scholarly journals Oscillation of arbitrary-order derivatives of solutions to the higher order non-homogeneous linear differential equations taking small functions in the unit disc

2021 ◽  
Vol 6 (12) ◽  
pp. 13746-13757
Author(s):  
Pan Gong ◽  
◽  
Hong Yan Xu
Keyword(s):  
Unit Disc ◽  
Arbitrary Order ◽  
Higher Order ◽  
Small Function ◽  
Linear Differential ◽  
Oscillation Theorems ◽  
The Relationship ◽  
Derivatives Of ◽  
Homogeneous Linear ◽  
Homogeneous Linear Differential Equation

<abstract><p>In this article, we study the relationship between solutions and their arbitrary-order derivatives of the higher order non-homogeneous linear differential equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f = F(z) \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>in the unit disc $ \bigtriangleup $ with analytic or meromorphic coefficients of finite $ [p, q] $-order. We obtain some oscillation theorems for $ f^{(j)}(z)-\varphi(z) $, where $ f $ is a solution and $ \varphi(z) $ is a small function.</p></abstract>

scholarly journals Growth and fixed points of solutions and their arbitrary-order derivatives of higher-order linear differential equations in the unit disc

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yu Chen ◽  
Guan-Tie Deng ◽  
Zhan-Mei Chen ◽  
Wei-Wei Wang
Keyword(s):  
Differential Equations ◽  
Fixed Points ◽  
Unit Disc ◽  
Arbitrary Order ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Iterated Order ◽  
Linear Differential ◽  
Derivatives Of

AbstractIn this paper, we investigate the growth and fixed points of solutions of higher-order linear differential equations in the unit disc. We extend the coefficient conditions to a type of one-constant-control coefficient comparison and obtain the same estimates of iterated order of solutions. We also obtain better estimates by providing a precise value of iterated order of solution instead of a range of that in the case of coefficient characteristic function comparison. Moreover, we utilize iteration to investigate and estimate the fixed points of solutions’ arbitrary-order derivatives with higher-order equations $f^{(k)}+A_{k-1}(z)f^{(k-1)}+{\cdots }+A_{1}(z)f'+A_{0}(z)f=0$ f ( k ) + A k − 1 ( z ) f ( k − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = 0 and provide a concise method to judge if the items generated by the iteration do not vanish identically and ensure the iteration proceeds. Our results are an improvement over those by B. Belaïdi, T. B. Cao, G. W. Zhang and A. Chen.

scholarly journals Value Distribution and Arbitrary-Order Derivatives of Meromorphic Solutions of Complex Linear Differential Equations in the Unit Disc

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 352
Author(s):  
Hai-Ying Chen ◽  
Xiu-Min Zheng
Keyword(s):  
Differential Equation ◽  
Unit Disc ◽  
Arbitrary Order ◽  
Value Distribution ◽  
Small Function ◽  
Meromorphic Solutions ◽  
Exponent Of Convergence ◽  
Complex Linear ◽  
Linear Differential ◽  
Derivatives Of

In this paper, we investigate the value distribution of meromorphic solutions and their arbitrary-order derivatives of the complex linear differential equation f ′ ′ + A ( z ) f ′ + B ( z ) f = F ( z ) in Δ with analytic or meromorphic coefficients of finite iterated p-order, and obtain some results on the estimates of the iterated exponent of convergence of meromorphic solutions and their arbitrary-order derivatives taking small function values.

scholarly journals The J-Generalized P - K Mittag-Leffler Function

2020 ◽  
Author(s):  
Kuldeep Singh Gehlot ◽  
Anjana Bhandari
Keyword(s):  
Mellin Transform ◽  
Integral Representations ◽  
Fractional Integrals ◽  
Linear Differential ◽  
Euler Transform ◽  
Relationship Of ◽  
The Relationship ◽  
Homogeneous Linear ◽  
Homogeneous Linear Differential Equation ◽  
Differential And Integral Equations

We know that the classical Mittag-Leffler function play an important role as solution of fractional order differential and integral equations. We introduce the j-generalized p - k Mittag-Leffler function. We evaluate the second order differential recurrence relation and four different integral representations and introduce a homogeneous linear differential equation whose one of the solution is the j-generalized p-k Mittag-Leffler function. Also we evaluate the certain relations that exist between j-generalized p - k Mittag-Leffler function and Riemann-Liouville fractional integrals and derivatives. We evaluate Mellin-Barnes integral representation of j-generalized p-k Mittag-Le er Function. The relationship of j-generalized p-k Mittag-Leffler Function with Fox H-Function and Wright hypergeometric function is also establish. we obtained its Euler transform, Laplace Transform and Mellin transform. Finally we derive some particular cases.

scholarly journals Value Distribution of Meromorphic Solutions and Their Derivatives of Complex Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Abdallah El Farissi
Keyword(s):  
Differential Equations ◽  
Meromorphic Functions ◽  
Higher Order ◽  
Value Distribution ◽  
Meromorphic Solutions ◽  
Small Functions ◽  
Complex Differential Equations ◽  
Linear Differential ◽  
The Relationship ◽  
Derivatives Of

We deal with the relationship between the small functions and the derivatives of solutions of higher-order linear differential equations f(k)+Ak-1f(k-1)+⋯+A0f=0,   k≥2, where Aj(z)  (j=0,1,…,k-1) are meromorphic functions. The theorems of this paper improve the previous results given by El Farissi, Belaïdi, Wang, Lu, Liu, and Zhang.

Pinned-Pinned Slender Solid Struts with Parabolic Taper

1942 ◽  
Vol 46 (378) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Turton
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Linear Equation ◽  
Elastic Stability ◽  
Formal Integration ◽  
Linear Differential ◽  
Homogeneous Linear ◽  
Homogeneous Linear Differential Equation

In 1917–19, Barling and Webb, Berry, Cowley and Levy, and Webb and Lang discussed the elastic stability of struts of various tapers, but it appears to have escaped notice that one of the few cases in which formal integration is possible is that in which the tapered profile of axial longitudinal sections is part of a parabola; this gives a “ homogeneous linear “ differential equation, i.e., a linear equation of the form f (xd/dx) y = F (x).

Linear differential equations with constant coefficients

2019 ◽  
Vol 103 (557) ◽  
pp. 257-264
Author(s):  
Bethany Fralick ◽  
Reginald Koo
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Auxiliary Equation ◽  
Real Solutions ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Complex Solutions ◽  
Image Position ◽  
Homogeneous Linear ◽  
Homogeneous Linear Differential Equation

We consider the second order homogeneous linear differential equation (H) $${ ay'' + by' + cy = 0 }$$ with real coefficients a, b, c, and a ≠ 0. The function y = emx is a solution if, and only if, m satisfies the auxiliary equation am2 + bm + c = 0. When the roots of this are the complex conjugates m = p ± iq, then y = e(p ± iq)x are complex solutions of (H). Nevertheless, real solutions are given by y = c1epx cos qx + c2epx sin qx.

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