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1
Content available Fast growing solutions to linear differential equations with entire coefficients having the same ρϕ-order
Belaϊdi Benharrat
Journal of Mathematics and Applications
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2019
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Vol. 42
63--77
EN
This paper deals with the growth of solutions of a class of higher order linear differential equations f(k)+Ak-1(z)f(k-1)+ … +A1(z)f’+A0(z)f=0; k≥2 when most coefficients Aj (z) (j = 0, ..., k-1) have the same ρϕ-order with each other. By using the concept of τϕ-type, we obtain some results which indicate growth estimate of every non-trivial entire solution of the above equations by the growth estimate of the coefficient A0 (z). We improve and generalize some recent results due to Chyzhykov-Semochko and the author.
2
Content available On the hyper-order of transcendental meromorphic solutions of certain higher order linear differential equations
Hamani K., Belaidi B.
Opuscula Mathematica
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2017
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Vol. 37, no. 6
853--874
EN
In this paper, we investigate the growth of meromorphic solutions of the linear differential equation [formula] where k ≥ 2 is an integer, Pj(z) (j = 0,1,... , k — 1) are nonconstant polynomials and hj(z) are meromorphic functions. Under some conditions, we determine the hyper-order of these solutions. We also consider nonhomogeneous linear differential equations.
3
Admissible pair of spaces for non-correctly solvable linear differential equations
Chernyavskaya N. A., Dorel L. S., Shuster L. A.
Journal of Applied Analysis
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2016
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Vol. 22, nr 1
1--14
EN
We consider the differential equation −y′(x)+q(x)y(x)=f(x),x∈R, where f∈Lp (R), p∈[1,∞), and 0≤q∈Lloc1 (R), ∫0−∞q(t)dt=∫∞0q(t)dt=∞, q0(a)=infx∈R∫x+ax−aq(t)dt=0 for any a∈(0,∞). Under these conditions, the above equation is not correctly solvable in Lp (R) for any p∈[1,∞). Let q∗(x) be the Otelbaev-type average of the function q(t), t∈R, at the point t=x; let θ(x) be a continuous positive function for x∈R, and Lp,θ (R)={f∈Llocp (R):∫ ∞−∞|θ(x)f(x)|pdx<∞},∥f∥p,θ:=∥f∥Lp,θ(R) =(∫ ∞−∞|θ(x)f(x)| pdx) 1/p. We show that if there exists a constant c∈[1,∞) such that the inequality c−1q∗(x)≤θ(x)≤cq∗(x) holds for all x∈R, then under some additional conditions for q the pair of spaces {Lp,θ (R);Lp (R)} is admissible for the considered equation.
4
Content available Growth and oscillation of some polynomials generated by solutions of complex differential equations
Latreuch Z., Belaidi B.
Opuscula Mathematica
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2015
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Vol. 35, no. 1
85--98
EN
In this paper, we continue the study of some properties on the growth and oscillation of solutions of linear differential equations with entire coefficients of the type [formula] and [formula].
5
Content available remote Properties of higher order differential polynomials generated by solutions of complex differential equations in the unit disc
Latreuch Z., Belaidi B.
Journal of Mathematics and Applications
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2014
|
Vol. 37
67--84
EN
The main purpose of this paper is to study the controllability of solutions of the differential equation [...] In fact, we study the growth and oscillation of higher order differential polynomial with meromorphic coefficients in the unit disc [...] generated by solutions of the above kth order differential equation.
6
Content available remote The frequency of the zeros of some differential polynomials
Belaïdi B.
Demonstratio Mathematica
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2013
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Vol. 46, nr 1
75--86
EN
Let ρp(ƒ) and σp(ƒ) denote respectively the iterated p-order and the iterated p-type of an entire function ƒ. In this paper, we study the iterated order and the fixed points of some differential polynomials generated by solutions of the differential equation f''+A1(z)f'+A0(z)f=0 where A1(z), A0(z) are entire functions of finite iterated p-order such that ρp(A1) = ρp(A0) = ρ(0< ρ <+∞) and σp(A1)< σp(A0) =σ(0< σ <+∞).
7
Content available Oscillation on the left and on the right
Ohriska J.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2009
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Vol. 14
87-95
EN
In the present paper we derive sufficient conditions for the linear differential equation (r(t)y′(t))′+p(t)y(t) = 0 to be either oscillatory or non-oscillatory on the left and eventually on the right. Some estimations of count of zero point s for solutions to considered equation on an interval are also presented.
8
Content available remote A time-step insensitive recurrent approach to analyze non-stationary random responses
Chen K. F., Zhang Q., Zhang S. W.
Computer Assisted Mechanics and Engineering Sciences
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2009
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Vol. 16, No. 1
1-9
EN
The recurrent approach constructed via the stochastic central difference (SCD) is a very fast method for analyzing non-stationary random responses. However, the computational results depend to a great extent upon the discrete time-step size. A new recurrent approach is proposed in this report. It is based on the theory of linear differential equations. Theoretical analysis shows that this algorithm is unconditionally stable for all damped systems. Two examples show that the proposed approach is not sensitive to the time-step.
9
Differential polynomials generated by second order linear differential equations
Belaidi B., El Farissi A.
Journal of Applied Analysis
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2008
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Vol. 14, nr 2
259-271
EN
In this paper, we study fixed points of solutions of the differential equation f" + A1 (z) f' + A0 (z) f = 0, where Aj (z) ( ≡ ≠ 0) (j = 0,1) are transcendental meromorphic functions with finite order. Instead of looking at the zeros of f (z) - z, we proceed to a slight generalization by considering zeros of g (z) -φ(z), where g is a differential polynomial in f with polynomial coefficients,φ is a small meromorphic function relative to f, while the solution f is of infinite order.
10
Content available remote On solvability of linear differential equations in Rn
Shkarin S. A.
Demonstratio Mathematica
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2005
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Vol. 38, nr 1
85--99
EN
We construct xo is an element of RN and a row-finite matrix T = {Ti,j(t)}i,j is an element of N of polynomials of one real variable t such that the Cauchy problem x(t) = Ttx(t), x{0) = xo in the Frechet space RN has no solutions. We also construct a row-finite matrix A = {Aij(t)}ij is an element of N of C°°(R) functions such that the Cauchy problem x{t) = Atx;(t), x(0) = xo in RN has no solutions for any xo infinity RN\ {0}. We provide some sufficient condition of solvability and unique solvability for linear ordinary differential equations x(t) = Ttx(t) with matrix elements Ti,j(t) analytically dependent on t.
11
Content available remote Asymptotic expansion of the solution for nonlinear wave equation with mixed non-homogeneous conditions
Long N. T., Dinh A. P. N., Diem T. N.
Demonstratio Mathematica
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2003
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Vol. 36, nr 3
683--695
12
Heat-balance integral method in solidification problems involving small spatial perturbations
Pauk V.
Journal of Technical Physics
|
2000
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Vol. 41, no 1
65-74
EN
The heat-balance integral method is applied to solve the solidification problem involving a small spatial periodic fluctuations in the mold temperature. These initial irregularities lead to perturbations in the temperature field and the planar solidification line. The problem is reduced to linear differential equations for amplitudes of the interface perturbations and solutions are presented in closed forms. The numerical analysis is performed for a sinusoidal perturbation. Presented results show the oscillations in the solidification front as a function of time and parameters of the problem. The stability of an interface between solid and liquid phases is investigated.
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