Generalized Riemann Integral in the analysis of dynamic systems

Renata Masarova, Julia Kurnatova Institute of Applied Informatics, Automation and Mathematics

Faculty of Materials Science and Technology STU Trnava

renata.masarova@stuba.sk, julia.kurnatova@stuba.sk

Abstract—This paper shows one off the possible tutoring in

mathematic lecturing in the first semester on MTF . Gives the

idea of introducing the latest knowledge of theoretical research

into teaching and their use.

Keywords-dynamic system; differential equation; Generalized

Riemann integral

I. INTRODUCTION

The basic course on every college with a focus on technology or engineering includes mathematics. Its subject is mostly based on the fundamental algebra and mathematical analysis. This foundation has been taught without (or only with small) changes for decades. The main change lies in using computers for calculations. New directions in mathematical research and findings that have been published in the last few decades are missing. In this paper I want to show how easy it is, in several cases, to expand the subject of these courses with new findings.

II. DYNAMIC SYSTEMS

The starting point, which let to this problem, is following question: how to express solution of the dynamic system

( )xtfx ,=•

by the integral ( ) ( )∫=t

t

dxfxtF

0

,, ττ .

Generalized differential equations are defined in the paper [ ]3 .

Let nEG ⊂ and ( ) [ ] nRTGtxF →× ,0:, . Then a function

[ ] nRbax →,: is a solution of ( )txDF

d

dx,=

τ if and only if is

( )( ) [ ]TGttx ,0, ×∈ for any [ ]bat ,∈ and the difference

( ) ( )21 sxsx − for all [ ]bass ,, 21 ∈ may be approximated by the sum

( )( ) ( )( )( )∑=

−−k

iiiii xFxF

11,, ατατ (1)

where 2101 ... ss k =<<<= ααα is the partition of the

interval [ ]21 , ss and [ ]iii αατ ,1−∈ .

Remark. The previous point said that the solution of the

equation ( )xtfx ,=•

satisfies the equality

( ) ( ) ( )( ) [ ]bassdtttxfsxsxs

s

,, ,, 2112

2

1

∈=− ∫

The sum (1) directs Kurzweil to his definition of a new type of integral.

III. GENERALIZED RIEMANN INTEGRAL

Kurzweil and Henstoch defined the generalized Riemann integral in [ ] [ ]1 ,3 . It is also called Kurzweil – Henstoch integral.

We introduce definition of this integral:

A. Definition

Definition 1. Let [ ] Rbaf →,: , bxxxa n =<<<= ...10 .

The set ( ) ( ) ( ){ }nnIIID ξξξ ,,...,,,, 2211= is said to be a partition

of the interval [ ]ba, , [ ] kkkkk IxxI ∈= − ξ ,,1 for every

nk ,...2,1= . Let function [ ] ( )∞→ ,0,: baδ (this function is

called calibration). A partition ( ) ( ) ( ){ }nnIIID ξξξ ,,...,,,, 2211= is said to be compatible with the calibration δ if

( ) ( )( )kkkkkI ξδξξδξ +−⊂ , for every nk ,...,2,1= .

A function ( )xf is generalized Riemann integrable over

[ ]ba, if and only if RA ∈∃ , 0>∀ε there exists a function [ ] ( )∞→ ,0,: baδ such that for each partition D of [ ]ba, ,

which is compatible with the calibration δ , we have

( ) ε<− ADfR , , where ( ) ( )( )∑=

−−=n

kkkk xxfDfR

11., ξ .

978-1-4673-2427-4/12/$31.00 ©2012 IEEE

The number A is called a generalized Riemann Integral

(or Kurzweil – Henstoch integral) and denoted ( ) ( )∫b

a

dxxfGR .

B. Properties

In the paper [ ]6 the authors said a significant theorem to calculate this integral:

Theorem 1. Let function [ ] RbaF →,: is differentiable on [ ]ba, , then a function F ′ is GR – integrable on [ ]ba, and

( ) ( ) ( ) ( )aFbFdxxFGRb

a

−=′∫ .

C. Examples

Example 1. Integrating Dirichlet function

Let [ ] { }1,01,0: →f

( )[ ]

( ) [ ]1,0 if ,0

1,0 if ,1

∩−∈

∩∈⟨=

QRx

Qxxf .

Prove that ( ) ( ) 01

0

=∫ dxxfGR .

We choose following function ( )xδ :

( ) ,1=xδ if [ ] Qxx ∉∧∈ 1,0

( )22 +

=jjrε

δ , if [ ] ,...2,1,1,0 =∩∈ jQrj

Let a partition ( ) ( ){ }nn tJtJD ,,...,, 11= is compatible with

the calibration ( )xδ .

If Qti ∉ then ( ) 0=itf .

If Qti ∈ then ji rt = and

( )( ) ( )121

2222

++− ==≤−jjjiii rxxtfεε

δ .

Then ( )( ) εε

=<−− ∑∑∞

=+

=−

11

11

220

jj

n

iiii xxtf .

Example 2. Calculate the integral ( )∫1

04

1dx

xGR .

The function ( )

4

3

4

3

xxF = is continuous on [ ]1,0 and

differentiable. Then the function ( ) ( ) 4

1

xxFxG =′= is

integrable on [ ]1,0 by the theorem 1 and

( )3

4

4

30

4

311 4

31

04

=−=∫ dxx

GR .

IV. USE IN TEACHING

A. Riemann integral

The basic course of mathematical analysis defines the Riemann integral as the following:

Definition 2. Let [ ] Rbaf →,: be a bounded function. A

set { }nn xxxD ,...,, 10= , where bxxxa n =<<<= ...10 is a

partition on [ ]ba, , let 1max −−= iii

n xxD . For arbitrary

[ ]iii xxz ,1−∈ we define ( ) ( )( )∑=

−−=n

iiiin xxzfDfR

11, .

A function ( )xf is called Riemann integrable over [ ]ba, if and only if exists RA∈ that for arbitrary 0>ε and for arbitrary partition nD on [ ]ba, such that

∞→→ nDn for 0 and any [ ]iii xxz ,1−∈ we have

( ) ε<− ADfR n, .

Remark. We note the difference between definition 1. and definition 2. If we use the calibration ( ) tt =δ in the definition 1. we obtain the Riemann integral. The calculation theorem is also similar.

Theorem 2. (Newton-Leibnitz)

Let for any [ ]bax ,∈ ; ( ) ( )xfxF =′ , let ( )xf be

continuous on [ ]ba, . Then ( ) ( ) ( )aFbFdxxfb

a

−=∫ .

Remark. The functions in the example 1 and 2 are not

Riemann integrable. The function ( )4

1

xxf = is not Riemann

integrable, because it is not a bounded (and a continuous) function on [ ]1,0 . However, there exists its infinite integral.

Example 3. Calculate infinite integral dxx

∫1

04

1.

=

==++ →

−

→∫∫

1

4

3

0

14

1

0

1

04

4

3limlim

1

k

kkk

xdxxdx

x

3

4

4

3

4

31

lim0

=

−=+→

k

k

B. Benefits of introducing the GR integral

The generalized Riemann integral is a simple extension of the Riemann integral. It is easy to understand for students without deeper mathematical knowledge . The resultant tool is very strong. Every Riemann integrable function is also generalized Riemann integrable with the same result. Moreover there are a lot of functions, that are GR integrable but not Riemann integrable (example 1). By the same method we can calculate the infinite integral as well (example 2. and example3.).

CONCLUSION

In mathematic we know several types of integrals which are stronger than Riemann integral (integrate more classes of functions), i.e. Lebesque integral or Perron integral. The definitions of these integrals require a deeper knowledge of mathematics and therefore are inappropriate for students of the basic course of mathematic.

REFERENCES [1] R. Henstoch, “A Riemann type integral of Lebesque pover,” Canad. J.

Math., vol. 20, pp. 79–87, 1968.

[2] S. Hu, and V. Lakshmikanthan “Some remarks on generalized Riemann integral,” J. Math. Anal. Appl., vol. 137, pp. 515–527, 1989.

[3] J. Kurzweil, “Generalized ordinary differential equations and continuous dependenceon a parameter,” Czechoslovak Math. J., vol. 82, pp. 418–446, 1957.

[4] R. Masarova, “Zovšeobecnenie určitého integrálu,” Infomat, pp. 96–99, 2008.

[5] T. Salat, “Operations with derivatives, generalized Riemann integration, and the structure of some function spaces,” Rev. Roumaine math. Pures appl., vol. 38, pp. 443–457, 1993.

[6] C. Schwartz, and B. Thomson “More on the fundamental theorem of Calculus,” AMM, pp. 644–650, 1988.

[7] S. Schwabik, “Český příspěvek k moderním teoriím integrálu,” Konference o matematice a fyzice na VŠT, vol.5, pp. 11–20, 2007.