论文信息 - Mathematical proof of closed form expressions for finite difference approximations based on Taylor series

Mathematical proof of closed form expressions for finite difference approximations based on Taylor series

Taylor series based finite difference approximations of derivatives of a function have already been presented in closed forms, with explicit formulas for their coefficients, However, those formulas were not derived mathematically and were based on observation of numerical results. In this paper, we provide a mathematical proof of those formulas by deriving them mathematically from the Taylor series.

[1] L. Collatz. The numerical treatment of differential equations , 1961 .

[2] Thomas L. Harman,et al. Advanced Engineering Mathematics Using Matlab V.4 , 1996 .

[3] Åke Björck,et al. Numerical Methods , 2021, Markov Renewal and Piecewise Deterministic Processes.

[4] I. R. Khan,et al. Closed-form expressions for the finite approximations of first and higher derivatives based on Taylor series , 1999 .

[5] F. B. Hildebrand,et al. Introduction To Numerical Analysis , 1957 .

[6] Curtis F. Gerald,et al. APPLIED NUMERICAL ANALYSIS , 1972, The Mathematical Gazette.

[7] A. Booth. Numerical Methods , 1957, Nature.

[8] Erwin Kreyszig,et al. Advanced Engineering Mathematics, Maple Computer Guide , 2000 .

[9] F. W. Kellaway,et al. Advanced Engineering Mathematics , 1969, The Mathematical Gazette.

[10] Ishtiaq Rasool Khan,et al. Digital Differentiators Based on Taylor Series , 1999 .

[11] Begnaud Francis Hildebrand,et al. Introduction to numerical analysis: 2nd edition , 1987 .

[12] J. Miller. Numerical Analysis , 1966, Nature.

[13] J. Douglas Faires,et al. Numerical Analysis , 1981 .

[14] H. E. Salzer,et al. Table errata: The numerical treatment of differential equations (third edition, Springer, Berlin, 1960) by L. Collatz , 1972 .