Diferansiyel Denklemlerin Geometrik Yaklaşımı Üzerine

Öz

Çevremizdeki fiziksel gerçekleri ifade etmek, matematiksel modelleri hem teoride hem de uygulamada kullanmayı gerektirir. Bu modeller genellikle diferansiyel denklemleri içerdiğinden, açılabilir regle yüzeyler ve doğru kongrüanslarını kullanarak bu modellere yeni bir geometrik yaklaşım sunuyoruz. Bu makalede, birinci mertebeden diferansiyel denkleme göre teğet açılabilir bir regle yüzey ve çözüm fonksiyonu sunulmaktadır. Ayrıca, tam diferansiyel denklemin çözümüne ve çözüm fonksiyonuna dayalı bir doğru kongrüansı ifade edilmektedir. Ayrıca, teğet açılabilir regle yüzeyler ve doğru kongrüanslar ile ilgili bazı örnekler verilmektedir.

Anahtar Kelimeler

• Açılabilir regle yüzey
• teğet açılabilir regle yüzey
• birinci mertebeden diferansiyel denklem
• tam diferansiyel denklem
• doğru kongrüansı.

On The Geometric Approach of Differential Equations

Abstract

Expressing physical facts around us possesses to use mathematical models in both theory and application. Since these models usually involves differential equations, we present a novel geometric approach to these models using developable ruled surfaces and line congruences. In this paper, a tangent developable ruled surface according to first order differential equation and its solution function is presented. Moreover, we expressed a line congrunce based on the solution of exact differential equation and its solution function. Also, some examples of tangent developable ruled surfaces and line congruences are given.

Keywords

• Developable ruled surface
• tangent developable ruled surface
• first-order differential equation
• exact differential equation
• line congruence.

Kaynakça

1. Bottema, O. and Roth, B. (1979) Theoretical Kinematics, North- Holland Press, New York.
2. Chen, M. and Tang, K. (2010) A fully geometric approach for developable cloth deformation simulation. Vis. Computer 26 (2010), 853–863.
3. Frey, W. and Bindschadler, D. (1993) Computer aided design of a class of developable Bézier surfaces, vol. 8057, General Motors R & D Publication.
4. Nagle, K. R., Saff, E. B. and Snider, A. D. (2012) Fundamentals of Differential Equations. Boston: Addison-Wesley.
5. Kreyszig, E. (1991) Differential geometry, Dover Publications.
6. Odehnal, B. and Pottmann, H. (2001) Computing with discrete models of ruled surfaces and line congruences, Proceedings of the 2nd workshop on computational kinematics, Seoul.
7. P´erez, F. and Su´arez, J. A. (2007) Quasi-developable Bspline surfaces in ship hull design. Comp. Aided Geom. Design 39(2007), 853–862.
8. Pottman, H. and Wallner, J. (2001) Computational Line Geometry, Springer-Verlag, Berlin, Heidelberg.

9. Zill, D. G. (2001) A first course in differential equations with modeling applications. Pacific Grove, CA: Brooks/Cole Thomson Learning.