Truncated V-fractional Taylor's Formula with Applications

José Vanterler da Costa Sousa, Edmundo Capelas de Oliveira

Abstract


In this paper, we present and prove a new truncated V-fractional Taylor's formula using the truncated V-fractional variation of constants formula. In this sense, we present the truncated V-fractional Taylor's remainder by means of V-fractional integral, essential for analyzing and comparing the error, when approaching functions by polynomials. From these new results, some applications were made involving some inequalities, specifically, we generalize the Cauchy-Schwartz inequality.

Keywords


Truncated V-fractional derivative, multivariable truncated V-fractional derivative, truncated V-fractional partial derivative, truncated V-fractional Jacobian matrix, truncated V-fractional Green

Full Text:

PDF

References


J. Stoer and R. Bulirsch, Introduction to numerical analysis, vol.12. Springer Science & Business Media, New York, 2013.

A. Ralston and P. Rabinowitz, A first course in numerical analysis. Courier Corporation, New York, 2001.

R. Courant and F. John, Introduction to calculus and analysis I. Springer Science & Business Media, New York, 2012.

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, vol. 198. 1999.

E. Capelas de Oliveira and J. A. Tenreiro Machado, A review of definitions for fractional derivatives and integral, Math. Probl. Eng., vol. 2014,, p. (238459), 2014.

R. Herrmann, Fractional calculus: An Introduction for Physicists, World Scientific Publishing Company, Singapore. 2011.

J. A. Tenreiro Machado, And I say to myself: What a fractional world!, Frac. Calc. Appl. Anal., vol. 14,, pp. 635--654, 2011.

M. D. Ortigueira and J. A. Tenreiro Machado, What is a fractional derivative?,'' J. Comput. Phys., vol. 293., pp. 4--13, 2015.

J. Vanterler da C. Sousa and E. Capelas de Oliveira, Mittag-Leffler functions and the truncated V-fractional derivative,'' submitted, (2017).}

U. N. Katugampola, A new fractional derivative with classical properties, arXiv preprint arXiv:1410.6535, 2014.

R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, A new definition of fractional derivative, J. Comput. and Appl. Math. , vol. 264, pp.65--70, 2014.

J. Vanterler da C. Sousa and E. Capelas de Oliveira, A new truncated M-fractional derivative unifying some fractional derivatives with classical properties,'' submitted, (2017).

J. Vanterler da C. Sousa and E. Capelas de Oliveira, M-fractional derivative with classical properties,'' submitted, (2017).

D. R. Anderson, Taylor's formula and integral inequalities for conformable fractional derivatives, in Contributions in Mathematics and Engineering, in Honor of Constantin Caratheodory, Springer, Berlin 25-43, 2016.

D. R. Anderson, Taylor's formula and integral inequalities for conformable fractional derivatives, arxiv.org/pdf/1409.5888v1, (2014).

W. G. Kelley and A. C. Peterson, The theory of differential equations: classical and qualitative}. Springer Science & Business Media, New York, 2010.

J. M. Steele, The Cauchy-Schwartz master class: an introduction to the art

of mathematical inequalities. Cambridge University Press, New York, 2004.

M. Z. Sarikaya and H. Budak, ``New inequalities of opial type for conformable fractional integrals,'' RGMIA Rearch Report Collection, vol. 19, p.10 pages, 2016.

J. Vanterler da C. Sousa and E. Capelas de Oliveira, ``A truncated V-fractional derivative in R^n,'' submitted, (2017).




DOI: https://doi.org/10.5540/tema.2018.019.03.525

Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM

Refbacks

  • There are currently no refbacks.


Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

 

Indexed in:

                       

         

 

Desenvolvido por:

Logomarca da Lepidus Tecnologia