scholarly journals On Symmetric Solutions to Linear Matrix Time-Varying Differential Equations

2021 ◽  
Vol 2090 (1) ◽  
pp. 012134
Author(s):  
Dmitry A. Fetisov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Finite Order ◽  
Initial Value Problem ◽  
Symmetric Solution ◽  
Linear Matrix ◽  
Sufficient Condition ◽  
Time Varying ◽  
Initial Value

Abstract In this paper, we discuss when the solution to the initial value problem for a linear matrix time-varying differential equation is symmetric on a given interval. By symmetry, we mean that the solution does not change when transposed. Throughout the paper, we assume that the equation has coefficients of finite order of smoothness. We demonstrate that, in order to verify whether the solution to the initial value problem is symmetric on a given interval, it can be useful to construct two matrix sequences associated to the equation. Using these sequences, we prove a sufficient condition for the solution symmetry on a given interval. Assuming that the initial value problem for a linear matrix time-varying differential equation satisfies this condition, we derive a formula for a symmetric solution to this problem.

scholarly journals Methods for solving the initial value problem with a two-sided estimate of the local error

2021 ◽  
pp. 88-92
Author(s):  
Yaroslav Pelekh ◽  
Andrii Kunynets ◽  
Halyna Beregova ◽  
Tatiana Magerovska
Keyword(s):  
Differential Equation ◽  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Local Error ◽  
Initial Value ◽  
Order Of Accuracy ◽  
Embedded Methods ◽  
The Right

Numerical methods for solving the initial value problem for ordinary differential equations are proposed. Embedded methods of order of accuracy 2(1), 3(2) and 4(3) are constructed. To estimate the local error, two-sided calculation formulas were used, which give estimates of the main terms of the error without additional calculations of the right-hand side of the differential equation, which favorably distinguishes them from traditional two-sided methods of the Runge- Kutta type.

scholarly journals MODELLING OF URBAN TRAFFIC FLOW

2017 ◽  
Vol 1 ◽  
pp. 109 ◽  
Author(s):  
Sharif E. Guseynov ◽  
Alexander V. Berezhnoy
Keyword(s):  
Differential Equation ◽  
Mathematical Models ◽  
Traffic Flow ◽  
Initial Value Problem ◽  
Planck Equation ◽  
Continuous Model ◽  
Urban Traffic ◽  
Sufficient Condition ◽  
Initial Value ◽  
Integro Differential Equation

In this paper non-deterministic motion of urban traffic is studied under certain assumptions. Based on those assumptions discrete and continuous mathematical models are developed: continuous model is written as the Cauchy initial-value problem for the integro-differential equation, whence among other things it is obtained the Fokker-Planck equation. Besides, the sufficient condition ensuring the mathematical legitimacy of the developed continuous model is formulated.

Solvability of Ordinary Differential Equations Near Singular Points: An Analytic Case

1967 ◽  
Vol 19 ◽  
pp. 1303-1313
Author(s):  
Homer G. Ellis
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Singular Point ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Singular Points ◽  
Initial Value ◽  
Analytic Case ◽  
Image Position

The question of solvability of the differential equation1with x ranging over an interval (0, a], and with the boundary condition ƒ(0+) = 0, can be investigated as an initial-value problem at 0, which may be a singular point for the equation.

Convergent and asymptotic expansions of solutions of second-order differential equations with a large parameter

2014 ◽  
Vol 12 (05) ◽  
pp. 523-536 ◽  
Author(s):  
Chelo Ferreira ◽  
José L. López ◽  
Ester Pérez Sinusía
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Initial Conditions ◽  
Special Functions ◽  
Auxiliary Problem ◽  
Second Order ◽  
Large Parameter ◽  
Initial Value

We consider the second-order linear differential equation [Formula: see text] where x ∈ [0, X], X > 0, α ∈ (-∞, 2), Λ is a large complex parameter and g is a continuous function. The asymptotic method designed by Olver [Asymptotics and Special Functions (Academic Press, New York, 1974)] gives the Poincaré-type asymptotic expansion of two independent solutions of the equation in inverse powers of Λ. We add initial conditions to the differential equation and consider the corresponding initial value problem. By using the Green's function of an auxiliary problem, we transform the initial value problem into a Volterra integral equation of the second kind. Then using a fixed point theorem, we construct a sequence of functions that converges to the unique solution of the problem. This sequence has also the property of being an asymptotic expansion for large Λ (not of Poincaré-type) of the solution of the problem. Moreover, we show that the idea may be applied also to nonlinear differential equations with a large parameter.

scholarly journals Monotone-iterative technique of Lakshmikantham for the initial value problem for a differential equation with a step function

2002 ◽  
Vol 30 (1) ◽  
pp. 57-62
Author(s):  
S. G. Hristova ◽  
M. A. Rangelova
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Special Kind ◽  
Step Function ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Initial Value ◽  
Monotone Iterative ◽  
The Given

The initial value problem for a special kind of differential equations with a step function is studied. The monotone-iterative technique of Lakshmikantham for approximate finding of the solutions of the given problem is well grounded.

scholarly journals The System of Differential Equations Associated With Involutions

2021 ◽  
Author(s):  
Farrukh Nuriddin ugli Dekhkonov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Initial Value Problem ◽  
Initial Conditions ◽  
Higher Order ◽  
System Of Differential Equations ◽  
Initial Value ◽  
Order Ordinary Differential Equation

In this paper, we consider with a class of system of differential equations whose argument transforms are involution. In this an initial value problem for a differential equation with involution is reduced to an initial value problem for a higher order ordinary differential equation. Than either two initial conditions are necessary for a solution, the equation is then reduced to a boundary value problem for a higher order ODE.

scholarly journals Explicit Solutions of Initial Value Problems for Fractional Generalized Proportional Differential Equations with and without Impulses

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 996
Author(s):  
Snezhana Hristova ◽  
Mohamed I. Abbas
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Explicit Solution ◽  
Explicit Solutions ◽  
Initial Value ◽  
Impulsive Conditions ◽  
Fractional Differential ◽  
Set Up ◽  
Two Parameters

The object of investigation in this paper is a scalar linear fractional differential equation with generalized proportional derivative of Riemann–Liouville type (LFDEGD). The main goal is the obtaining an explicit solution of the initial value problem of the studied equation. Note that the locally solvability, being the same as the existence of solutions to the initial value problem, is connected with the symmetry of a transformation of a system of differential equations. At the same time, several criteria for existence of the initial value problem for nonlinear fractional differential equations with generalized proportional derivative are connected with the linear ones. It leads to the necessity of obtaining an explicit solution of LFDEGD. In this paper two cases are studied: the case of no impulses in the differential equation are presented and the case when instantaneous impulses at initially given points are involved. All obtained formulas are based on the application of Mittag–Leffler function with two parameters. In the case of impulses, initially the appropriate impulsive conditions are set up and later the explicit solutions are obtained.

Initial Value Problem of Fractional Integro-Differential Equations in Banach Space

2015 ◽  
Vol 18 (1) ◽  
Author(s):  
Adel Jawahdou
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Initial Value Problem ◽  
Measure Of Noncompactness ◽  
Existence Of Solutions ◽  
Initial Value

AbstractThis paper is devoted to study the existence of solutions of nonlinear fractional integro-differential equation, via the techniques of measure of noncompactness. The investigation is based on a Schauder's fixed point theorem. The main result is less restrictive than those given in the literature. An illustrative example is given.

scholarly journals On the boundedness of solution of the second order ordinary differential equation with damping term and involution

2021 ◽  
Vol 102 (2) ◽  
pp. 16-24
Author(s):  
A. Ashyralyev ◽  
◽  
M. Ashyralyyeva ◽  
O. Batyrova ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Second Order ◽  
Stability Estimates ◽  
Damping Term ◽  
Initial Value ◽  
Order Ordinary Differential Equation ◽  
Linear Differential

In the present paper the initial value problem for the second order ordinary differential equation with damping term and involution is investigated. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with damping term and involution. Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with damping term and involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for second order ordinary nonlinear differential equation with damping term and involution is established.

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