Take part in involution take part in differential equations

 

Definition:If at stubborning equality done this stubborning be named to involution.

This as functions organize involution.

Look at following equation

(1)

Here involution that is free term.

On the bottom (1) well, we will show made unraweling (1) inequality involution been make a change:

(2)

For extense get. This is not delivery hurt for totall. We, in the bottom (2) from equality will found together . That, made up this equality.

(3)

This equal then, we will increasing.

In that

(4)

be crop this equality.

Third and fourth equality spearated. In result:

(5)

be crop this equality. Well, (5) equality (1) this shown. That equality to be help, then simple differencial made up equation.

Examples:

1.                been. n=2, m=1 we use this values then made up equation.

(n-m)=1, (2n –m)=3.Well, there are to (5).The result:

(6)

be crop this equalition.

this equality total value its one of own value and its one gender part composed of total together.

We will find one fender part’s total unravelling:

,made up carasteristic equality

from than

Well, the total unravelling’s one gender of part composed of this:

b) (6) we will take off one own unraveling this equality:

searching. In shown it is to (6) take put in

from

, d- voluntary number.

Well (6) unraveling private equality.

Well, (6) in equality total unraveling.

(7)

(7) untying (6) the total unravelling.

2.. We (n-m)=2,

(2n-m)=5,

There are to (5) put on. Result:

(8)

be crop this equality.

(8) this equality unraveling same before the xample. We could set free this equality from involution.

Conclusion: With us, well in (1) equality stage of numbers to tied funcsions should,came too.

But, unraveling this kind of examples came to difficult direction.

But, this kind of equality:

(9)

we could unraveling. Well, been position . In this , are to dendent on funcsions.(9) equality (1) we should look at this unraveling as equality’s state.

 

References:

 

1.                Винер И. Я. Дифференциальные уравнения с инволюциями. // Дифференциальные уравнения. Том 5, 1969.
2.                Хромов А. П. Смешанная задача для дифференциального уравнения с инволюцией и потенциалом специального вида. // Известия Саратовского университета, Нов.сер. Математика, Механика, Информатика. 2010,,т.10. вып № 4,с.17–22.