New approaches to numerical differentiation for second and third order

Öz

In this article, new numerical methods for calculation of second and third order derivatives are designed by using basic finite difference methods; forward, central and backward finite difference approaches. Those approaches are originally derived from the well-known Taylor series. Main advantage of new numerical formulas (named as Improved Backward Finite Difference Method, Improved Forward Finite Difference Method) is that they produce more accurate numerical results with smaller step size than the well-known backward and forward finite difference methods. For this purpose, some numerical examples are given to compare these new formulas with the traditional finite difference methods; backward and forward. The performance of the new methods in terms of error analysis and elapsed time for both second and third order derivative computations is also presented.

Anahtar Kelimeler

• Central Finite Difference Method
• Forward Finite Difference Method
• Backward Finite Difference Approaches
• Improved Backward Finite Difference Method
• Improved Forward Finite Difference Method.

İkinci ve üçüncü mertebeden türev almada yeni yaklaşımlar

Öz

Bu makalede, temel fark; geri, merkezi ve ileri yöntemler kullanılarak, 2. ve 3. mertebeden türev almak için yeni yöntemler geliştirilmiştir. Bu yöntemler; iyi bilinen Taylor serilerinden baz alınarak, oluşturulmuştur. Yeni yöntemlerin temel yöntemlere göre ana avantajı; küçük adım boylarında daha kesin ve hassas sayısal sonuçlar üretmektir. Bu amaçla, sayısal örnekler verilmiş. Burada geleneksel yöntemler ile yeni yöntemler karşılaştırılmıştır. Yeni yöntemlerin performans analizleri hata analizi ve hesaplamada geçen süre cinsinden 2. ve 3. mertebelerde türev almak için sunulmuştur.

Anahtar Kelimeler

• Merkezi Sonlu Fark Yöntemi,
• İleri Sonlu Fark Yöntemi
• Geri Sonlu Fark Yöntemi
• İyileştirilmiş Geri Sonlu Fark Yöntemi
• İyileştirilmiş İleri Sonlu Fark Yöntemi

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