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The Super-Diffusive Singular Perturbation Problem

Author

Listed:
  • Edgardo Alvarez

    (Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla, Colombia
    These authors contributed equally to this work.)

  • Carlos Lizama

    (Departamento de Matemática y Ciencia de la Computación, Facultad de Ciencia, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
    These authors contributed equally to this work.)

Abstract

In this paper we study a class of singularly perturbed defined abstract Cauchy problems. We investigate the singular perturbation problem ( P ϵ ) ϵ α D t α u ϵ ( t ) + u ϵ ′ ( t ) = A u ϵ ( t ) , t ∈ [ 0 , T ] , 1 < α < 2 , ϵ > 0 , for the parabolic equation ( P ) u 0 ′ ( t ) = A u 0 ( t ) , t ∈ [ 0 , T ] , in a Banach space, as the singular parameter goes to zero. Under the assumption that A is the generator of a bounded analytic semigroup and under some regularity conditions we show that problem ( P ϵ ) has a unique solution u ϵ ( t ) for each small ϵ > 0 . Moreover u ϵ ( t ) converges to u 0 ( t ) as ϵ → 0 + , the unique solution of equation ( P ) .

Suggested Citation

  • Edgardo Alvarez & Carlos Lizama, 2020. "The Super-Diffusive Singular Perturbation Problem," Mathematics, MDPI, vol. 8(3), pages 1-14, March.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:403-:d:331407
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    References listed on IDEAS

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    1. J. F. Gómez Aguilar & T. Córdova-Fraga & J. Tórres-Jiménez & R. F. Escobar-Jiménez & V. H. Olivares-Peregrino & G. V. Guerrero-Ramírez, 2016. "Nonlocal Transport Processes and the Fractional Cattaneo-Vernotte Equation," Mathematical Problems in Engineering, Hindawi, vol. 2016, pages 1-15, June.
    2. H. J. Haubold & A. M. Mathai & R. K. Saxena, 2011. "Mittag-Leffler Functions and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-51, May.
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