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Exact Finite-Difference Calculus: Beyond Set of Entire Functions

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  • Vasily E. Tarasov

    (Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia
    Department of Physics, 915, Moscow Aviation Institute, National Research University, Moscow 125993, Russia)

Abstract

In this paper, a short review of the calculus of exact finite-differences of integer order is proposed. The finite-difference operators are called the exact finite-differences of integer orders, if these operators satisfy the same characteristic algebraic relations as standard differential operators of the same order on some function space. In this paper, we prove theorem that this property of the exact finite-differences is satisfies for the space of simple entire functions on the real axis (i.e., functions that can be expanded into power series on the real axis). In addition, new results that describe the exact finite-differences beyond the set of entire functions are proposed. A generalized expression of exact finite-differences for non-entire functions is suggested. As an example, the exact finite-differences of the square root function is considered. The use of exact finite-differences for numerical and computer simulations is not discussed in this paper. Exact finite-differences are considered as an algebraic analog of standard derivatives of integer order.

Suggested Citation

  • Vasily E. Tarasov, 2024. "Exact Finite-Difference Calculus: Beyond Set of Entire Functions," Mathematics, MDPI, vol. 12(7), pages 1-37, March.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:7:p:972-:d:1363519
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    References listed on IDEAS

    as
    1. Mohammad Mehdizadeh Khalsaraei & Ali Shokri & Samad Noeiaghdam & Maryam Molayi, 2021. "Nonstandard Finite Difference Schemes for an SIR Epidemic Model," Mathematics, MDPI, vol. 9(23), pages 1-13, November.
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    5. Vasily E. Tarasov, 2016. "Exact Discrete Analogs of Canonical Commutation and Uncertainty Relations," Mathematics, MDPI, vol. 4(3), pages 1-13, June.
    6. Tarasov, Vasily E., 2015. "Lattice fractional calculus," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 12-33.
    7. Tarasov, Vasily E., 2015. "Fractional Liouville equation on lattice phase-space," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 421(C), pages 330-342.
    8. Garba, S.M. & Gumel, A.B. & Hassan, A.S. & Lubuma, J.M.-S., 2015. "Switching from exact scheme to nonstandard finite difference scheme for linear delay differential equation," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 388-403.
    9. Laskin, N. & Zaslavsky, G., 2006. "Nonlinear fractional dynamics on a lattice with long range interactions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 368(1), pages 38-54.
    10. Vasily E. Tarasov, 2015. "Exact Discrete Analogs of Derivatives of Integer Orders: Differences as Infinite Series," Journal of Mathematics, Hindawi, vol. 2015, pages 1-8, November.
    11. Anguelov, Roumen & Lubuma, Jean M.-S., 2003. "Nonstandard finite difference method by nonlocal approximation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 61(3), pages 465-475.
    12. Chen-Charpentier, Benito M. & Dimitrov, Dobromir T. & Kojouharov, Hristo V., 2006. "Combined nonstandard numerical methods for ODEs with polynomial right-hand sides," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 73(1), pages 105-113.
    13. María Ángeles Castro & Miguel Antonio García & José Antonio Martín & Francisco Rodríguez, 2019. "Exact and Nonstandard Finite Difference Schemes for Coupled Linear Delay Differential Systems," Mathematics, MDPI, vol. 7(11), pages 1-14, November.
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