Properties of the solutions non-linear differential equation by the model of the technological process.

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

1. INTRODUCTION

Let the system differential equations is in the form

[x'.sub.1](t) = [g.sub.1] (t, x(t))

[x'.sub.2](t) = [g.sub.2] (t, x(t)), (1)

[g.sub.i] (t, [x.sub.1] (t), [x.sub.2] (t)) [member of] [C.sub.0] (D [equivalent to] J x [R.sup.2], R), i = 1,2 where [C.sub.0] is the place of the continuous real functions with three variables t, [x.sub.1] (t), [x.sub.2] (t) defined on set J x R x R. If non empty set [D.sub.0] [subset] D is open, [partial derivative][g.sub.1] (t,[x.sub.1] (t), [x.sub.2] (t)) / [partial derivative[x.sub.j], [partial derivative][g.sub.2] (t,[x.sub.1] (t), [x.sub.2] (t)) / [partial derivative] [x.sub.j] the derivatives are continuous functions on the domain D for every j [member of] {1,2}, than the one and only one integral curves [??] [member of] D of the system passes through every point ([t.sub.0], [x.sup.0.sub.1], [x.sup.0.sub.2]) [member of] [D.sub.0].

Let [bar.x](t) = [([x.sub.1](t), [x.sub.2](t)).sup.T] is general solution of the system (1). About each the solutions we suppose [bar.x](t), [x.sub.1]([t.sub.0]) = [x.sup.0.sub.1], [x.sub.2] ([t.sub.0]) = [x.sup.0.sub.2], [t.sub.0] [member of] J that exist on the interval J. Right terminal of the interval J is denoted h > [t.sub.0] > 0 and [J.sub.0] = ([t.sub.0], h). The system (1) is represented by the system differential equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

The significance of the quantities of the system (2) is following x = x(t) represented trajectory of the weight with the mass, K(x), B(x') are the properties of the materiel, M(t) is the mass of the weight for the motion (Hrubina et al., 2002). Let B([x.sub.2]) = 0, u(t) = 0, in the system (2), than

[x'.sub.1] = [x.sub.2]

[x'.sub.2] = -K([x.sub.1])/M(t) [x.sub.1], (3)

where

- K([x.sub.1])/M(t) [member of] [C.sub.0] (D, R) = [C.sub.0] (D [equivalent to] J x [R.sup.2], R).

Let in the system (3) is K([x.sub.1]) = 0 then all solutions x(t), t [member of] [J.sub.0] of the system (3) [x.sub.2] (t) = c, c [member of] R is the constant and [x.sub.1] (t) = c(t - [t.sub.0]), c [member of] R and x(t) is [x.sub.2] (t) - trivial solution for c = 0 and x(t) is [x.sub.1] (t) - constant solution if c = 0 .

Let in the system (3) K([x.sub.1]) [not equal to] 0. Let the functions r(t) > 0 and u(t) exist for every nontrivial solution x(t), t [member of] [J.sub.0] of the system (2), u(t) [member of] [C.sub.1] (J, R), u(t) [not equal to] 0, where [C.sub.1] (J, R) is the space of the derivative functions with one real variable t defined at the interval J.

2. PRELIMINARIES

The solution x(t) = ([x.sub.1](t), [x.sub.2](t)), t [member of] [J.sub.0] of the system (1) is called [x.sub.i]--trivial ([x.sub.i]--nontrivial, [x.sub.i]--constant, [x.sub.i]--bounded, [x.sub.i]--non-bounded) i = 1,2 is the certain point, if [x.sub.i]--is the trivial (nontrivial, constant, bounded, non-bounded) function at the interval [J.sub.0]. If [x.sub.i] (t) is the bounded function at the interval [J.sub.0] that is called [x.sub.i]--from above ([x.sub.i]--from underarm) non-bounded, i = 1,2 is the certain point, if [x.sub.i] (t) is the from above (from underarm) non-bounded function at the interval [J.sub.0].

The solution [bar.x](t) of the system (1) is called trivially, if it [x.sub.i] (t) = 0, i = 1,2 on the interval [J.sub.0]. [bar.x](t) is non-trivially, i = 1,2 (shortly is called nontrivially) in the others cases (Hrubina & Macurova, 2003).

The solution [bar.x](t) of the system (1) is called constant, if it [x.sub.1] (t) = [x.sub.1] ([t.sub.0]), [x.sub.2] (t) = [x.sub.2] ([t.sub.0]) on the interval [J.sub.0]. In the others cases x(t) is non-constant solution.

The solution [bar.x](t) of the system (1) is called bounded, if [x.sub.i]--bounded, i = 1,2 functions on [J.sub.0] the interval. In the others cases [bar.x](t) is non-bounded function, i = 1,2 (shortly is called non-bounded).

If (0,0), t [member of] [J.sub.0] is [x.sub.12]--trivially solution of the system (1), [g.sub.i] (t,0,0) = 0, i = 1,2 for every t [member of] [J.sub.0].

If t [member of] [J.sub.0] is [x.sub.1](t)--trivially solution (1), [g.sub.1] (t,0, [x.sub.2](t)) = 0, [x'.sub.2](t) = [g.sub.2] (t,0,[x.sub.2](t)).

Similarly for [x.sub.2] (t)--trivially solution of the system (1) hold true [x'.sub.1](t) = g (t,[x.sub.1](t),0,), 0 = [g.sub.2] (t, [x.sub.1] (t),0) (Macura, 2003).

For system (1) hold true, if [x.sub.2] (t)--is trivially solution, need not be [x.sub.1] (t)--trivially.

Let in the system (3) is K([x.sub.1]) = 0 then all solutions x(t), t [member of] [J.sub.0] of the system (3) are [x.sub.2] (t) = c, c [member of] R is the constant and [x.sub.1] (t) = c(t - [t.sub.0]), c [member of] R and x(t) is [x.sub.2] (t)--trivial solution for c = 0 and x(t) is [x.sub.1] (t)--constant solution for c = 0. Let in the system (3) is K([x.sub.1]) [not equal to] 0.

3. TRANSFORMATION EQUATIONS

Let the functions r(t) > 0 and u(t) exist for every nontrivial solution x(t), t [member of] [J.sub.0] of the system (2) and u(t) [member of] [C.sub.1] (J, R),u(t)[not equal to] 0, where [C.sub.1] (J, R) is the space of the derivative functions with one real variable t defined at the interval J (Macurova, 2007). Let for the solutions [x.sub.i] (t), t [member of] [J.sub.0], i = 1,2 be valid

[x.sub.1](t)=r(t)cosv(t) [x.sub.2](t)=r(t)sinv(t) (5)

The system (3) is represented by the equations (5) in the form

r'(t)cosv(t)-r(t)sinv(t)v'(t)=r(t)sinv(t) r'(t)sinv(t)+r(t)cosv(t)v'(t)= - K(t,r(t)cosv(t))/M(t) r(t)cosv(t) (6)

After finishing at the system (6) are the equations in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

We denoted with the symbols [I.sub.i], i = 1,2 the next integrals following

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

where y(t) t [member of] [J.sub.0] represented the continuous function.

Let for all continuous functions y(t), t [member of] [J.sub.0] exist the integrals [I.sub.1], [I.sub.2] according to contentions a) to f). That about every nontrivial solution x(t), t [member of] [J.sub.0] of the system (2) deals that is (Macura, 2003)

a) [x.sub.1], [x.sub.2]--non-bounded if [I.sub.1] = [infinity], [I.sub.2] = [+ or -] [infinity],

b) [x.sub.1], [x.sub.2]--non-bounded if [I.sub.1] = [infinity], [I.sub.2] = K, K [member of] R is the constant, K + u([t.sub.0]) [not equal to] k[pi]/2], k [member of] Z is the integer number,

c) [x.sub.1], [x.sub.2]--bounded that [x.sub.1](t) [right arrow] 0 if [I.sub.1] = -[infinity], [I.sub.2] = [+ or -] [infinity]

d) [x.sub.1], [x.sub.2]--bounded that [x.sub.1](t) [right arrow] 0 if [I.sub.1] = -[infinity], [I.sub.2] = K, when K [member of] R is the constant, K + u([t.sub.0]) [not equal to] k[pi]/2, k [member of] Z is the integer number,

e) [x.sub.1], [x.sub.2]--bounded if [I.sub.1] = L, L > 0, [I.sub.2] = [+ or -] [infinity],

f) [x.sub.1], [x.sub.2]--bounded if [I.sub.1] = L, L > 0, [I.sub.2] = K, K [member of] R is the constant K + u([t.sub.0]) [not equal to] k[pi]/2], k [member of] Z is the integer number (Macura, 2005).

When we integrate (7) in the interval <[t.sub.0], h> we have (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If exist limited (improper) value r(h)> 0 (r(h) = [infinity]), very if the polar function r(t) is bounded (non-bounded), that every non-trivial solution x(t), t [member of] [J.sub.0] of the system (2) is [x.sub.1], [x.sub.2]--bounded ([x.sub.1], [x.sub.2]--non-bounded). The situation [x.sub.1] (t) [right arrow] 0, [x.sub.2] (t) [right arrow] 0 is if and only if [I.sub.1] = -[infinity] or if r(h) = 0.

4. CONCLUSION

Apply to the transformation by the polar coordinates on the nonlinear differential equation of the second order we have the integral system equations for the expressed asymptotic properties of the solution of the differential equation in the implicit form (Hrubina & Jadlovska, 2002). The properties described the technological process and the possibility of the motion in process (Hrubina, 2000).

5. REFERENCES

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

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

Hrubina, K.; Jadlovska A. & Hrehova, S. (2002). Methods and Tasks of the Operation Analysis Solution by Computer, Informatech, Ltd., ISBN 80-88941-19-9, Kosice

Hrubina, K. & Macurova, A. (2003). On the approximation method applied to the solution of mathematical model expressed by differential equation of the second order. Transactions of the Universities of Kosice, 2003, pg. 36-47. ISSN 1335-2334

Macura, D. (2005). Function of Multivariable, University of Presov, ISBN 80 -8068-321-2, Presov

Macura, D. (2003). Ordinary Differential Equations, University of Presov, ISBN 80-8068-175-9, Presov

Macurova, A. (2007). About solution of the non-linear differential equations, Technical University of Kosice. ISBN 978-80-8073-910-2.Kosice