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Controllability theorem for nonlinear dynamical systems

100%
Kisielewicz M.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2002
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tom 22
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nr 2
225-232
EN
Some sufficient conditions for controllability of nonlinear systems described by differential equation ẋ = f(t,x(t),u(t)) are given.
2
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Dynamics of the tumor-immune system competition - the effect of time delay

100%
Galach M.
International Journal of Applied Mathematics and Computer Science
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2003
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tom 13
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nr 3
395-406
EN
The model analyzed in this paper is based on the model set forth by V.A. Kuznetsov and M.A. Taylor, which describes a competition between the tumor and immune cells. Kuznetsov and Taylor assumed that tumor-immune interactions can be described by a Michaelis-Menten function. In the present paper a simplified version of the Kuznetsov-Taylor model (where immune reactions are described by a bilinear term) is studied. On the other hand, the effect of time delay is taken into account in order to achieve a better compatibility with reality.
3
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New method to solve certain differential equations

100%
Rajchel K., Szczęsny J.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
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2016
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tom 15
EN
A new method to solve stationary one-dimensional Schroedinger equation is investigated. Solutions are described by means of representation of circles with multiple winding number. The results are demonstrated using the well-known analytical solutions of the Schroedinger equation.
4
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Sur la méthode de l’intégrale particulière et sur ses conséquences pour l’équation de Riccati et pour les équations différentielles linéaires et homogènes d’ordre supérieur

51%
Kapcia A.
Commentationes Mathematicae
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2007
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tom 47
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nr 2
EN
This paper presents the method of particular solution for solving the Riccati equation and linear homogenous equations of second and third order, as well as its certain application to linear homogenous equations of n-th order. The conditions of effective integrability for equations (0.1) and (0.2) are expressed in symbolic (operator) form and also for equation (0.3) in fully expanded form. There have been proved three theorems which state the following: for any subclass of differential equations of the form (0.1), (0.2), (0.3), if there are known, respectively: a particular solution \(y_0\), a particular solution u 0 , two linearly independent particular solutions \(u_1 , u_2\), then it is possible to construct superclasses of differential equations of the given class, using classes cited in [6, 7, 8, 9]. Moreover, one may obtain their effectively integrable generalizations. Numerous examples provided illustrate the above results. The article presents also a practical way of applying the method of particular solution to linear equations of n-th order. This method enables us to integrate more general equations than those described in [4, 5, 14] of the form (0.1), (0.2), (0.3), (0.4) for which the particular solutions are cited therein.
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