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A monotone iterative technique combined to finite element method for solving reaction-diffusion problems pertaining to non-integer derivative

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Abstract

This paper focuses on some mathematical and numerical aspects of reaction-diffusion problems pertaining to non-integer time derivatives using the well-known method of lower and upper solutions combined with the monotone iterative technique. First, we study the existence and uniqueness of weak solutions of the proposed models, then we prove some comparison results. Besides, linear finite element spaces on triangles are used to discretize the problem in space, whereas the generalized backward-Euler method is adopted to approximate the time non-integer derivative. Furthermore, the idea of this method is to construct two sequences of solutions of a linear initial value problem which are easier to compute and converge to the solution of the nonlinear problem. We show numerically through two examples that this convergence requires only few iterations. Some well-known examples with exact solutions and numerical results based on the finite element method in 2D are provided to validate the theoretical results. As a result, we confirm that the proposed method is efficient and easy to use to overcome the convergence and stability difficulties.

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References

  1. Adolfsson K, Enelund M, Olsson P (2005) On the fractional order model of viscoelasticity. Mech Time-dependent Mater 9(1):15–34

    Google Scholar 

  2. Kumar D, Baleanu D (2019) Fractional calculus and its applications in physics. Front Phys 7:81

    Google Scholar 

  3. Hilfer R (2000) Applications of fractional calculus in physics. World Scientific

    MATH  Google Scholar 

  4. Tarasov VE (2013) Review of some promising fractional physical models. Int J Mod Phys B 27(09):1330005

    MathSciNet  MATH  Google Scholar 

  5. Zhou H, Yang S, Zhang S (2018) Conformable derivative approach to anomalous diffusion. Physica A 491:1001–1013

    MathSciNet  MATH  Google Scholar 

  6. Tuan NH, Ngoc TB, Baleanu D, O’Regan D (2020) On well-posedness of the sub-diffusion equation with conformable derivative model. Commun Nonlinear Sci Numer Simul 89:105332

    MathSciNet  MATH  Google Scholar 

  7. Drapaca C, Sivaloganathan S (2012) A fractional model of continuum mechanics. J Elast 107(2):105–123

    MathSciNet  MATH  Google Scholar 

  8. Atanacković TM, Pilipović S, Stanković B, Zorica D (2014) Fractional calculus with applications in mechanics. Wiley Online Library, Hoboken

    MATH  Google Scholar 

  9. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, vol 204. Elsevier, Amsterdam

    MATH  Google Scholar 

  10. Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, Amsterdam

    MATH  Google Scholar 

  11. Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley

    MATH  Google Scholar 

  12. Zhou Y, Wang J, Zhang L (2016) Basic theory of fractional differential equations. World scientific

    Google Scholar 

  13. Ma X, Wu W, Zeng B, Wang Y, Wu X (2020) The conformable fractional grey system model. ISA Trans 96:255–271

    Google Scholar 

  14. Boccaletti S, Ditto W, Mindlin G, Atangana A (2020) Modeling and forecasting of epidemic spreading: the case of COVID-19 and beyond. Chaos Solitons Fract 135:109794

    Google Scholar 

  15. Singh J, Kumar D, Hammouch Z, Atangana A (2018) A fractional epidemiological model for computer viruses pertaining to a new fractional derivative. Appl Math Comput 316:504–515

    MathSciNet  MATH  Google Scholar 

  16. Almeida R, da Cruz AMB, Martins N, Monteiro MTT (2019) An epidemiological MSEIR model described by the Caputo fractional derivative. Int J Dyn Control 7(2):776–784

    MathSciNet  Google Scholar 

  17. Royston P, Ambler G, Sauerbrei W (1999) The use of fractional polynomials to model continuous risk variables in epidemiology. Int J Epidemiol 28(5):964–974

    Google Scholar 

  18. Goufo EFD, Maritz R, Munganga J (2014) Some properties of the Kermack-McKendrick epidemic model with fractional derivative and nonlinear incidence. Adv Differ Equ 2014(1):1–9

    MathSciNet  MATH  Google Scholar 

  19. Alla Hamou A, Azroul E, Hammouch Z, Alaoui AL (2021) A Fractional Multi-Order Model to Predict the COVID-19 Outbreak in Morocco. Appl Comput Math 20(1):177–203

    MathSciNet  MATH  Google Scholar 

  20. Meerschaert MM, Sikorskii A (2011) Stochastic models for fractional calculus, vol 43. Walter de Gruyter

  21. Çenesiz Y, Kurt A, Nane E (2017) Stochastic solutions of conformable fractional Cauchy problems. Stat Probab Lett 124:126–131

    MathSciNet  MATH  Google Scholar 

  22. Yang Q, Chen D, Zhao T, Chen Y (2016) Fractional calculus in image processing: a review. Fract Calc Appl Anal 19(5):1222–1249

    MathSciNet  MATH  Google Scholar 

  23. Larnier S, Mecca R (2012) Fractional-order diffusion for image reconstruction, In: 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, 1057–1060

  24. Bai J, Feng X-C (2007) Fractional-order anisotropic diffusion for image denoising. IEEE Trans Image Process 16(10):2492–2502

    MathSciNet  Google Scholar 

  25. Cruz-Duarte JM, Rosales-Garcia J, Correa-Cely CR, Garcia-Perez A, Avina-Cervantes JG (2018) A closed form expression for the Gaussian-based Caputo-Fabrizio fractional derivative for signal processing applications. Commun Nonlinear Sci Numer Simul 61:138–148

    MathSciNet  MATH  Google Scholar 

  26. Epstein CL (2007) Introduction to the mathematics of medical imaging. SIAM

  27. Deng W, Li C (2005) Chaos synchronization of the fractional Lü system. Physica A 353:61–72

    Google Scholar 

  28. Hartley TT, Lorenzo CF, Qammer HK (1995) Chaos in a fractional order Chua’s system. IEEE Trans Circ Syst I 42(8):485–490

    Google Scholar 

  29. Li C, Chen G (2004) Chaos in the fractional order Chen system and its control. Chaos Solitons Fract 22(3):549–554

    MATH  Google Scholar 

  30. Owolabi KM, Karaagac B (2020) Chaotic and spatiotemporal oscillations in fractional reaction-diffusion system. Chaos Solitons Fract 141:110302. ISSN 0960–0779. https://doi.org/10.1016/j.chaos.2020.110302, https://www.sciencedirect.com/science/article/pii/S0960077920306986

  31. Sweilam NH, El-Sayed AAE, Boulaaras S (2021) Fractional-order advection-dispersion problem solution via the spectral collocation method and the non-standard finite difference technique. Chaos Solitons Fract 144:110736

    MathSciNet  MATH  Google Scholar 

  32. Kumar S, Pandey P (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 Fract 130:109402

    MathSciNet  MATH  Google Scholar 

  33. Hu Y, Li C, Li H (2017) The finite difference method for Caputo-type parabolic equation with fractional Laplacian: one-dimension case. Chaos Solitons Fract 102:319–326

    MathSciNet  MATH  Google Scholar 

  34. Jin B, Lazarov R, Zhou Z (2013) Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J Numer Anal 51(1):445–466

    MathSciNet  MATH  Google Scholar 

  35. Zhao X, Hu X, Cai W, Karniadakis GE (2017) Adaptive finite element method for fractional differential equations using hierarchical matrices. Comput Methods Appl Mech Eng 325:56–76

    MathSciNet  MATH  Google Scholar 

  36. Deng W (2009) Finite element method for the space and time fractional Fokker-Planck equation. SIAM J Numer Anal 47(1):204–226

    MathSciNet  MATH  Google Scholar 

  37. Jin B, Lazarov R, Zhou Z (2016) A Petrov-Galerkin finite element method for fractional convection-diffusion equations. SIAM J Numer Anal 54(1):481–503

    MathSciNet  MATH  Google Scholar 

  38. Zheng Y, Zhao Z (2020) The time discontinuous space-time finite element method for fractional diffusion-wave equation. Appl Numer Math 150:105–116

    MathSciNet  MATH  Google Scholar 

  39. Deng W (2009) Finite element method for the space and time fractional Fokker-Planck equation. SIAM J Numer Anal 47(1):204–226. https://doi.org/10.1137/080714130

    Article  MathSciNet  MATH  Google Scholar 

  40. Kumar D, Chaudhary S, Kumar VS (2019) Finite element analysis for coupled time-fractional nonlinear diffusion system. Comput Math Appl 78(6):1919–1936

    MathSciNet  MATH  Google Scholar 

  41. Gao F, Wang X (2014) A modified weak Galerkin finite element method for a class of parabolic problems. J Comput Appl Math 271:1–19

    MathSciNet  MATH  Google Scholar 

  42. Zheng Y, Zhao Z (2017) The discontinuous Galerkin finite element method for fractional cable equation. Appl Numer Math 115:32–41. ISSN 0168-9274, https://doi.org/10.1016/j.apnum.2016.12.006, https://www.sciencedirect.com/science/article/pii/S0168927417300053

  43. Jin B, Lazarov R, Zhou Z (2013) Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J Numer Anal 51(1):445–466. https://doi.org/10.1137/120873984

    Article  MathSciNet  MATH  Google Scholar 

  44. Xu T, Liu F, Lü S, Anh VV (2020) Finite difference/finite element method for two-dimensional time-space fractional Bloch-Torrey equations with variable coefficients on irregular convex domains. Comput Math Appl 80(12):3173–3192

    MathSciNet  MATH  Google Scholar 

  45. Liu X, Yang X (2021) Mixed finite element method for the nonlinear time-fractional stochastic fourth-order reaction-diffusion equation. Comput Math Appl 84:39–55

    MathSciNet  MATH  Google Scholar 

  46. Jia J, Wang H, Zheng X (2021) A preconditioned fast finite element approximation to variable-order time-fractional diffusion equations in multiple space dimensions. Appl Numer Math 163:15–29

    MathSciNet  MATH  Google Scholar 

  47. Li D, Liao H-L, Sun W, Wang J, Zhang J, Analysis of \(L1\)-Galerkin FEMs for time-fractional nonlinear parabolic problems, arXiv preprint arXiv:1612.00562

  48. Pao C (1993) Positive solutions and dynamics of a finite difference reaction-diffusion system. Num Methods Part Differ Equ 9(3):285–311

    MathSciNet  MATH  Google Scholar 

  49. Pao C (1996) Blowing-up and asymptotic behaviour of solutions for a finite difference system. Appl Anal 62(1–2):29–38

    MathSciNet  MATH  Google Scholar 

  50. Bellman R, Juncosa ML, Kalaba R (1961) Some numerical experiments using newton’s method for nonlinear parabolic and elliptic boundary-value problems. Commun ACM 4(4):187–191, ISSN 0001-0782, https://doi.org/10.1145/355578.366508

  51. Parter SV (1964) Mildly nonlinear elliptic partial differential equations and their numerical solution. I, Tech. Rep., Wisconsin Univ Madison Mathematics Research Center R

  52. Greenspan D, Parter SV (1964) Mildly nonlinear elliptic partial differential equations and their numerical solution. Tech. Rep., Wisconsin Univ Madison Mathematics Research Center, II

  53. Walter W (1968) Die Linienmethode bei nichtlinearen parabolischen Differentialgleichungen. Numer Math 12(4):307–321

    MathSciNet  MATH  Google Scholar 

  54. Pao C (2002) Finite difference reaction-diffusion systems with coupled boundary conditions and time delays. J Math Anal Appl 272(2):407–434

    MathSciNet  MATH  Google Scholar 

  55. Pao C (2001) Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions. J Comput Appl Math 136(1–2):227–243

    MathSciNet  MATH  Google Scholar 

  56. Pao C-V (2012) Nonlinear parabolic and elliptic equations. Springer Science & Business Media

    Google Scholar 

  57. Pao C-V (1982) On nonlinear reaction-diffusion systems. J Math Anal Appl 87(1):165–198

    MathSciNet  MATH  Google Scholar 

  58. Pao C (1985) Monotone iterative methods for finite difference system of reaction-diffusion equations. Numer Math 46(4):571–586

    MathSciNet  MATH  Google Scholar 

  59. Khalil R, Al Horani M, Yousef A, Sababheh M (2014) A new definition of fractional derivative. J Comput Appl Math 264:65–70. https://doi.org/10.1016/j.cam.2014.01.002

    Article  MathSciNet  MATH  Google Scholar 

  60. Abdeljawad T (2015) On conformable fractional calculus. J Comput Appl Math 279:57–66. https://doi.org/10.1016/j.cam.2014.10.016

    Article  MathSciNet  MATH  Google Scholar 

  61. Wang Y, Zhou J, Li Y (2016) Fractional Sobolev’s spaces on time scales via conformable fractional calculus and their application to a fractional differential equation on time scales. Adv Math Phys 2016:1–22. https://doi.org/10.1155/2016/9636491

    Article  MathSciNet  MATH  Google Scholar 

  62. Benkhettou N, Hassani S, Torres DF (2016) A conformable fractional calculus on arbitrary time scales. J King Saud Univ-Sci 28(1):93–98. https://doi.org/10.1016/j.jksus.2015.05.003

    Article  Google Scholar 

  63. Abdeljawad T (2015) On conformable fractional calculus. J Comput Appl Math 279:57–66

    MathSciNet  MATH  Google Scholar 

  64. Alla Hamou A, Azroul E, Alaoui A (2020) Monotone iterative technique for nonlinear periodic time fractional parabolic problems. Adv Theory Nonlinear Anal Appl 4(3):194–213

    Google Scholar 

  65. Brezis H, Ciarlet PG, Lions JL (1999) Analyse fonctionnelle: théorie et applications, vol 91. Dunod Paris

    Google Scholar 

  66. Adams RA, Fournier JJ (2003) Sobolev spaces. Elsevier

    MATH  Google Scholar 

  67. Alla hamou A, Azroul E, Hammouch Z, Lamrani alaoui A, Modeling and numerical investigation of a Conformable co-infection model for describing Hantavirus of the European moles, math. meth. app. sci. Preprint

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences Dhar Elmahraz, Sidi Mohamed Ben Abdellah University, B.P. 1796, Atlas, 30000, Fez, Morocco

    Abdelouahed Alla Hamou & El Houssine Azroul

  2. Division of Applied Mathematics, Thu Dau Mot University, Thu Dau Mot, Binh Duong Province, Vietnam

    Zakia Hammouch

  3. Department of Medical Research, China Medical University Hospital, Taichung, Taiwan

    Zakia Hammouch

  4. Department of Sciences, École normale supérieure, Moulay Ismail University of Meknes, Meknes, Morocco

    Zakia Hammouch

  5. Department of Mathematics, Regional Center of Education and Professional Training, B.P. 49, 30000, Fez, Morocco

    Abdelilah Lamrani Alaoui

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  1. Abdelouahed Alla Hamou
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  2. El Houssine Azroul
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  3. Zakia Hammouch
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  4. Abdelilah Lamrani Alaoui
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Correspondence to Zakia Hammouch.

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Hamou, A.A., Azroul, E., Hammouch, Z. et al. A monotone iterative technique combined to finite element method for solving reaction-diffusion problems pertaining to non-integer derivative. Engineering with Computers 39, 2515–2541 (2023). https://doi.org/10.1007/s00366-022-01635-4

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