Truncated V-fractional Taylor's Formula with Applications

Authors

  • José Vanterler da Costa Sousa Imecc-Unicamp - Universidade Estadual de Campinas
  • Edmundo Capelas de Oliveira Imecc-Unicamp

DOI:

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

Keywords:

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

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.

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).

Downloads

Published

2018-12-17

How to Cite

Sousa, J. V. da C., & Oliveira, E. C. de. (2018). Truncated V-fractional Taylor’s Formula with Applications. Trends in Computational and Applied Mathematics, 19(3), 525. https://doi.org/10.5540/tema.2018.019.03.525

Issue

Vol. 19 No. 3 (2018)

Section

Original Article

License

Copyright

Authors of articles published in the journal Trends in Computational and Applied Mathematics retain the copyright of their work. The journal uses Creative Commons Attribution (CC-BY) in published articles. The authors grant the TCAM journal the right to first publish the article.

Intellectual Property and Terms of Use

The content of the articles is the exclusive responsibility of the authors. The journal uses Creative Commons Attribution (CC-BY) in published articles. This license allows published articles to be reused without permission for any purpose as long as the original work is correctly cited.

The journal encourages Authors to self-archive their accepted manuscripts, publishing them on personal blogs, institutional repositories, and social media, as long as the full citation is included in the journal's website version.