# First-order systems of linear partial differential equations: normal forms, canonical systems, transform methods

• Volume: 13, Issue: 1, page 109-132
• ISSN: 2300-133X

top

## Abstract

top
In this paper we consider first-order systems with constant coefficients for two real-valued functions of two real variables. This is both a problem in itself, as well as an alternative view of the classical linear partial differential equations of second order with constant coefficients. The classification of the systems is done using elementary methods of linear algebra. Each type presents its special canonical form in the associated characteristic coordinate system. Then you can formulate initial value problems in appropriate basic areas, and you can try to achieve a solution of these problems by means of transform methods.

## How to cite

top

Heinz Toparkus. "First-order systems of linear partial differential equations: normal forms, canonical systems, transform methods." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 13.1 (2014): 109-132. <http://eudml.org/doc/268779>.

@article{HeinzToparkus2014,
abstract = {In this paper we consider first-order systems with constant coefficients for two real-valued functions of two real variables. This is both a problem in itself, as well as an alternative view of the classical linear partial differential equations of second order with constant coefficients. The classification of the systems is done using elementary methods of linear algebra. Each type presents its special canonical form in the associated characteristic coordinate system. Then you can formulate initial value problems in appropriate basic areas, and you can try to achieve a solution of these problems by means of transform methods.},
author = {Heinz Toparkus},
journal = {Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica},
keywords = {characteristic coordinate system},
language = {eng},
number = {1},
pages = {109-132},
title = {First-order systems of linear partial differential equations: normal forms, canonical systems, transform methods},
url = {http://eudml.org/doc/268779},
volume = {13},
year = {2014},
}

TY - JOUR
AU - Heinz Toparkus
TI - First-order systems of linear partial differential equations: normal forms, canonical systems, transform methods
JO - Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
PY - 2014
VL - 13
IS - 1
SP - 109
EP - 132
AB - In this paper we consider first-order systems with constant coefficients for two real-valued functions of two real variables. This is both a problem in itself, as well as an alternative view of the classical linear partial differential equations of second order with constant coefficients. The classification of the systems is done using elementary methods of linear algebra. Each type presents its special canonical form in the associated characteristic coordinate system. Then you can formulate initial value problems in appropriate basic areas, and you can try to achieve a solution of these problems by means of transform methods.
LA - eng
KW - characteristic coordinate system
UR - http://eudml.org/doc/268779
ER -

## References

top
1. [1] W.F. Ames, Numerical methods for partial differential equations, Second edition, Computer Science and Applied Mathematics. Applications of Mathematics Series. Academic Press, New York, Thomas Nelson & Sons, London-Lagos-Melbourne, 1977. Cited on 110. Zbl0577.65077
2. [2] C. Babovsky, H. Toparkus, Revisited: the explicit solution of the characteristical initial value problem for hyperbolic first-order systems, Integral Transforms Spec. Funct. 20 (2009), no. 1-2, 79-81. Cited on 130.[WoS] Zbl1175.35082
3. [3] H. Berndt, H. Toparkus, Erprobung von Runge-Kutta-Verfahren zur Lösung von charakteristischen Anfangswertproblemen für hyperbolische Differentialgleichungssysteme erster Ordnung, Wissensch. Beitr., Friedrich-Schiller-Univ., Jena, 1984, 20-35. Cited on 130. Zbl0545.65067
4. [4] B. Davies, Integral transforms and their applications, Second edition, Applied Mathematical Sciences 25, Springer-Verlag, New York, 1985. Cited on 125.
5. [5] L. Debnath, Integral transforms and their applications, CRC Press, Boca Raton, FL, 1995. Cited on 125 and 128. Zbl0920.44001
6. [6] E. Dietzel, H. Toparkus, The explicit solution of the characteristical initial value problem for canonical hyperbolic systems by means of the two-dimensional Laplace transform, Integral Transform. Spec. Funct. 7 (1998), no. 3-4, 225-236. Cited on 130. Zbl0931.35025
7. [7] G. Doetsch, Einführung in Theorie und Anwendung der Laplace-Transformation, Ein Lehrbuch für Studierende der Mathematik, Physik und Ingenieurwissenschaft, Zweite, Mathematische Reihe, Band 24, Birkhäuser Verlag, Basel-Stuttgart, 1970. Cited on 118.
8. [8] D.G. Duffy, Transform methods for solving partial differential equations, Second edition, Chapman & Hall/CRC, Boca Raton, FL, 2004. Cited on 125 and 127. Zbl1073.35001
9. [9] G.B. Folland, Fourier analysis and its applications, The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. Cited on 125.
10. [10] I.M. Gelfand, G.E. Schilov, Verallgemeinerte Funktionen (Distributionen). III. Einige Fragen zur Theorie der Differentialgleichungen, Hochschulbücher für Mathematik, Band 49, VEB Deutscher Verlag der Wissenschaften, Berlin 1964. Cited on 125. Zbl0133.37503
11. [11] W. Haack, W. Wendland, Vorlesungen über Partielle und Pfaffsche Differentialgleichungen, Mathematische Reihe, Band 39, Birkhäuser Verlag, Basel-Stuttgart, 1969. Cited on 110.
12. [12] G. Hellwig, Bemerkungen zu der Satzgruppe von Hilbert über Systeme elliptischer Differentialgleichungen, Math. Z. 55 (1952), 276-283. Cited on 110. Zbl0046.10202
13. [13] G. Hellwig, Partial differential equations. An introduction, Second edition, Mathematische Leitfäden. B. G. Teubner, Stuttgart, 1977. Cited on 110 and 111.
14. [14] M. Hermann, Numerik gewöhnlicher Differentialgleichungen, Anfangs-und Randwertprobleme, Oldenbourg Verlag, Munich, 2004. Cited on 119.
15. [15] E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen. Teil I: Gewöhnliche Differentialgleichungen, Mathematik und ihre Anwendungen in Physik und Technik, Reihe A, Bd. 18, Akademische Verlagsgesellschaft, Geest & Portig K.-G., Leipzig 1959. Cited on 123.
16. [16] J. Kevorkian, Partial differential equations. Analytical solution techniques, Second edition, Texts in Applied Mathematics 35, Springer-Verlag, New York, 2000. Cited on 110 and 114. Zbl0937.35001
17. [17] A. Kratzer, W. Franz, Transzendente Funktionen, Mathematik und ihre Anwendungen in Physik und Technik, Reihe A, Bd. 28, Akademische Verlagsgesellschaft, Geest & Portig K.-G., Leipzig 1960. Cited on 129.
18. [18] F. Oberhettinger, L. Badii, Tables of Laplace transforms, Springer-Verlag, New York-Heidelberg, 1973. Cited on 129. Zbl0285.65079
19. [19] F. Oberhettinger, Tabellen zur Fourier Transformation, Springer-Verlag, Berlin- Göttingen-Heidelberg, 1957. Cited on 125. Zbl0077.12002
20. [20] A.D. Polyanin, V.I. Zaitsev, Handbook of exact solutions for ordinary differential equations, CRC Press, Boca Raton, FL, 1995. Cited on 125. Zbl0855.34001
21. [21] A.D. Poularikas, The transforms and applications handbook, The Electrical Engineering Handbook Series, CRC Press, Boca Raton, FL, IEEE Press, New York, 1996. Cited on 121 and 125. Zbl0851.44001
22. [22] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and series. Vol. 5. Inverse Laplace transforms, Gordon and Breach Science Publishers, New York, 1992. Cited on 129. Zbl0781.44002
23. [23] W.I. Smirnow, Lehrgang der höheren Mathematik. Teil IV, Dritte, berichtigte Auflage. Hochschulbücher für Mathematik, Band 5, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963. Cited on 110 and 114.
24. [24] I.N. Sneddon, Fourier transforms, (Reprint of the 1951 original) Dover Publications, Inc., New York, 1995. Cited on 121 and 125. Zbl0038.26801
25. [25] H. Toparkus, Zur Lösung des charakteristischen Anfangswertproblems bei hyperbolischen Systemen erster Ordnung im skalaren linearen Fall, Sem. Inst. Prikl. Mat. Dokl. No. 21 (1990), 5-7, 65, 67, 69. Cited on 114 and 130.
26. [26] H. Toparkus, The one-dimensional heat equation as a first-order system: formal solutions by means of the Laplace transform, Bull. Math. Anal. Appl. 4 (2012), no. 2, 174-185, http://www.bmathaa.org. Cited on 117, 118, 119 and 120. Zbl1314.35018
27. [27] A.N. Tychonoff, A.A. Samarskii˘, Differentialgleichungen der mathematischen Physik, Hochschulbücher für Mathematik, Bd. 39, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959. Cited on 114.
28. [28] I.N. Vekua, Systeme von Differentialgleichungen erster Ordnung vom elliptischen Typus und Randwertaufgaben; mit einer Anwendung in der Theorie der Schalen, Mathematische Forschungsberichte, II. VEB Deutscher Verlag der Wissenschaften, Berlin, 1956. Cited on 110. Zbl0072.10503
29. [29] I.N. Vekua, Verallgemeinerte analytische Funktionen, Herausgegeben von Wolfgang Schmidt Akademie-Verlag, Berlin 1963. Cited on 110.
30. [30] D. Voelker, G. Doetsch, Die zweidimensionale Laplace-Transformation. Eine Einführung in ihre Anwendung zur Lösung von Randwertproblemen nebst Tabellen von Korrespondenzen, Verlag Birkhäuser, Basel, 1950. Cited on 111. Zbl0040.05902
31. [31] R. Zurmühl, Matrizen und ihre technischen Anwendungen, Vierte neubearbeitete Auflage Springer-Verlag, Berlin, 1964. Cited on 112. Zbl0123.32201

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.