Generalized Bernoulli Wick differential equation

2021 ◽  
Vol 24 (01) ◽  
pp. 2150008
Author(s):  
Sami H. Altoum ◽  
Aymen Ettaieb ◽  
Hafedh Rguigui
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Exponential Growth ◽  
Present Method ◽  
Dual Space ◽  
Entire Functions ◽  
Wick Product ◽  
Derivation Operator ◽  
New Type ◽  
Minimal Type

Based on the distributions space on [Formula: see text] (denoted by [Formula: see text]) which is the topological dual space of the space of entire functions with exponential growth of order [Formula: see text] and of minimal type, we introduce a new type of differential equations using the Wick derivation operator and the Wick product of elements in [Formula: see text]. These equations are called generalized Bernoulli Wick differential equations which are the analogue of the classical Bernoulli differential equations. We solve these generalized Wick differential equations. The present method is exemplified by several examples.

scholarly journals On the Growth of Solutions of Some Second Order Linear Differential Equations with Entire Coefficients

2013 ◽  
Vol 21 (2) ◽  
pp. 35-52
Author(s):  
Benharrat Belaïdi ◽  
Habib Habib
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Infinite Order ◽  
Entire Functions ◽  
Complex Numbers ◽  
Order Of Growth ◽  
Hyper Order ◽  
Linear Differential ◽  
Growth Of Solutions

Abstract In this paper, we investigate the order and the hyper-order of growth of solutions of the linear differential equation where n≥2 is an integer, Aj (z) (≢0) (j = 1,2) are entire functions with max {σ A(j) : (j = 1,2} < 1, Q (z) = qmzm + ... + q1z + q0 is a nonoonstant polynomial and a1, a2 are complex numbers. Under some conditions, we prove that every solution f (z) ≢ 0 of the above equation is of infinite order and hyper-order 1.

A note on meromorphic functions with finite order and of bounded l-index

2021 ◽  
Vol 18 (1) ◽  
pp. 1-11
Author(s):  
Andriy Bandura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Finite Order ◽  
General Problem ◽  
Meromorphic Functions ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Riccati Differential Equation

We present a generalization of concept of bounded $l$-index for meromorphic functions of finite order. Using known results for entire functions of bounded $l$-index we obtain similar propositions for meromorphic functions. There are presented analogs of Hayman's theorem and logarithmic criterion for this class. The propositions are widely used to investigate $l$-index boundedness of entire solutions of differential equations. Taking this into account we raise a general problem of generalization of some results from theory of entire functions of bounded $l$-index by meromorphic functions of finite order and their applications to meromorphic solutions of differential equations. There are deduced sufficient conditions providing $l$-index boundedness of meromoprhic solutions of finite order for the Riccati differential equation. Also we proved that the Weierstrass $\wp$-function has bounded $l$-index with $l(z)=|z|.$

scholarly journals Relation between Small Functions with Differential Polynomials Generated by Solutions of Linear Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Zhigang Huang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Entire Functions ◽  
Linear Differential Equations ◽  
Differential Polynomials ◽  
Small Functions ◽  
Linear Differential ◽  
The Relationship ◽  
Growth Of Solutions

This paper is devoted to studying the growth of solutions of second-order nonhomogeneous linear differential equation with meromorphic coefficients. We also discuss the relationship between small functions and differential polynomialsL(f)=d2f″+d1f′+d0fgenerated by solutions of the above equation, whered0(z),d1(z),andd2(z)are entire functions that are not all equal to zero.

scholarly journals Nonlocal Symmetries for Time-Dependent Order Differential Equations

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 771
Author(s):  
Andrei Ludu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Variable Coefficients ◽  
Time Dependent ◽  
Harmonic Numbers ◽  
Nonlocal Symmetries ◽  
The Riemann Zeta Function ◽  
Symmetry Of The Solution ◽  
New Type ◽  
Riemann Zeta

A new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equation represent deformations of the solutions of the classical (integer order) differential equations, mapping them into one-another as limiting cases. This equation can also move, remove or generate singularities without involving variable coefficients. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers is observed.

scholarly journals III. On linear differential equations.— No. II

1870 ◽  
Vol 18 (114-122) ◽  
pp. 210-212
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Entire Functions ◽  
Linear Differential Equations ◽  
General Linear ◽  
Linear Differential

The principles laid down in my former paper will enable us to integrate a proposed differential equation, when the solution can be expressed in the form—P/Q, where P, Q, to are rational and entire functions of ( x ). Let (α 0 +α 1 x +α 2 x 2 + ... +α m x m ) d n y / dx n + (β 0 +β 1 +β 2 x 2 + ... +α m x m ) d n-1 y / dx n-1 + (γ 0 +γ 1 x +γ 2 x 2 + ... +γ m x m ) d n-2 y / dx n-2 +.... +(λ 0 +λ 1 +λ 2 x 2 + ... +λ m x m ) y =0 be the general linear differential equation of the nth order, where none of the indices of ( x ) in the coefficients of the succeeding terms are greater than those in the coefficients of the two first.

scholarly journals Improved Numerical Computation to Solve Lane-Emden type Equations

2021 ◽  
Vol 10 (3) ◽  
pp. 1980-1984
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Differential Equations ◽  
Present Method ◽  
Mathematical Physics ◽  
Equilibrium Density ◽  
Series Solutions ◽  
Maclaurin Series ◽  
Emden Equation ◽  
Power Series Solutions

Lane-Emden equation is also of fundamental importance in mathematical physics, celestial mechanics,and computer science. It can be used to describe stellar structures, equilibrium density distribution in polytrophicisothermal gas, thermal behavior in mutual attraction of its molecules. An improved numerical method is developed for solving Lane-Emden type differential equations. The method is based on power series solutions of differential equations and Maclaurin series expansion. A python program is written to carry out numerical calculations. Five examples are solved and shown in this paper, the solutions obtained by the program are compared with the exact solutions of differential equation, an excellent agreement is found between them. The present method improves runtime.

Linear directional differential equations in the unit ball: solutions of bounded L-index

2019 ◽  
Vol 69 (5) ◽  
pp. 1089-1098 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Unit Ball ◽  
Analytic Functions ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Analytic Solutions ◽  
Directional Derivatives ◽  
Several Variables

Abstract We study sufficient conditions of boundedness of L-index in a direction b ∈ ℂn ∖ {0} for analytic solutions in the unit ball of a linear higher order non-homogeneous differential equation with directional derivatives. These conditions are restrictions by the analytic coefficients in the unit ball of the equation. Also we investigate asymptotic behavior of analytic functions of bounded L-index in the direction and estimate its growth. The results are generalizations of known propositions for entire functions of several variables.

scholarly journals Some Properties of Solutions of Second-Order Linear Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Zinelaâbidine Latreuch ◽  
Benharrat Belaïdi
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Finite Order ◽  
Entire Functions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Linearly Independent ◽  
Linear Differential

We study the growth and oscillation of gf=d1f1+d2f2, where d1 and d2 are entire functions of finite order not all vanishing identically and f1 and f2 are two linearly independent solutions of the linear differential equation f′′+A(z)f=0.

scholarly journals Multivalued backward stochastic differential equations with time delayed generators

2014 ◽  
Vol 12 (11) ◽  
Author(s):  
Bakarime Diomande ◽  
Lucian Maticiuc
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Convex Function ◽  
Stochastic Differential Equations ◽  
Existence Theorem ◽  
Backward Stochastic Differential Equation ◽  
Backward Stochastic Differential Equations ◽  
The Past ◽  
New Type

AbstractOur aim is to study the following new type of multivalued backward stochastic differential equation: $$\left\{ \begin{gathered} - dY\left( t \right) + \partial \phi \left( {Y\left( t \right)} \right)dt \ni F\left( {t,Y\left( t \right),Z\left( t \right),Y_t ,Z_t } \right)dt + Z\left( t \right)dW\left( t \right), 0 \leqslant t \leqslant T, \hfill \\ Y\left( T \right) = \xi , \hfill \\ \end{gathered} \right.$$ where ∂φ is the subdifferential of a convex function and (Y t, Z t):= (Y(t + θ), Z(t + θ))θ∈[−T,0] represent the past values of the solution over the interval [0, t]. Our results are based on the existence theorem from Delong & Imkeller, Ann. Appl. Probab., 2010, concerning backward stochastic differential equations with time delayed generators.

Chebyshev wavelet method for numerical solutions of integro-differential form of Lane–Emden type differential equations

2017 ◽  
Vol 15 (02) ◽  
pp. 1750015 ◽  
Author(s):  
P. K. Sahu ◽  
S. Saha Ray
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Differential Form ◽  
Present Method ◽  
Numerical Solutions ◽  
Algebraic Equations ◽  
Singular Differential Equation ◽  
Wavelet Method ◽  
Chebyshev Wavelets ◽  
Chebyshev Wavelet

In this paper, Chebyshev wavelet method (CWM) has been applied to solve the second-order singular differential equations of Lane–Emden type. Firstly, the singular differential equation has been converted to Volterra integro-differential equation and then solved by the CWM. The properties of Chebyshev wavelets were first presented. The properties of Chebyshev wavelets via Gauss–Legendre rule were used to reduce the integral equations to a system of algebraic equations which can be solved numerically by Newton’s method. Convergence analysis of CWM has been discussed. Illustrative examples have been provided to demonstrate the validity and applicability of the present method.

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