1. bookVolume 223 (2020): Issue 4 (December 2020)
Journal Details
License
Format
Journal
eISSN
2720-4286
First Published
30 Mar 2016
Publication timeframe
1 time per year
Languages
English
access type Open Access

An Operational Calculus Model with the (m,n) - Symmetric Difference

Published Online: 31 Dec 2020
Volume & Issue: Volume 223 (2020) - Issue 4 (December 2020)
Page range: 67 - 82
Journal Details
License
Format
Journal
eISSN
2720-4286
First Published
30 Mar 2016
Publication timeframe
1 time per year
Languages
English
Abstract

The paper determines a non-classical Bittner operational calculus model, in which the derivative is understood as an -symmetric difference Dm,n{x (k)} := {x (k+m)− x (kn). By considering an operation Dm,n,b{x (k)} := {x (k+m)− b (k)x (kn), the formulated model has been generalized.

Keywords

[1] Bittner R., On certain axiomatics for the operational calculus, ‘Bull. Acad. Polon. Sci.’, Cl. III, 1959, 7(1), pp. 1-9.Search in Google Scholar

[2] Bittner R., Operational calculus in linear spaces, ‘Studia Math.’, 1961, 20, pp. 1–18.10.4064/sm-20-1-1-18Search in Google Scholar

[3] Bittner R., Algebraic and analytic properties of solutions of abstract differential equations, ‘Rozprawy Matematyczne’ [‘Dissertationes Math.’], 42, PWN, Warszawa 1964.Search in Google Scholar

[4] Bittner R., Rachunek operatorów w przestrzeniach liniowych, PWN, Warszawa 1974 [Operational Calculus in Linear Spaces — available in Polish].Search in Google Scholar

[5] Brito da Cruz A. M. C., Symmetric Quantum Calculus, PHD thesis, University of Aveiro, Department of Mathematics, 2012 [online], https://arxiv.org/pdf/1306.1327 [access 29.12.2020].Search in Google Scholar

[6] Brito da Cruz A. M. C., Martins N., Torres D. F. M., A Symmetric Quantum Calculus, In: Pinelas S., Chipot M., Dosla Z. (eds), ‘Differential and Difference Equations with Applications’, Springer Proceedings in Mathematics & Statistics, vol. 47, pp. 359-366. Springer, New York 2013 [online], https://arxiv.org/pdf/1112.6133 [access 29.12.2020]10.1007/978-1-4614-7333-6_29Search in Google Scholar

[7] Elaydi S., An Introduction to Difference Equations, Springer Sci. and Business Media, New York 2005.Search in Google Scholar

[8] Levy H., Lessman F., Finite Difference Equations, Pitman & Sons, London 1959.Search in Google Scholar

[9] Mickens R. E., Difference Equations: Theory, Applications and Advanced Topics, Chapman & Hall/CRC Press, Boca Raton 2005.Search in Google Scholar

[10] Mieloszyk E., Example of operational calculus, ‘Zeszyty Naukowe Politechniki Gdańskiej, Mate-matyka XIII’, 1985, 383, pp. 151-157.Search in Google Scholar

[11] Mikusiński J., Operational Calculus, Pergamon Press 1959.Search in Google Scholar

[12] Spiegel M. R., Schaum’s Outline of Calculus of Finite Differences and Difference Equations, McGraw-Hill, New York 1971.Search in Google Scholar

[13] Wysocki H., The nabla difference model of the operational calculus, ‘Demonstratio Math.’, 2013, Vol. 46, No. 2, pp. 315-326.10.1515/dema-2013-0461Search in Google Scholar

[14] Wysocki H., Model rachunku operatorów dla różnicy wstecznej przy podstawach a, b, ‘Zeszyty Naukowe Akademii Marynarki Wojennej’, 2013, 1(192), pp. 109-120 [The backward a, b-difference operational calculus model – available in Polish].Search in Google Scholar

[15] Wysocki H., The operational calculus model for the nth-order backward difference, ‘Zeszyty Naukowe Akademii Marynarki Wojennej — Scientific Journal of PNA’, 2015, 3(202), pp. 75–88.10.5604/0860889X.1178573Search in Google Scholar

[16] Wysocki H., An operational calculus model for the nth-order forward difference, ‘Zeszyty Naukowe Akademii Marynarki Wojennej — Scientific Journal of PNA’, 2017, 3(210), pp. 107–117.10.5604/01.3001.0010.6593Search in Google Scholar

[17] Wysocki H., An operational calculus model for the central difference and exponential-trigonometric and hyperbolic Fibonacci sequences, ‘Scientific Journal of Polish Naval Academy — Zeszyty Naukowe AMW’, 2018, 3(214), pp. 39-62.10.2478/sjpna-2018-0018Search in Google Scholar

[18] Wysocki H., A generalization of difference models of the Bittner operational calculus, ‘Scientific Journal of Polish Naval Academy’, 2019, 4(219), pp. 41-52.10.2478/sjpna-2019-0024Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo