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

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.

Keywords

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

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.

Keywords

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

References

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