Mathematical modeling of wave propagation in viscoelastic media with the fractional Zener model

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dc.citation.epage615
dc.citation.issue4
dc.citation.spage601
dc.contributor.affiliationУніверситет Хасана II Касабланки
dc.contributor.affiliationHassan II University of Casablanca
dc.contributor.authorАіт Ічоу, М.
dc.contributor.authorЕль Амрі, Х.
dc.contributor.authorЕцціані, А.
dc.contributor.authorAit Ichou, M.
dc.contributor.authorEl Amri, H.
dc.contributor.authorEzziani, A.
dc.coverage.placenameЛьвів
dc.coverage.placenameLviv
dc.date.accessioned2023-11-01T07:49:43Z
dc.date.available2023-11-01T07:49:43Z
dc.date.created2021-03-01
dc.date.issued2021-03-01
dc.description.abstractУ цій роботі розглянута задача математичого моделювання поширення хвилі в дисипативних середовищах. Розглянуто узагальнену дробову модель Зенера вимірності d (d = 1, 2, 3). Ця робота присвячена математичному аналізу такої моделі, а саме: існування та єдиність сильного та слабкого розв’язку та загасання енергії, що забезпечує розсіювання хвиль. Також подаються апріорні оцінки розв’язків, що допомагають показати існування слабкого розв’язку.
dc.description.abstractThe question of interest for the presented study is the mathematical modeling of wave propagation in dissipative media. The generalized fractional Zener model in the case of dimension d (d = 1, 2, 3) is considered. This work is devoted to the mathematical analysis of such model: existence and uniqueness of the strong and weak solution and energy decay result which guarantees the wave dissipation. The existence of the weak solution is shown using a priori estimates for solutions which are also presented.
dc.format.extent601-615
dc.format.pages15
dc.identifier.citationAit Ichou M. Mathematical modeling of wave propagation in viscoelastic media with the fractional Zener model / M. Ait Ichou, H. El Amri, A. Ezziani // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2021. — Vol 8. — No 4. — P. 601–615.
dc.identifier.citationenAit Ichou M. Mathematical modeling of wave propagation in viscoelastic media with the fractional Zener model / M. Ait Ichou, H. El Amri, A. Ezziani // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2021. — Vol 8. — No 4. — P. 601–615.
dc.identifier.doi10.23939/mmc2021.04.601
dc.identifier.urihttps://ena.lpnu.ua/handle/ntb/60450
dc.language.isoen
dc.publisherВидавництво Львівської політехніки
dc.publisherLviv Politechnic Publishing House
dc.relation.ispartofMathematical Modeling and Computing, 4 (8), 2021
dc.relation.references[1] Mainardi F. Fractional Calculus. In: Carpinteri A., Mainardi F. (eds) Fractals and Fractional Calculus in Continuum Mechanics. International Centre for Mechanical Sciences (Courses and Lectures), vol. 378. Springer, Vienna (1997).
dc.relation.references[2] Mainardi F., Gorenflo R. Time-fractional derivatives in relaxation processes: a tutorial survey. An International Journal for Theory and Applications. 10, 269–308 (2008).
dc.relation.references[3] Podlubny I. Fractional differential equation. Acadimic Press (1999).
dc.relation.references[4] Heymans N., Bauwens J.-C. Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheologica Acta. 33, 210–219 (1994).
dc.relation.references[5] Bachraoui M., Ait Ichou M., Hattaf K., Yousfi N. Spatiotemporal dynamics of a fractional model for hepatitis b virus infection with cellular immunity. Mathematical Modelling of Natural Phenomena. 16, 5 (2021).
dc.relation.references[6] Ait-Bella F., El Rhabi M., Hakim A., Laghrib A. Analysis of the nonlocal wave propagation problem with volume constraints. Mathematical Modeling and Computing. 7 (2), 334–344 (2020).
dc.relation.references[7] Atanackovic T. M., Janev M., Oparnica Lj., Pilipovic S., Zorica D. Space-time fractional Zener wave equation. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 471 (2174), 20140614 (2015).
dc.relation.references[8] Metalert R., Shiessel H., Blument A., Nonnemachert T. F. Generalized viscoelastic models: Their fractional equation with solution. Journal of Physics A: Mathematical and General. 28, 6567–6584 (1995).
dc.relation.references[9] Diethelm K., Ford N. J., Freed A. D., Luchko Yu. Algorithms for the fractional calculus: A selection of numerical methods. Computer Methods in Applied Mechanics and Engineering. 194 (6–8), 743–773 (2005).
dc.relation.references[10] Ford N. J., Simpson A. C. The numerical solution of fractional differential equations: Speed versus accuracy. Numerical Algorithms. 26, 333–346 (2001).
dc.relation.references[11] Mainardi F., Gorenflo R. Time-fractional derivatives in relaxation processes: a tutorial survey. Fractional Calculus and Applied Analysis. 10, 269–308 (2007).
dc.relation.references[12] Moczo P., Kristek J. On the rheological models used for time-domain methods of seismic wave propagation. Geophysical Research Letters. 32 (1), (2005).
dc.relation.references[13] Emmerich H., Korn M. Incorporation of attenuation into time-domain computations of seismic wave fields. Geophysic. 52 (9), 1252–1264 (1987).
dc.relation.references[14] Konjik S., Oparnica L., Zorica D. Waves in fractional zener type viscoelastic media. Journal of Mathematical Analysis and Applications. 365 (10), 259–268 (2010).
dc.relation.references[15] Ait Ichou M., El Amri H., Ezziani A. On existence and uniqueness of solution for space-time fractional Zener model. Acta Applicandae Mathematicae. 170, 593–609 (2020).
dc.relation.references[16] Atanackovi´c T. M., Janev M., Konjik S., Pilipovi´c S. Wave equation for generalized Zener model containing complex order fractional derivatives. Continuum Mechanics and Thermodynamics. 29, 569–583 (2017).
dc.relation.references[17] Ezziani A. Mod´elisation math´ematique et num´erique de la propagation d’ondes dans les milieux visco´elastiques et poro´elastiques. PhD thesis, Universit´e Paris Dauphine (2005).
dc.relation.references[18] Haddar H., Li J.-R., Matignon D. Efficient solution of a wave equation with fractional-order-dissipative terms. Journal of Computational and Applied Mathematics. 234 (6), 2003–2010 (2010).
dc.relation.references[19] Duvaut G., Lions J. L. In´equations en m`ecanique et en physique. Dunod (1972).
dc.relation.references[20] Nitsche J. A. On Korn’s second inequality. R.A.I.R.O Numerical Analysis (1981).
dc.relation.referencesen[1] Mainardi F. Fractional Calculus. In: Carpinteri A., Mainardi F. (eds) Fractals and Fractional Calculus in Continuum Mechanics. International Centre for Mechanical Sciences (Courses and Lectures), vol. 378. Springer, Vienna (1997).
dc.relation.referencesen[2] Mainardi F., Gorenflo R. Time-fractional derivatives in relaxation processes: a tutorial survey. An International Journal for Theory and Applications. 10, 269–308 (2008).
dc.relation.referencesen[3] Podlubny I. Fractional differential equation. Acadimic Press (1999).
dc.relation.referencesen[4] Heymans N., Bauwens J.-C. Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheologica Acta. 33, 210–219 (1994).
dc.relation.referencesen[5] Bachraoui M., Ait Ichou M., Hattaf K., Yousfi N. Spatiotemporal dynamics of a fractional model for hepatitis b virus infection with cellular immunity. Mathematical Modelling of Natural Phenomena. 16, 5 (2021).
dc.relation.referencesen[6] Ait-Bella F., El Rhabi M., Hakim A., Laghrib A. Analysis of the nonlocal wave propagation problem with volume constraints. Mathematical Modeling and Computing. 7 (2), 334–344 (2020).
dc.relation.referencesen[7] Atanackovic T. M., Janev M., Oparnica Lj., Pilipovic S., Zorica D. Space-time fractional Zener wave equation. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 471 (2174), 20140614 (2015).
dc.relation.referencesen[8] Metalert R., Shiessel H., Blument A., Nonnemachert T. F. Generalized viscoelastic models: Their fractional equation with solution. Journal of Physics A: Mathematical and General. 28, 6567–6584 (1995).
dc.relation.referencesen[9] Diethelm K., Ford N. J., Freed A. D., Luchko Yu. Algorithms for the fractional calculus: A selection of numerical methods. Computer Methods in Applied Mechanics and Engineering. 194 (6–8), 743–773 (2005).
dc.relation.referencesen[10] Ford N. J., Simpson A. C. The numerical solution of fractional differential equations: Speed versus accuracy. Numerical Algorithms. 26, 333–346 (2001).
dc.relation.referencesen[11] Mainardi F., Gorenflo R. Time-fractional derivatives in relaxation processes: a tutorial survey. Fractional Calculus and Applied Analysis. 10, 269–308 (2007).
dc.relation.referencesen[12] Moczo P., Kristek J. On the rheological models used for time-domain methods of seismic wave propagation. Geophysical Research Letters. 32 (1), (2005).
dc.relation.referencesen[13] Emmerich H., Korn M. Incorporation of attenuation into time-domain computations of seismic wave fields. Geophysic. 52 (9), 1252–1264 (1987).
dc.relation.referencesen[14] Konjik S., Oparnica L., Zorica D. Waves in fractional zener type viscoelastic media. Journal of Mathematical Analysis and Applications. 365 (10), 259–268 (2010).
dc.relation.referencesen[15] Ait Ichou M., El Amri H., Ezziani A. On existence and uniqueness of solution for space-time fractional Zener model. Acta Applicandae Mathematicae. 170, 593–609 (2020).
dc.relation.referencesen[16] Atanackovi´c T. M., Janev M., Konjik S., Pilipovi´c S. Wave equation for generalized Zener model containing complex order fractional derivatives. Continuum Mechanics and Thermodynamics. 29, 569–583 (2017).
dc.relation.referencesen[17] Ezziani A. Mod´elisation math´ematique et num´erique de la propagation d’ondes dans les milieux visco´elastiques et poro´elastiques. PhD thesis, Universit´e Paris Dauphine (2005).
dc.relation.referencesen[18] Haddar H., Li J.-R., Matignon D. Efficient solution of a wave equation with fractional-order-dissipative terms. Journal of Computational and Applied Mathematics. 234 (6), 2003–2010 (2010).
dc.relation.referencesen[19] Duvaut G., Lions J. L. In´equations en m`ecanique et en physique. Dunod (1972).
dc.relation.referencesen[20] Nitsche J. A. On Korn’s second inequality. R.A.I.R.O Numerical Analysis (1981).
dc.rights.holder© Національний університет “Львівська політехніка”, 2021
dc.subjectдробова похідна
dc.subjectсильний розв’язок
dc.subjectслабкий розв’язок
dc.subjectзагасання енергії
dc.subjectплоскі хвилі
dc.subjectв’язкопружні хвилі
dc.subjectмодель Зенера
dc.subjectfractional derivative
dc.subjectstrong solution
dc.subjectweak solution
dc.subjectenergy decay
dc.subjectplane waves
dc.subjectviscoelastic waves
dc.subjectZener’s model
dc.titleMathematical modeling of wave propagation in viscoelastic media with the fractional Zener model
dc.title.alternativeМатематичне моделювання поширення хвиль у в’язкопружних середовищах за допомогою дробової моделі Зенера
dc.typeArticle

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Mathematical Modeling And Computing. – 2021. – Vol. 8, No. 4