional differential equations are rather recent beginning with a paper by Webb [17]. Since then a considerable number of papers deal with problems in this direction. A common feature of the majority of these papers is that as a first step with the functional differential equation there is associated a non-linear operator A in a suitable Banach-space. Then appropriate conditions are imposed on the problem such that the conditions of the Crandall-Liggett-Theorem [5] hold for the operator A. This gives a non-linear semigroup. Finally the connection of this semigroup tothe solutions of the original differential equation has to be investigated [c.f. 8, 15, 18]. To solve thislast problem in general is the most difficult part of this approach.In the present paper we consider the given functional differential equation as a perturbation of the simple equationx = 0. The solutions of this equation generate a very simple C1-semigroup. The solutions of the original functional differential equation generate solutions of an integral equation which is the variation of constants formula for the abstract Cauchy problem associated with the equation x = 0. Under very mild conditions we can prove a one-to-one correspondence between solutions of the given functional differential equation and solutions of the integral equation in the Lp-space setting. In the C-space setting the integral equation inthe state space has to be replaced by a ‘pointwise’ integral equation. Using the pointwise integral equation together with a theorem which guarantees continuous dependence of fixed points on parameters we show under rather weak hypotheses that the original functional differential equation can be approximated by a sequence of ordinary differential equations. Using 1st order spline functions we finally get results which are very similar to those obtained in [1 and 11] in the L2-space setting.

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