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A new derivative with normal distribution kernel: Theory, methods and applications

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  • Atangana, Abdon
  • Gómez-Aguilar, J.F.

Abstract

New approach of fractional derivative with a new local kernel is suggested in this paper. The kernel introduced in this work is the well-known normal distribution that is a very common continuous probability distribution. This distribution is very important in statistics and also highly used in natural science and social sciences to portray real-valued random variables whose distributions are not known. Two definitions are suggested namely Atangana–Gómez Averaging in Liouville–Caputo and Riemann–Liouville sense. We presented some relationship with existing integrals transform operators. Numerical approximations for first and second order approximation are derived in detail. Some Applications of the new mathematical tools to describe some real world problems are presented in detail. This is a new door opened the field of statistics, natural and socials sciences.

Suggested Citation

  • Atangana, Abdon & Gómez-Aguilar, J.F., 2017. "A new derivative with normal distribution kernel: Theory, methods and applications," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 476(C), pages 1-14.
    • Handle: RePEc:eee:phsmap:v:476:y:2017:i:c:p:1-14
    DOI: 10.1016/j.physa.2017.02.016
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    Cited by:

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    8. Owolabi, Kolade M., 2019. "Mathematical modelling and analysis of love dynamics: A fractional approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 849-865.
    9. Kumar, Sachin & Pandey, Prashant, 2020. "A Legendre spectral finite difference method for the solution of non-linear space-time fractional Burger’s–Huxley and reaction-diffusion equation with Atangana–Baleanu derivative," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    10. Liang, Jiarui & Song, Songhe & Zhou, Weien & Fu, Hao, 2018. "Analysis of the damped nonlinear space-fractional Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 495-511.
    11. Owolabi, Kolade M. & Pindza, Edson, 2019. "Modeling and simulation of nonlinear dynamical system in the frame of nonlocal and non-singular derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 146-157.
    12. Sene, Ndolane, 2020. "Second-grade fluid model with Caputo–Liouville generalized fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    13. Addai, Emmanuel & Zhang, Lingling & Ackora-Prah, Joseph & Gordon, Joseph Frank & Asamoah, Joshua Kiddy K. & Essel, John Fiifi, 2022. "Fractal-fractional order dynamics and numerical simulations of a Zika epidemic model with insecticide-treated nets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 603(C).
    14. Dubey, Ved Prakash & Dubey, Sarvesh & Kumar, Devendra & Singh, Jagdev, 2021. "A computational study of fractional model of atmospheric dynamics of carbon dioxide gas," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    15. Owolabi, Kolade M. & Atangana, Abdon, 2018. "Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations," Chaos, Solitons & Fractals, Elsevier, vol. 111(C), pages 119-127.
    16. Khennaoui, Amina-Aicha & Ouannas, Adel & Bendoukha, Samir & Grassi, Giuseppe & Lozi, René Pierre & Pham, Viet-Thanh, 2019. "On fractional–order discrete–time systems: Chaos, stabilization and synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 150-162.
    17. Paul Hauseux & Jack S Hale & Stéphane P A Bordas, 2017. "Calculating the Malliavin derivative of some stochastic mechanics problems," PLOS ONE, Public Library of Science, vol. 12(12), pages 1-18, December.
    18. Atangana, Abdon & Gómez-Aguilar, J.F., 2018. "Fractional derivatives with no-index law property: Application to chaos and statistics," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 516-535.
    19. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.
    20. Al-khedhairi, A. & Elsadany, A.A. & Elsonbaty, A., 2019. "Modelling immune systems based on Atangana–Baleanu fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 129(C), pages 25-39.
    21. Yavuz, Mehmet & Bonyah, Ebenezer, 2019. "New approaches to the fractional dynamics of schistosomiasis disease model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 373-393.
    22. Khan, Muhammad Altaf & Khan, Yasir & Islam, Saeed, 2018. "Complex dynamics of an SEIR epidemic model with saturated incidence rate and treatment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 493(C), pages 210-227.
    23. Saqib, Muhammad & Khan, Ilyas & Shafie, Sharidan, 2018. "Application of Atangana–Baleanu fractional derivative to MHD channel flow of CMC-based-CNT's nanofluid through a porous medium," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 79-85.
    24. Shah, Syed Azhar Ali & Khan, Muhammad Altaf & Farooq, Muhammad & Ullah, Saif & Alzahrani, Ebraheem O., 2020. "A fractional order model for Hepatitis B virus with treatment via Atangana–Baleanu derivative," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 538(C).

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