When we derive differential equations for flow, i.e. the mass, momentum, and energy conservations, we use Taylor's series to determine the relationship between the 2 points. Are there no other series that produce a more precise equations?
Dear respective researchers, please give your reply.
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Dear Prof. Tapan K. Sengupta
,Thank you for your reply.
What do you think if we use this series to derive the Navier Stokes equation?
Please help me to recommend and comment my all article, I will do the same for you. Advance Thanks
Yes, Taylor's Polynomials have produced a more precise equation than Taylor's series.
Good question I will follow to find an answer
Spectral plane should be checked !!?
I feel that I need to revise Taylor series itself first
Thanks for checking my knowledge however it is a tough check for practitioner like myself :)
Dear Nazaruddin Sinaga
I agree with Isam Alkhalifawi
Best regards
Dear Nazaruddin Sinaga and Readers,
Please consult
A GENERALIZATION OF TAYLOR'S SERIES
by
D. V. WIDDERt
https://www.ams.org/journals/tran/1928-030-01/S0002-9947-1928-1501425-4/S0002-9947-1928-1501425-4.pdf
Dear Janusz Pudykiewicz ,
Thank you for your reply and valuable contributions.
Best regards,
Nazar
https://www.quora.com/Are-there-no-other-series-that-produce-a-more-precise-equation-than-Taylors-series
@Faraed Salman
Please consider the paper mentioned in my comment last year, in the first section we can read:
”In view of the great importance of Taylor's series in analysis, it may be regarded as extremely surprising that so few attempts at generalization have been made. The problem of the representation of an arbitrary function by means of linear combinations of prescribed functions has received no small amount of attention. It is well known that one phase of this problem leads directly to Taylor's series, the prescribed functions in this case being polynomials. It is the purpose of the present paper to discuss this same phase of the problem when the prescribed functions are of a more general nature“.
and:
“Of course this problem becomes definite only after a precise definition of the phrase "best approximation" has been given. Various methods have been used, of which we mention the following:
(A) The method of least squares; (B) The method of Tchebycheff; (C) The method of Taylor.”
The answer provided in the Quora link above is not sufficiently precise.
Taylor series are exact, it has nothing to do with being precise. Maybe what you're asking is if some other series could produce a better approximation with fewer terms?
The answer is No!
Please consider Padé approximation:
The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons Padé approximants are used extensively in computer calculations.
https://en.wikipedia.org/wiki/Padé_approximant
https://mathworld.wolfram.com/PadeApproximant.html