İkinci ve üçüncü mertebeden türev almada yeni yaklaşımlar

Yıl 2025, , 158 - 175, 31.01.2025

Çiğdem Dinçkal

https://doi.org/10.61112/jiens.1576464

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

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.

Kaynakça

• Groetsch CW (1984) The theory of Tikhonov regularization for Fredholm equations of the first kind. Research Notes in Mathematics, Pitman, Vol. 105 (Boston, Mass.–London: Advanced Publishing Program)
• Hanke M. and Scherzer O (2001) Inverse problems light: numerical differentiation. American Mathematical Monthly, 108(6):512–521.
• Tikhonov AN and Arsenin VY (1977) Solutions of Ill-posed Problems (Washington: Winston & Sons).
• Murio DA (1993) The Mollification Method and the Numerical Solution of Ill-posed Problems (New York: A Wiley-Interscience Publication, John Wiley & Sons Inc.).
• Wang YB Jia XZ and Cheng J (2000) A numerical differentiation method and its application to reconstruction of discontinuity. Inverse Problems 18:1461–1476.
• Jia XZ Wang YB and Cheng J (2003) The numerical differentiation of scattered data and it’s error estimate. Mathematics, A Journal of Chinese Universities 25:81–90.
• Chapra SC and Canale RP (2010) Numerical methods for engineers, Sixth Edition. McGraw Hill, New York.
• Turner PR (1994). Numerical Differentiation. In: Numerical Analysis. Macmillan College Work Out Series. Palgrave, London.
• Zhang Y Jin L Guo D Yin Y and Chou Y (2015) Taylor-type 1-step-ahead numerical differentiation rule for first-order derivative approximation and ZNN discretization. Journal of Computational and Applied Mathematics 273(1):29-40.
• Qu R (1996) A new approach to numerical differentiation and integration. Mathematical and Computer Modelling 24(10):55-68.
• Khan IR and Ohba R (2000) New finite difference formulas for numerical differentiation. Journal of Computational and Applied Mathematics 126(1-2):269-276.
• Abdul-Hassan NY Ali HA and Park C (2021) A new fifth-order iterative method free from second derivative for solving nonlinear equations. Journal of Applied Mathematics and Computing 68(1):2877-2886.
• Hyman JM Larrouturou (1982) The numerical differentiation of discrete functions using polynomial interpolation methods. Applied Mathematics and Computation 10(11):487-506.
• Jianping L (2005) General explicit difference formulas for numerical differentiation. Journal of Computational and Applied Mathematics 183(1):29-52.
• Zhang X Xiong C Ding Y and Ding H (2016) Prediction of chatter stability in high speed milling using the numerical differentiation method. The International Journal of Advanced Manufacturing Technology 89(1):2535–2544.
• Zhau Z Meng Z and He G (2009) A new approach to numerical differentiation. Journal of Computational and Applied Mathematics 232(2):227-239.
• Xu H and Liu J (2010) Stable numerical differentiation for the second order derivatives. Advances in Computational Mathematics 33(1):431-447.
• Ekaterinaris JA (2005) High-order accurate, low numerical diffusion methods for aerodynamics. Progress in Aerospace Sciences 41(3-4):192-300.
• Leonetti L and Mukhametzhanov MS (2022) Novel first and second order numerical differentiation techniques and their application to nonlinear analysis of Kirchhoff–Love shells. Computational Mechanics 70:29-47
• Yang HQ Chen ZJ Przekwas A and Dudley J (2015) A high-order CFD method using successive differentiation. Joıurnal of Computational Physics 281:690-707.