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 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>

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.

A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations

1950 ◽  
Vol 46 (4) ◽  
pp. 570-580 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Bessel Functions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Zeros Of Bessel Functions ◽  
Zeros Of Solutions ◽  
Linear Differential ◽  
Homogeneous Linear ◽  
Homogeneous Linear Differential Equation

The zeros of solutions of the general second-order homogeneous linear differential equation are shown to satisfy a certain non-linear differential equation. The method here proposed for their determination is the numerical integration of this differential equation. It has the advantage of being independent of tabulated values of the actual functions whose zeros are being sought. As an example of the application of the method the Bessel functions Jn(x), Yn(x) are considered. Numerical techniques for integrating the differential equation for the zeros of these Bessel functions are described in detail.

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