Abstract
İkinci mertebeden dalga
denklemini çözümü için ünlü D'Alembert formülünün, dalgaların dinamiğini
incelemek açısından çok önemli bir araç olduğu iyi bilinmektedir. Yüksek
mertebeden kısmi türevli diferansiyel denklemler için de D'Alembert tipinden
çözümlerin elde edilmesinin büyük önem taşıdığı açıktır. Bu makalede üçüncü
mertebeye göre homojen sabit katsayılı lineer kısmi diferansiyel denklemler
için Cauchy probleminin D'Alembert çözümleri ele alınmıştır. Son olarak, elde edilen
çözümler kullanılarak, üç farklı kök durumunda bazı bilgisayar testleri
yapılmıştır. Bulunan sonuçlar belli başlangıç profile sahip dalgaların dağılım
dinamiklerini açıkça ifade etmektedir.
References
- Courant, K., Hilbert, D. Methoden der Mathematischen Physik, Springer, Berlin, 1937.
- Courant, K., Lax, A., Remarks on Cauchy's Problem for Hyperbolic Partial Differential Equations with Constant Coefficients in Several Independent Variables, Comm. Pure Appl. Math., 8 (4), 1955, 497-502.
- Garding, L., Linear Hyperbolic Partial Differential Equations with Constant Coefficients, Acta Math., 85, 1950, 1-62.
- Hadamard, J., Le Probleme de Cauchy et les Equations aux Dérivées Partielies Linéaires Hyperboliques, Hermann, Paris, 1932.
- John, F., Special Topics in Partial Differential Equations, Lecture Notes, Institute of Mathematical Sciences, New York University, 1952.
- Lax, A., On Cauchy’s Problem for Partial Differential Equations with Multiple Characteristics, Comm. Pure Appl. Math., 9, 1956, 135-169.
- Leray, J., Hyperbolic Differential Equations, Institute for Advanced Study, Princeton, 1953.
- Mizohata, S., Lectures on Cauchy Problem, Tata Institute of Fundamental Research, Bombay, 1965.
- Petrovski, I. G., On the Cauchy Problem for Systems of Linear Partial Differential Equations in the Class of Non Analytic Functions, Bul. Mosk. Gos. Univ. Mat. Mekh. 7, 1938.
- Rasulov, M.L., Methods of Contour Integration, North Holland, Amsterdam, 1967.
- Sneddon, I.N., Elements of Partial Differential Equations, Mc.Grav-Hill Book Company Inc., 1957.
- Sobolev, S.L., Applications of Functional Analysis to Equations of Mathematical Physics, American Mathematical Society, Providence, 1963.
- Tikhonov, A.N., Samarskii, A.A., Equations of Mathematical Physics, Pergamon Press, Oxford, 1963. [Translated by Robson, A.R.M, and Basu, P.; Translation Edited by Brink, D.M.]
Abstract
It is well known that
the famous D'Alembert formula for solving the wave equation of second-order is
a very important instrument in the study of the dynamics of waves. It is also
obvious that D'Alembert's solutions for higher-order partial differential
equations are of great importance. In this paper, the D'Alembert solutions of
the Cauchy problem for linear partial differential equations with homogeneous
constant coefficients of the third-order are obtained. Finally, using the obtained
solutions, some computer tests on three distinct roots have been carried out. The results clearly indicate
the dispersion dynamics of waves with some initial profile.
References
- Courant, K., Hilbert, D. Methoden der Mathematischen Physik, Springer, Berlin, 1937.
- Courant, K., Lax, A., Remarks on Cauchy's Problem for Hyperbolic Partial Differential Equations with Constant Coefficients in Several Independent Variables, Comm. Pure Appl. Math., 8 (4), 1955, 497-502.
- Garding, L., Linear Hyperbolic Partial Differential Equations with Constant Coefficients, Acta Math., 85, 1950, 1-62.
- Hadamard, J., Le Probleme de Cauchy et les Equations aux Dérivées Partielies Linéaires Hyperboliques, Hermann, Paris, 1932.
- John, F., Special Topics in Partial Differential Equations, Lecture Notes, Institute of Mathematical Sciences, New York University, 1952.
- Lax, A., On Cauchy’s Problem for Partial Differential Equations with Multiple Characteristics, Comm. Pure Appl. Math., 9, 1956, 135-169.
- Leray, J., Hyperbolic Differential Equations, Institute for Advanced Study, Princeton, 1953.
- Mizohata, S., Lectures on Cauchy Problem, Tata Institute of Fundamental Research, Bombay, 1965.
- Petrovski, I. G., On the Cauchy Problem for Systems of Linear Partial Differential Equations in the Class of Non Analytic Functions, Bul. Mosk. Gos. Univ. Mat. Mekh. 7, 1938.
- Rasulov, M.L., Methods of Contour Integration, North Holland, Amsterdam, 1967.
- Sneddon, I.N., Elements of Partial Differential Equations, Mc.Grav-Hill Book Company Inc., 1957.
- Sobolev, S.L., Applications of Functional Analysis to Equations of Mathematical Physics, American Mathematical Society, Providence, 1963.
- Tikhonov, A.N., Samarskii, A.A., Equations of Mathematical Physics, Pergamon Press, Oxford, 1963. [Translated by Robson, A.R.M, and Basu, P.; Translation Edited by Brink, D.M.]
Details
| Primary Language | English |
|---|---|
| Subjects | Engineering |
| Journal Section | Articles |
| Authors | |
| Publication Date | August 5, 2019 |
| Published in Issue | Year 2019 Volume: 12 Issue: 1 |
Cite
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