SOLUTION OF HOMOGENEOUS DIFFERENTIAL EQUATIONS WITH ALGEBRAIC COEFFICIENT BY FINITE DIFFERENCE EQUATIONS

Öz

There is no a general method for evaluating the implicit solutions of homogeneous differential equations since they generally have individual properties. In this study, the implicit solutions for some types of these differential equations are obtained by using finite difference equations. The implicit solution obtained by reducing some differential equations with algebraic coefficient to finite difference equations with constant coefficient is also considered and numerical examples are presented

Anahtar Kelimeler

• Ordinary differential equations, Difference equation, Algebra tic equations.

CEBİRSEL KATSAYILI HOMOJEN DİFERANSİYEL DENKLEMLERİN FARK DENKLEMLERİ İLE ÇÖZÜMÜ

Öz

Genel olarak, değişken katsayılı homojen diferansiyel denklemler, kendine has özellikler içerdiklerinden kapalı çözümlerinin elde edilebilmeleri için genel bir yöntem yoktur. Bu çalışmada bu tür diferansiyel denklemlerin bazı tiplerinin kapalı çözümleri fark denklemleri kullanılarak elde edilmiştir. Ayrıca çalışmada bazı cebirsel katsayılı diferansiyel denklemlerin sabit katsayılı fark denklemine indirgenmesi halinde elde edilen kapalı çözümü üzerinde durulmuş ve sayısal örnekler sunulmuştur.

Anahtar Kelimeler

• Adi diferansiyel denklemler, Fark denklemi, Cebirsel denklemler

Kaynakça

1. Alku S. (1992): “A Solution of Homegeneous Differential Equations with Variable Coefficients by Finite Difference Equations”, D.E.Ü. Fen Bilimleri Enstitüsü, Yüksek Lisans Tezi, İzmir.
2. Abramov S.A. (1989): “Rational Solutions of of Linear Differantial and Difference Equations with Polynomial Coefficients”, Zh.Vychisl. Mat. i Mat. Fiz. 29, no.11, 1611-1620, 1787.
3. Artzrouni M. (1987): “Conditions for Asymptotically Exponential Solutions of Linear Difference Equations with Variable Coefficients”, J. Math. Anal. Appl. 121, no.1, 160- 172.
4. Hooker J.W. (1987): “Oscillatory Second Order Linear Difference Equations and Riccati Equations”, Siam J. Math. Anal., 18, no.1, 54-63.
5. Kaczorek T. (1985): “Extension of the Method of Continuants for n-order Linear Difference Equations with Variable Coefficients”, Bull.Polish.Acad.Sci.Tech.Sci. 33, no.7-8, 395- 400.
6. Levy H., Lessman F. (1959): “Finite Difference Equations”, Sir Isaac Pitman&Sons Ltd., London.
7. Popenda J. (1987a): “Oscilation and Nonoscilation Theorems for Second Order Difference Equations”, J.Math.Anal.Appl. 123, no.1, 34-38.
8. Popenda J. (1987b): “One Expression for The Solutions of Second Order Difference Equations”, Proc.Amer.Math.Soc. 100, no.1, 87-93.

9. Tuzik A.I. (1989): “Solvability of a Discrete Equations of Convolution Type with Variable Coefficients”, Differentsial’nye Urauneniya 25, no.8, 1462-1464, 1472.