Model mathematical expressed by differential equation of second order.

Hrubina, Kamil ; Wessely, Emil ; Macurova, Anna 等

1. INTRODUCTION

The presented paper is aimed at the solution of the evolutionary differential equation of the second order by the approximation method which represents a mathematical model of the process control. Actually, the principal objective consists in searching for the solution to the numerical problem that substitutes the solution of the original problem of the mathematical analysis.

2. THEORETICAL BACKGROUND AND THE PROBLEM FORMULATION

2.1 Theoretical background

Let V and H be the two Hilbert's spaces within R in which we indicate the norms [[parallel]*[parallel].sub.V] and [[parallel]*[parallel].sub.H] and scalar products [((*,*)).sub.V], [((*,*)).sub.H] corresponding to them (Hrubina ,2000).

Let V [subset] H, injective representation from V to H is continuous and V is dense in H.

The space H is identified with its dual space. Let space V' indicate the as dual to V. Then H can be identified with some subspace within V', V [subset] H [subset] V'

Let be given a set of bilinear forms a(t; u, v) within V x V, (t [member of]<0, T>) which satisfies the conditions:

* [there exists]k > 0, k[[parallel]u[parallel].sup.2] [less than or equal to] a(t;u,v) [for all]u [member of] V

* a(t; u, v) = a(t; v, u), [for all]u, v [member of] V

* the representation t [right arrow] a(t; u, v) is differentiable of which it follows that there exist such numbers M and N that

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2.2 Problem formulation

Let be given the differential equation

u"(t) + A(t) x u(t) = f (t) u(0) = [u.sub.0], u'(0) = [u.sup.1] (1)

f (t) is given in [L.sub.2](0,T;H), [u.sub.0] is given within V and [u.sub.1] is given within H. The task is to find the solution u [member of] [L.sub.[infinity]] (0, T; V) and u' [member of] [L.sub.[infinity]](0,T; H) which satisfies (1). Then the problem solution is a unique one. Let [alpha] > 0, the task is to find the solution [u.sub.a] [member of] [L.sub.2] (0,T;V) and [u'.sub.a] [member of] [L.sub.2] (0,T;V') which satisfies

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[u.sub.0] be given within V, f (t) is given in [L.sub.2](- T,T; H), v(t) properly selected function, where it is supposed that v(t) [member of] [L.sub.2] (-T,O;H), v'(t)[member of] [L.sub.2] (-T,O;H), further on we suppose that the expression t [right arrow] a(t; u, v) can be expanded also for t < 0 (continuing differentiability). First of all we have to define one solution of the defined problem (2) and then to show the convergence of this solution to the solution of the differential equation (1) for ([alpha] [right arrow] 0) [alpha] converging to zero.

3. EXISTENCE AND UNEQUIVOCAL OF THE SOLUTION [u.sub.[alpha]]

We know that [u.sup.[alpha]] is the solution to the problem (2), where [alpha] is really given (Hrubina. & Macurova, 2003).

a) Let t [member of] <0, [alpha]>, we are searching for [u.sup.(0).sub.[alpha]] [member of] [L.sub.2] (O, [alpha]; V) and [u'.sup.0).sub.[alpha]] [member of] [L.sub.2](O, [alpha]; V') which is the solution to the differential equation

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Similarly as with the previous example, we can show the existence and unequivocal of the solution [u.sub.(1).sub.[alpha]] and thus proceeding step by step until the upper interval boundary is reached, i.e.

b) On the solution result regularity

We consider the following differential equation in formed

u'(t)+A(t) u(t) = f (t) u(0) = [u.sub.0] (3)

4. ON THE PROBLEM OF THE STRONG CONVERGENCE

First of all, the following suppositions are expressed. Let [OMEGA] be an open limited space and a regular one. Let the operator A defined within the interval be independent on T. Further on, let us express H = [L.sub.2] ([OMEGA])

[L.sub.2] (O,T;V) = [L.sub.2](Q), Q = [OMEGA] x <0, T>

We can prove that the sequence {[u.sub.[alpha]]} is compact. Therefore, it is sufficient to show that

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2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Based on the results of the third part of this work

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Applying a similar method, the decomposition of the first member on the right hand side is provided. Finally, we come to the conclusion that the sequence {[u.sub.[alpha]]} is compact in [L.sub.2](Q) alternatively we can choose the sequence that converges strong in [L.sub.2](Q), thus within the limit it converges to the solution u(t). As a result, each sequence {[u.sub.[alpha]]} converges strong to the solution u(t) in [L.sub.2](Q).

Applying a similar method, the decomposition of the first member on the right hand side is provided. The member becomes majority by the following relation:

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[u.sub.[alpha]] [member of] [L.sub.2](Q), in addition we require [partial derivative][u.sub.[alpha]]/[partial derivative] [L.sub.2](Q); [u.sub.[alpha]] is continuous. Then

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thus,

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In order to investigate the last member of the inequality, we define the function [w.sub.[alpha]b] in [L.sub.2], thus [w.sub.[alpha]b](t) = [u.sub.[alpha]](f)-[u.sub.[alpha]](t - b), b > 0.

Let us define the task:

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Let [w.sub.a,b] be the solution to the problem and let it satisfy the following inequality

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Finally, we come to the conclusion that the sequence {[u.sub.[alpha]} is compact in [L.sub.2] (Q) ( Macurova, 2007).

Moreover, we can choose the sequence that converges strong in [L.sub.2] (Q), thus within the limit it converges to the solution u(t). As a result, each sequence {[u.sub.[alpha]} converges strong to the solution u(t) in [L.sub.2](Q).

5. CONCLUSION

In general, we investigated the issue of the existence and unequivocal of the solution u(t) of the numerical problem which substitutes the previous problem of the mathematical analysis, i.e. the differential equation of the second order. Moreover, the problems of weak and strong convergence of the sequence {[u.sub.[alpha]} of the numerical problem which converges to the solution u(t) of the original problem [L.sub.2](Q). The paper presents the approximation method applied to the solution of the evolutionary differential equation of the second order. The contribution of the paper lies in the consideration of the problem which is approached to by means of the functional analysis and the theoretical results obtained to create the background to the creation of algorithms applied to the solution of the defined problem (Hrubina & Jadlovska, 2002). The theoretical results presented above represent the background to the creation and the implementation of algorithm applied to the solution of problems of processes optimum control as the solution of the differential equation of the second order by the approximation method is in general the solution of the mathematical model which is used to describe the process of control.

6. REFERENCES

Hrubina, K. (2000). Mathematical modeling of technical. Processes, Informatech Kosice, ISBN 80-88941-12-1, Kosice

Macurova, A. (2007). About solution of the non-linear differential equations, Technical University of Kosice, ISBN 978-80-8073-9i0-2, Kosice

Hrubina, K. & Jadlovska, A. (2002). Optimal control and approximation of variation inequalities. The international journal of system & cybernetics, Vol. 31, No. 9/10, 2002, pp. 1401-1414. ISSN: 0368-492X