A robust stability criterion in the heat equation with a conformable fractional derivative defined on a radially symmetric sphere

• Temoltzi-Ávila, Raúl ^[1] ; Ávila-Pozos, Roberto ^[1] ; Cruz-Castillo, Ricardo ^[1] ; Jiménez-Munguía, Ronald R. ^[1] ; Santillán-Hernández, Alma S. ^[1]
  1. [1] Universidad Autónoma del Estado de Hidalgo

     Universidad Autónoma del Estado de Hidalgo

     México

     
• Localización: Revista de Matemática: Teoría y Aplicaciones, ISSN 2215-3373, ISSN-e 2215-3373, Vol. 32, Nº. 1, 2025 (Ejemplar dedicado a: Revista de Matemática: Teoría y Aplicaciones), págs. 35-53
• Idioma: inglés
• DOI: 10.15517/rmta.v32i1.57678
• Títulos paralelos:
  • Un criterio de estabilidad robusta en la ecuación de calor con una derivada fraccionaria conformable general definida sobre una esfera radialmente simétrica
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    En este artículo, presentamos un criterio de estabilidad robusta para una ecuación de calor con simetría axial y con derivada fraccionaria general conformable en el tiempo definida en una esfera. Se supone que la ecuación de calor admite una fuente de calor externa que se representa como una serie de Fourier con coeficientes descritos por funciones continuas a trozos y acotadas. El criterio de estabilidad robusta establece condiciones para garantizar que la solución de la ecuación de calor, así como su derivada parcial con respecto al eje radial y su derivada fraccionaria conformable general en el tiempo, son funciones acotadas por un valor constante prefijado. El criterio de estabilidad robusta se obtiene por una extensión del concepto de estabilidad bajo perturbaciones de acción constante que se aplica a sistemas de ecuaciones diferenciales ordinarias. Los resultados se ilustran numéricamente.

  • English

    In this paper, we present a robust stability criterion for a heat equation with axial symmetry and with a general time-conformable fractional derivative defined on a sphere. The heat equation is assumed to have a heat source that is represented as a Fourier series with coefficients described by bounded, piecewise continuous functions. The robust stability criterion establishes conditions to guarantee that the solution of the heat equation, along with its partial derivative with respect to the radial axis and its general timeconformable fractional derivative, remains bounded by a predetermined value. The robust stability criterion is obtained by extending the concept of stability under constant-acting perturbations applied to systems of ordinary differential equations. The results are illustrated numerically.

• Referencias bibliográficas
  • T. Abdeljawad, On conformable fractional calculus. J. Comput. Appl. Math. 279(2015), 57–66. doi: 10.1016/j.cam.2014.10.016
  • M. Abu-Shady, M. K. A. Kaabar, A generalized definition of the fractional derivative with applications. Math. Probl. Eng. 2021(2021), 9444803,...
  • R. Almeida, M. Guzowska, T. Odzijewicz, A remark on local fractional calculus and ordinary derivatives. Open Math. 14(2016), no. 1, 1122–1124....
  • D. Avci, I. Eroglu, N. Ozdemir, Conformable heat problem in a cylinder. International Conference on Fractional Differentiation and its Applications....
  • D. Avci, E. Iskender, N. Ozdemir, Conformable heat equation on a radial symmetric plate. Therm. Sci. 21(2017), no. 2, 819–926. doi: 10.2298/TSCI160427302A
  • V. Barbu, Controllability and stabilization of parabolic equations. Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser,...
  • B. Bayour, D. F. M. Torres, Existence of solution to a local fractional nonlinear differential equation. J. Comput. Appl. Math. 312(2017),...
  • Y. Çenesiz, A. Kurt, The solutions of time and space conformable fractional heat equations with conformable Fourier transform. Acta Univ....
  • L. Elsgolts, Differential equations and the calculus of variations.Mir,Moscow, 1977. url: mirtitles . org / 2013 / 11 / 23 / differential...
  • P. M. Guzmán et al., A new definition of a fractional derivative of local type. J. Math. Anal. 9(2018), no. 2, 88–98. url: www.ilirias.com/jma/repository/docs/JMA9-2-9.pdf.
  • M. A. Hammad, R. Khalil, Conformable fractional heat differential equation. Int. J. Pure Appl. Math. 94(2014), no. 2, 215–221. doi: 10.12732/ijpam.v94i2.8
  • R. Khalil, M. A. Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative. J. Comput. Appl. Math. 264(2014), 65–70. doi:...
  • S. Li, S. Zhang, R. Liu, The existence of solution of diffusion equation with the general conformable derivative. J. Funct. Spaces 2020(2020),...
  • K. B. Oldham, J. Spanier, The fractional calculus: theory and applications of differentiation and integration to arbitrary order. Academic...
  • E. C. de Oliveira, J. A. T. Machado, A review of definitions for fractional derivatives and integral. Math. Probl. Eng. 2014(2014), 1–6. doi:...
  • M. R. S. Rahmat, A new definition of conformable fractional derivative on arbitrary time scales. Adv. Cont. Discr. Mod. 2019(2019), 354. doi:...
  • A. Souahi, A. B. Makhlouf, M. A. Hammami, Stability analysis of conformable fractional-order nonlinear systems. Indag. Math. 28(2017), no....
  • R. Temoltzi-Ávila, A robust stability criterion on the time-conformable fractional heat equation in a axisymmetric cylinder. SeMA Journal...
  • G. S. Teodoro, J. A. T. Machado, E. C. de Oliveira, A review of definitions of fractional derivatives and other operators. J. Comput. Phys....
  • G. N. Watson, A treatise on the theory of Bessel functions. Cambridge University Press, London, 1952.
  • X. Zhang, F. Gao, C. Chi, Improvement on conformable fractional derivative and its applications in fractional differential equations. J. Funct....
  • D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation. Calcolo 54(2017), no. 3, 903–917. doi: 10.1007/s10092-017-0213-8
  • V. N. Zhermolenko, R. Temoltzi-´Avila, Bulgakov problem for a hyperbolic equation and robust stability. Mosc. Univ. Mech. Bull. 76(2021),...
  • W. Zhong, L. Wang, Basic theory of initial value problems of conformable fractional differential equations. Adv. Cont. Discr. Mod. 2018(2018),...