EXPLICIT SOLUTIONS TO SOME VECTORIAL DIFFERENTIAL EQUATIONS I. General Results (original) (raw)
Bul. Inst. Polit. Iasi. 04/2001; Bul. Inst. Polit. Iasi, XLVII (LI)(1-2, s.I., 2001):pp. 315-325..
"This paper presents some applications of the procedures presented in [6] for determining the explicit solution to vectorial differential equations of a certain type. Two classic problems of theoretical mechanics are studied: the motion of an electric charged particle in a non-stationary electromagnetic field with respect to an inertial frame and that of a particle in uniform force fields with respect to a non-inertial frame. To these problems, approximate solutions are usually given."
Acta Marisiensis. Seria Technologica
Generalized solution of a Cauchy problem given by a nonhomogeneous linear differential system is recovered to this approach. It considers the case of the free term having at most countable number of discontinuity points. The method, called successive approach, uses the solution on the previous interval (except the first one) for the condition on the given interval. The sequence of commands for a computer algebra system to this method is given.
φA-algebrizable differential equations
2019
We introduce the φA-differentiability, the corresponding generalized Cauchy-Riemann equations (φA-CREs), the Cauchy-integral theorem, and consider the problem of when a given linear system of two first order partial differential equations results the φA-CREs for some function φ and a two dimensional algebra A. We show that the four dimensional vector fields associated with triangular billiards are φA-differentiable. Keyword: Vector fields, Lorch differentiability, Generalized Cauchy-Riemann equations MSC[2010]: 37C10, 58C20, 53C22. Introduction Consider a linear system of partial differential equations (PDEs) of the form a111ux + a121vx + a131wx + a112uy + a122vy + a132wy = 0 a211ux + a221vx + a231wx + a212uy + a222vy + a232wy = 0 a311ux + a321vx + a331wx + a312uy + a322vy + a332wy = 0 , (1) where aijk are functions of (x, y, z), ux = ∂u ∂x and so on. In this paper we define a type of differentiability for which in particular cases the “Generalized Cauchy-Riemann equations” are line...
Banach Journal of Mathematical Analysis, 2016
In this paper we show the unexpected property that extension from local to global without loss of regularity holds for the solutions of a wide class of vector-valued differential equations, in particular for the class of fractional abstract Cauchy problems in the subdiffusive case. The main technique is the use of the algebraic structure of these solutions, which are defined by new versions of functional equations defining solution families of bounded operators. The convolution product and the double Laplace transform for functions of two variables are useful tools which we apply also to extend these solutions. Finally we illustrate our results with different concrete examples.
Differential Equations Lecture Notes No.17
In this lecture, we will study some important applications of the previous material. Many real life problems concern vibrations of a mechanical or electrical nature. Differential equations provide an excellent setting to understand them.
ODEs together PDEs and Vector Fields in the SoftAge
International Journal on Engineering, Science and Technology, 2021
At this moment when we can employ software, mathematics education has to be reviewed. In this article, we point out that ODEs should be taught from a geometric and qualitative point of view together with an introduction to PDEs and vector fields. This would increase the skills of the future mathematics user, not only to obtain explicit solutions from a straight command like DSolve, but also in the situations where this command does not help. The geometric interpretation and the concept of direction fields with images generated by software will give us a good understanding of possible system evolutions.
On solutions of Rashevskii equation
2012
The solutions of Rashevskii equation for gonometric family of plane curves are considered. Their properties are discussed. The connection with the theory of duality for the second order ODE’s is discussed. 1 Gonometric family of plane curves Two parametrical family of plane curves is defined by the equation From the equation (1) and its differential at fixed (x, y) can be find the coordinates From the condition is followed expression for the angle θ x = x(ξ, η, dη dξ), F(x, y, ξ, η) = 0. (1)