D’ALEMBERT’S SOLUTION OF THE INITIAL VALUE PROBLEM FOR THE THIRD-ORDER LINEAR HYPERBOLIC EQUATION

Öz

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.

Anahtar Kelimeler

• D’alembert tipli çözüm,üçüncü mertebeden hiperbolik denklem,karakteristik denklem

Öz

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.

Anahtar Kelimeler

• D'Alembert's solution,third-order hyperbolic equation,characteristic equation

Kaynakça

1. Courant, K., Hilbert, D. Methoden der Mathematischen Physik, Springer, Berlin, 1937.
2. 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.
3. Garding, L., Linear Hyperbolic Partial Differential Equations with Constant Coefficients, Acta Math., 85, 1950, 1-62.
4. Hadamard, J., Le Probleme de Cauchy et les Equations aux Dérivées Partielies Linéaires Hyperboliques, Hermann, Paris, 1932.
5. John, F., Special Topics in Partial Differential Equations, Lecture Notes, Institute of Mathematical Sciences, New York University, 1952.
6. Lax, A., On Cauchy’s Problem for Partial Differential Equations with Multiple Characteristics, Comm. Pure Appl. Math., 9, 1956, 135-169.
7. Leray, J., Hyperbolic Differential Equations, Institute for Advanced Study, Princeton, 1953.
8. Mizohata, S., Lectures on Cauchy Problem, Tata Institute of Fundamental Research, Bombay, 1965.

9. 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.
10. Rasulov, M.L., Methods of Contour Integration, North Holland, Amsterdam, 1967.
11. Sneddon, I.N., Elements of Partial Differential Equations, Mc.Grav-Hill Book Company Inc., 1957.
12. Sobolev, S.L., Applications of Functional Analysis to Equations of Mathematical Physics, American Mathematical Society, Providence, 1963.
13. 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.]