Differential Equations with Degenerate Operators at the Derivative Depending on an Unknown Function

2018 ◽  
Vol 233 (6) ◽  
pp. 875-904
Author(s):  
B. V. Loginov ◽  
Yu. B. Rousak ◽  
L. R. Kim-Tyan
Keyword(s):  
Differential Equations ◽  
Unknown Function ◽  
Degenerate Operators

scholarly journals Numerical and Analytical Study for Fourth-Order Integro-Differential Equations Using a Pseudospectral Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
N. H. Sweilam ◽  
M. M. Khader ◽  
W. Y. Kota
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Upper Bound ◽  
Unknown Function ◽  
Approximate Formula ◽  
Analytical Study ◽  
Pseudospectral Method ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Collocation Points

A numerical method for solving fourth-order integro-differential equations is presented. This method is based on replacement of the unknown function by a truncated series of well-known shifted Chebyshev expansion of functions. An approximate formula of the integer derivative is introduced. The introduced method converts the proposed equation by means of collocation points to system of algebraic equations with shifted Chebyshev coefficients. Thus, by solving this system of equations, the shifted Chebyshev coefficients are obtained. Special attention is given to study the convergence analysis and derive an upper bound of the error of the presented approximate formula. Numerical results are performed in order to illustrate the usefulness and show the efficiency and the accuracy of the present work.

scholarly journals On the integrability problem for systems of partial differential equations in one unknown function, I

2019 ◽  
Vol 11 (4) ◽  
pp. 35-71 ◽  
Author(s):  
Antonio Kumpera
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Unknown Function ◽  
Contact Structures ◽  
Partial Differential ◽  
Practical Applications ◽  
First Order ◽  
Integrability Problem ◽  
Integration Problem ◽  
In The Beginning

We discuss the integration problem for systems of partial differential equations in one unknown function and special attention is given to the first order systems. The Grassmannian contact structures are the basic setting for our discussion and the major part of our considerations inquires on the nature of the Cauchy characteristics in view of obtaining the necessary criteria that assure the existence of solutions. In all the practical applications of partial differential equations, what is mostly needed and what in fact is hardest to obtains are the solutions of the system or, occasionally, some specific solutions. This work is based on four most enlightening Mémoires written by Élie Cartan in the beginning of the last century.

scholarly journals Analytic Solutions of an Iterative Functional Differential Equation near Resonance

2009 ◽  
Vol 2009 ◽  
pp. 1-14 ◽  
Author(s):  
Tongbo Liu ◽  
Hong Li
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Unknown Function ◽  
Analytic Solutions ◽  
Complex Field ◽  
Second Order Differential Equations ◽  
Near Resonance ◽  
Functional Differential ◽  
Iterative Functional Differential Equation

We investigate the existence of analytic solutions of a class of second-order differential equations involving iterates of the unknown function in the complex field . By reducing the equation with the Schröder transformation to the another functional differential equation without iteration of the unknown function + = , we get its local invertible analytic solutions.

Oscillatory and asymptotic behaviour of all solutions of differential equations with deviating arguments

1978 ◽  
Vol 81 (3-4) ◽  
pp. 195-210 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Unknown Function ◽  
Continuous Functions ◽  
Linear Differential Equations ◽  
Deviating Arguments ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

SynopsisThis paper deals with the oscillatory and asymptotic behaviour of all solutions of a class of nth order (n > 1) non-linear differential equations with deviating arguments involving the so called nth order r-derivative of the unknown function x defined bywhere r1, (i = 0,1,…, n – 1) are positive continuous functions on [t0, ∞). The results obtained extend and improve previous ones in [7 and 15] even in the usual case where r0 = r1 = … = rn–1 = 1.

scholarly journals Nonparametric estimation of trend function for stochastic differential equations driven by a bifractional Brownian motion

2020 ◽  
Vol 12 (1) ◽  
pp. 128-145
Author(s):  
Abdelmalik Keddi ◽  
Fethi Madani ◽  
Amina Angelika Bouchentouf
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Nonparametric Estimation ◽  
Unknown Function ◽  
Asymptotic Properties ◽  
Kernel Estimator ◽  
Trend Function ◽  
Bifractional Brownian Motion ◽  
Numerical Example

AbstractThe main objective of this paper is to investigate the problem of estimating the trend function St = S(xt) for process satisfying stochastic differential equations of the type {\rm{d}}{{\rm{X}}_{\rm{t}}} = {\rm{S}}\left( {{{\rm{X}}_{\rm{t}}}} \right){\rm{dt + }}\varepsilon {\rm{dB}}_{\rm{t}}^{{\rm{H,K}}},\,{{\rm{X}}_{\rm{0}}} = {{\rm{x}}_{\rm{0}}},\,0 \le {\rm{t}} \le {\rm{T,}}where {{\rm{B}}_{\rm{t}}^{{\rm{H,K}}},{\rm{t}} \ge {\rm{0}}} is a bifractional Brownian motion with known parameters H ∈ (0, 1), K ∈ (0, 1] and HK ∈ (1/2, 1). We estimate the unknown function S(xt) by a kernel estimator ̂St and obtain the asymptotic properties as ε → 0. Finally, a numerical example is provided.

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