Asymptotic behavior of the formal solution of a degenerate differential equation at a singular point

1978 ◽  
Vol 18 (1) ◽  
pp. 84-89
Author(s):  
B. Kvedaras
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Singular Point ◽  
Formal Solution ◽  
Degenerate Differential Equation

scholarly journals Asymptotic Solution Of Differential Equations In a Domain Containing a Regular Singular Point

1956 ◽  
Vol 8 ◽  
pp. 97-104 ◽  
Author(s):  
N. D. Kazarinoff ◽  
R. McKelvey
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Singular Point ◽  
Differential Equations ◽  
Asymptotic Solution ◽  
Connected Domain ◽  
Regular Singular Point ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

1. Introduction. In this paper we study the asymptotic behavior in λ of the solutions about the origin in the z-plane of the differential equation.Both the variable z and the parameter λ are complex. The coefficient P(z, λ) is assumed to be analytic and single-valued in λ at infinity and in z throughout a bounded, closed, simply connected domain D containing z = 0.

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Regular Variation ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Continuous Function ◽  
Linear Differential

AbstractWe study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

Applications of Erdélyi-Kober fractional integral for solving time-fractional Tricomi-Keldysh type equation

2020 ◽  
Vol 23 (5) ◽  
pp. 1381-1400 ◽  
Author(s):  
Kangqun Zhang
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Integral Operator ◽  
Nonlinear Equations ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Formal Solution ◽  
Type Equation ◽  
Multiplier Theorem ◽  
Problem Of Time

Abstract In this paper we consider Cauchy problem of time-fractional Tricomi-Keldysh type equation. Based on the theory of a Erdélyi-Kober fractional integral operator, the formal solution of the inhomogeneous differential equation involving hyper-Bessel operator is presented with Mittag-Leffler function, then nonlinear equations are considered by applying Gronwall-type inequalities. At last, we establish the existence and uniqueness of L p -solution of time-fractional Tricomi-Keldysh type equation by use of Mikhlin multiplier theorem.

scholarly journals On [p,q]-order of growth of solutions of complex linear differential equations near a singular point

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4013-4020
Author(s):  
Jianren Long ◽  
Sangui Zeng
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Linear Differential Equations ◽  
Order Of Growth ◽  
Complex Linear ◽  
Linear Differential ◽  
Growth Of Solutions

We investigate the [p,q]-order of growth of solutions of the following complex linear differential equation f(k)+Ak-1(z) f(k-1) + ...+ A1(z) f? + A0(z) f = 0, where Aj(z) are analytic in C? - {z0}, z0 ? C. Some estimations of [p,q]-order of growth of solutions of the equation are obtained, which is generalization of previous results from Fettouch-Hamouda.

scholarly journals Weighted fourth order elliptic problems in the unit ball

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zongming Guo ◽  
Fangshu Wan
Keyword(s):  
Asymptotic Behavior ◽  
Singular Point ◽  
Unit Ball ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Weighted Sobolev Spaces ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Uniqueness Results ◽  
Positive Radial Solutions

<p style='text-indent:20px;'>Existence and uniqueness of positive radial solutions of some weighted fourth order elliptic Navier and Dirichlet problems in the unit ball <inline-formula><tex-math id="M1">\begin{document}$ B $\end{document}</tex-math></inline-formula> are studied. The weights can be singular at <inline-formula><tex-math id="M2">\begin{document}$ x = 0 \in B $\end{document}</tex-math></inline-formula>. Existence of positive radial solutions of the problems is obtained via variational methods in the weighted Sobolev spaces. To obtain the uniqueness results, we need to know exactly the asymptotic behavior of the solutions at the singular point <inline-formula><tex-math id="M3">\begin{document}$ x = 0 $\end{document}</tex-math></inline-formula>.</p>

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.

MULTIPOLE MATRIX ELEMENTS 〈nl|rβ|El'〉v FOR H-LIKE ATOMS AND THEIR APPLICATIONS (AS β = 1, n ≤ 4, Enl < 0 ≤ E ≤ 1)

2008 ◽  
Vol 22 (08) ◽  
pp. 937-965 ◽  
Author(s):  
V. F. TARASOV
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Atomic Physics ◽  
Nuclear Charge ◽  
Dipole Moments ◽  
Oscillator Strengths ◽  
Matrix Elements ◽  
Auxiliary Parameter ◽  
Radial Functions ◽  
Line Intensities

This article deals with the connection between multipole matrix elements 〈nl|rβ|n'l'〉ν and 〈nl|rβ|El'〉ν for H-like atoms, where ν is the so-called "auxiliary" parameter of Heun's differential equation and [Formula: see text] is the "effective" nuclear charge, and new properties of Appell's function F2(x,y) to the vicinity of the singular point (1, 1) and in addition, here, first V. A. Fock's idea for the continuous spectrum is taken into consideration. Such an approach allows us to get the explicit expressions for squares of the dipole moments and the certain physical characteristics in atomic physics and also their exact numerical values, e.g., the average oscillator strengths [Formula: see text] and the line intensities J(nl, El'), etc., as n ≤ 4, l'= l ± 1 and 0 ≤ E ≤ 1 (see Tables 1–3). Besides, diagrams of certain radial functions for the discrete-continuous transitions are given here.

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