Szemináriumok

A new successive approximation method of general non-local boundary value problems

Időpont: 
2018. 10. 04. 10:15
Hely: 
H306
Előadó: 
Rontó Miklós (Miskolci Egyetem)

We propose a new successive approximation technique for the solvability analysis and approximate solution of general non-local boundary value  problems for non-linear systems of differential equations with locally Lipschitzian non-linearities. It will be studied the non-linear boundary value problem 
$\frac{dx(t)}{dt}=f(t,x(t)),\,t\in \left[ a,b\right] ,  \Phi (x)=d, $
where $\Phi :C(\left[ a,b\right] ,\mathbb{R}^{n})\rightarrow \mathbb{R}^{n}$ is a vector functional (possibly non-linear), $~f:\left[ a,b\right]
\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}~$is a function satisfying the Caratheodory conditions in a certain bounded set $D$, which will be concretized later, $d$ is a given vector and $f\in Lip(K,D),$ i.e. $f$ locally Lipschitzian 
$\left\vert f(t,u)-f(t,v)\right\vert \leq K\left\vert u-v\right\vert ,~\text{for all}\ \left\{ u,v\right\} \subset D~\text{ }a.e.t\in \left[ a,b\right] . $
By a solution of the problem, one means an absolutely continuous function satisfying the differential system almost everywhere on $\left[ a,b\right] .$ The analysis is constructive in the sense that it allows one to both study the solvability of the problem and approximately construct its solutions by operating with objects that are determined explicitly in finitely many steps of computation. The practical application of the technique is explained on a numerical example.

Two Limit Cycles in a Two-Species Reaction

Időpont: 
2018. 09. 27. 10:15
Hely: 
BME H. épület 306-os terem
Előadó: 
Ilona Nagy (BME, Department of Analysis)

Kinetic differential equations, being nonlinear, are capable of producing many kinds of exotic phenomena. However, the existence of multistationarity, oscillation or chaos is usually proved by numerical methods. Here we investigate a relatively simple reaction among two species consisting of five reaction steps, one of the third order. About this reaction we show the following facts (using symbolic methods): the kinetic differential equation of the reaction has two limit cycles surrounding the stationary point of focus type in the positive quadrant. The outer limit cycle is always stable and the inner one is always unstable. We also performed the search for partial integrals of the system and have found one such integral. Application of the methods needs computer help (Wolfram language and Singular) because the symbolic calculations to carry out are too complicated to do by hand. Even to make characteristic drawings is very far from trivial: the inner limit cycle is very small. Some of the methods we use are relatively new, and all of them have never been used in reaction kinetics, although they can be used to have similar exact results on other models.

 

An intrinsically adaptive formulation of multistep methods

Időpont: 
2018. 09. 20. 11:15
Hely: 
H607
Előadó: 
Carmen Arévalo

Multistep methods are important tools for solving ordinary differential equations with initial conditions. In order for these methods to be efficient they must be adaptive, that is, they must allow the choice of an appropriate step-size for each integration step. We present a comprehensive way of formulating multistep methods that is adaptive by construction and show how this methodology can be applied to particular situations. We also show the application to strong stability preserving methods, used to solve ODEs arising from the semi-discretization of time-dependent partial differential equations (PDEs), especially hyperbolic PDEs with shocks. We finally demonstrate how we apply the formulation to differential algebraic equations (DAEs), an ODE system coupled with algebraic constraints.

Adaptive Time-Stepping. Time transformations applied to reversible Hamiltonian dynamics and weakly dissipative systems

Időpont: 
2018. 09. 20. 10:15
Hely: 
H607
Előadó: 
Gustaf Söderlind

In the second talk on time step adaptivity, we focus on the special needs of conservative dynamical systems. This includes Hamiltonian problems, and weakly dissipative systems. In integrable Hamiltonian problems, the mathematical solution is time reversible, which precludes the use of classical controllers, which adapt the step size to manage the error observed in previous steps. Instead, a time reversible tracking algorithm is developed, which allows full reversibility of the adaptive computational process. This is shown to preserve first integrals over long times, and even improves the accuracy over constant step size symplectic integrators. We demonstrate the procedure in two examples from celestial mechanics, and then proceed to demonstrate how a similar approach can be combined with splitting methods in weakly dissipative systems. The latter approach has been put to effective use in rolling bearing dynamic simulation.

(Professors Gustaf Söderlind and Carmen Arévalo will present three talks during their visit to Budapest. All three talks deal with adaptive methods of ODEs. Professor Söderlind’s first talk will be held at the Seminar on Applied Analysis of the Department of Applied Analysis at ELTE university (details at the Miklós Farkas seminar), while his second talk together with Professor Arévalo’s talk will be presented at the Miklós Farkas Seminar. Although professor Söderlind’s talks form a complete presentation together, yet they can be followed separately.)

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