Sufficient conditions of stability of the solutions of a linear homogeneous system of differential equations with variable coefficients

In this article, we employ the notion of coupled fixed points on a complete b-metric space endowed with a graph to give sufficient conditions to guarantee a solution of system of differential equations with impulse effects. We derive recisely some new coupled fixed point theorems under some conditions and then apply our results to achieve our goal.

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Symmetry-based algorithms to relate partial differential equations: I. Local symmetries

European Journal of Applied Mathematics ◽

10.1017/s0956792500000176 ◽

1990 ◽

Vol 1 (3) ◽

pp. 189-216 ◽

Cited By ~ 41

Author(s):

G. W. Bluman ◽

S. Kumei

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Sufficient Conditions ◽

Linear Partial Differential Equation ◽

Variable Coefficients ◽

Necessary And Sufficient Conditions ◽

Partial Differential ◽

Flow Equations ◽

Local Symmetries ◽

Necessary And Sufficient

Simple and systematic algorithms for relating differential equations are given. They are based on comparing the local symmetries admitted by the equations. Comparisons of the infinitesimal generators and their Lie algebras of given and target equations lead to necessary conditions for the existence of mappings which relate them. Necessary and sufficient conditions are presented for the existence of invertible mappings from a given nonlinear system of partial differential equations to some linear system of equations with examples including the hodograph and Legendre transformations, and the linearizations of a nonlinear telegraph equation, a nonlinear diffusion equation, and nonlinear fluid flow equations. Necessary and sufficient conditions are also given for the existence of an invertible point transformation which maps a linear partial differential equation with variable coefficients to a linear equation with constant coefficients. Other types of mappings are also considered including the Miura transformation and the invertible mapping which relates the cylindrical KdV and the KdV equations.

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Some results of Watson, Plancherel type integral transforms related to the Hartley, Fourier convolutions and applications

10.22541/au.163257659.93199253/v1 ◽

2021 ◽

Author(s):

Tuan Trinh

Keyword(s):

Differential Equations ◽

Type Theorem ◽

Sufficient Conditions ◽

Integral Transforms ◽

System Of Differential Equations ◽

Symmetric Form ◽

Original Function ◽

Type Integral ◽

Necessary And Sufficient ◽

Sequence Of Functions

In this work, we study the Watson-type integral transforms for the convolutions related to the Hartley and Fourier transformations. We establish necessary and sufficient conditions for these operators to be unitary in the L 2 (R) space and get their inverse represented in the conjugate symmetric form. Furthermore, we also formulated the Plancherel-type theorem for the aforementioned operators and prove a sequence of functions that converge to the original function in the defined L 2 (R) norm. Next, we study the boundedness of the operators (T k ). Besides, showing the obtained results, we demonstrate how to use it to solve the class of integro-differential equations of Barbashin type, the differential equations, and the system of differential equations. And there are numerical examples given to illustrate these.

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On conditions of solvability of three-dimensional integralsystem in quadratures

Vestnik of Samara University Natural Science Series ◽

10.18287/2541-7525-2015-21-10-40-46 ◽

2017 ◽

Vol 21 (10) ◽

pp. 40-46

Author(s):

E.A. Sozontova

Keyword(s):

Differential Equations ◽

Dimensional Space ◽

Three Dimensional ◽

Sufficient Conditions ◽

Original System ◽

Goursat Problem ◽

System Of Differential Equations ◽

Third Order ◽

First Order ◽

Three Dimensional Space

In this paper we consider the system of equations with partial integrals in three-dimensional space. The purpose is to ﬁnd suﬃcient conditions of solvability of this system in quadratures. The proposed method is based on the reduction of the original system, ﬁrst, to the Goursat problem for a system of diﬀerential equations of the ﬁrst order, and after that to the three Goursat problems for diﬀerential equations of the third order. As a result, the suﬃcient conditions of solvability of the considering system in explicit form were obtained. The total number of cases discussing solvability is 16.

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Lagrangian descriptions of dissipative systems: a review

Mathematics and Mechanics of Solids ◽

10.1177/1081286520971834 ◽

2020 ◽

pp. 108128652097183

Author(s):

Alberto Maria Bersani ◽

Paolo Caressa

Keyword(s):

Differential Equations ◽

Equations Of Motion ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Dissipative Systems ◽

Lagrangian Formalism ◽

Lagrangian Description ◽

System Of Differential Equations ◽

Helmholtz Conditions ◽

Necessary And Sufficient

In this paper, we review classical and recent results on the Lagrangian description of dissipative systems. After having recalled Rayleigh extension of Lagrangian formalism to equations of motion with dissipative forces, we describe Helmholtz conditions, which represent necessary and sufficient conditions for the existence of a Lagrangian function for a system of differential equations. These conditions are presented in different formalisms, some of them published in the last decades. In particular, we state the necessary and sufficient conditions in terms of multiplier factors, discussing the conditions for the existence of equivalent Lagrangians for the same system of differential equations. Some examples are discussed, to show the application of the techniques described in the theorems stated in this paper.

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Some further results on oscillation of neutral differential equations

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700011758 ◽

1992 ◽

Vol 46 (1) ◽

pp. 149-157 ◽

Cited By ~ 32

Author(s):

Jianshe Yu ◽

Zhicheng Wang

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Sufficient Conditions ◽

Variable Coefficients ◽

Neutral Differential Equation ◽

Neutral Differential Equations ◽

All Solutions ◽

Image Position

We obtain new sufficient conditions for the oscillation of all solutions of the neutral differential equation with variable coefficientswhere P, Q, R ∈ C([t0, ∞), R+), r ∈ (0, ∞) and τ, σ ∈ [0, ∞). Our results improve several known results in papers by: Chuanxi and Ladas; Lalli and Zhang; Wei; Ruan.

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Minimizing of the quadratic functional on Hopfield networks

Electronic journal of qualitative theory of differential equations ◽

10.14232/ejqtde.2021.1.92 ◽

2021 ◽

pp. 1-20

Author(s):

Oleksandr Boichuk ◽

Dmytro Bihun ◽

Victor Feruk ◽

Oleksandr Pokutnyi

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Sufficient Conditions ◽

Quadratic Functional ◽

Problem Solution ◽

Linear Boundary ◽

System Of Differential Equations ◽

Hopfield Networks ◽

Necessary And Sufficient ◽

The Given

In this paper, we consider the continuous Hopfield model with a weak interaction of network neurons. This model is described by a system of differential equations with linear boundary conditions. Also, we consider the questions of finding necessary and sufficient conditions of solvability and constructive construction of solutions of the given problem, which turn into solutions of the linear generating problem, as the parameter $\varepsilon$ tends to zero. An iterative algorithm for finding solutions has been constructed. The problem of finding the extremum of the target functions on the given problem solution is considered. To minimize a functional, an accelerated method of conjugate gradients is used. Results are illustrated with examples for the case of three neurons.

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