We study two generalized versions of a system of equations which describe the time evolution of the hydrodynamic fluctuations of density and velocity in a linear viscoelastic fluid. In the first of these versions, the time derivatives are replaced by conformable derivatives, and in the second version left-handed Caputo’s derivatives are used. We show that the solutions obtained with these two types of derivatives exhibit significant similarities, which is an interesting (and somewhat surprising) result, taking into account that the conformable derivatives are local operators, while Caputo’s derivatives are nonlocal operators. We also show that the solutions of the generalized systems are similar to the solutions of the original system, if the order α of the new derivatives (conformable or Caputo) is less than one. On the other hand, when α is greater than one, the solutions of the generalized systems are qualitatively different from the solutions of the original system.
References
[1]
Pandey, V. and Holm, S. (2016) Connecting the Grain-Shearing Mechanism of Wave Propagation in Marine Sediments to Fractional Order Wave Equations. Journal of the Acoustical Society of America, 140, 4225-4236. https://doi.org/10.1121/1.4971289
[2]
Buckingham, M.J. (2007) On Pore-Fluid Viscosity and the Wave Properties of Saturated Granular Materials Including Marine Sediments. Journal of the Acoustical Society of America, 122, 1486-1501. https://doi.org/10.1121/1.2759167
[3]
Holm, S. and Sinkus, R. (2010) A Unifying Fractional Wave Equation for Compressional and Shear Waves. Journal of the Acoustical Society of America, 127, 542-548. https://doi.org/10.1121/1.3268508
[4]
Holm, S. and Näsholm, S.P. (2011) A Causal and Fractional All-Frequency Wave Equation for Lossy Media. Journal of the Acoustical Society of America, 130, 2195-2202. https://doi.org/10.1121/1.3631626
[5]
Chen, W., Hu, S. and Cai, W. (2016) A Causal Fractional Derivative Model for Acoustic Wave Propagation in Lossy Media. Archive of Applied Mechanics, 86, 529-539. https://doi.org/10.1007/s00419-015-1043-2
[6]
Fujioka, J., Espinosa, A. and Rodríguez, R.F. (2010) Fractional Optical Solitons. Physics Letters A, 374, 1126-1134. https://doi.org/10.1016/j.physleta.2009.12.051
[7]
Fujioka, J., Velasco, M. and Ramírez, A. (2016) Fractional Optical Solitons and Fractional Noether’s Theorem with Ortigueira’s Centered Derivatives. Applied Mathematics, 7, 1340-1352. https://doi.org/10.4236/am.2016.712118
[8]
Rodríguez, R.F., Fujioka, J. and Salinas-Rodríguez, E. (2013) Fractional Fluctuation Effects on the Light Scattered by a Viscoelastic Suspension. Physical Review E, 88, Article ID: 022154. https://doi.org/10.1103/PhysRevE.88.022154
[9]
Rodríguez, R.F., Fujioka, J. and Salinas-Rodríguez, E. (2015) Fractional Correlation Functions in Simple Viscoelastic Liquids. Physica A: Statistical Mechanics and Its Applications, 427, 326-340. https://doi.org/10.1016/j.physa.2015.01.060
[10]
Wang, X., Qi, H., Yu, B., Xiong, Z. and Xu, H. (2017) Analytical and Numerical Study of Electroosmotic Slip Flows of Fractional Second Grade Fluids. Communications in Nonlinear Science and Numerical Simulation, 50, 77-87. https://doi.org/10.1016/j.cnsns.2017.02.019
[11]
Rodríguez, R.F., Salinas-Rodríguez, E. and Fujioka, J. (2018) Fractional Time Fluctuations in Viscoelasticity: A Comparative Study of Correlations and Elastic Moduli. Entropy, 20, Article 28. https://doi.org/10.3390/e20010028
[12]
Pandey, V. and Holm, S. (2016) Linking the Fractional Derivative and the Lomnitz Creep Law to Non-Newtonian Time-Varying Viscosity. Physical Review E, 94, Article ID: 032606. https://doi.org/10.1103/PhysRevE.94.032606
[13]
Rodríguez, R.F., Fujioka, J. and Salinas-Rodríguez, E. (2021) Nonequilibrium Fractional Correlation Functions and Fluctuation-Dissipation in Linear Viscoelasticity. Physica A: Statistical Mechanics and Its Applications, 583, Article ID: 126352. https://doi.org/10.1016/j.physa.2021.126352
[14]
Suzuki, J.L., Gulian, M., Zayernouri, M. and D’Elia, M. (2022) Fractional Modeling in Action: A Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials. Journal of Peridynamics and Nonlocal Modeling. https://doi.org/10.1007/s42102-022-00085-2
[15]
Deng, W., Wu, X. and Wang, W. (2017) Mean Exit Time and Escape Probability for the Anomalous Processes with the Tempered Power-Law Waiting Times. Europhysics Letters, 117, Article ID: 10009. https://doi.org/10.1209/0295-5075/117/10009
[16]
Le Vot, F., Abad, E. and Yuste, S.B. (2017) Continuous-Time Random-Walk Model for Anomalous Diffusion in Expanding Media. Physical Review E, 96, Article ID: 032117. https://doi.org/10.1103/PhysRevE.96.032117
[17]
Flores-Calderón, R., Fujioka, J. and Espinosa-Cerón, A. (2021) Soliton Dynamics of a High-Density Bose-Einstein Condensate Subject to a Time Varying Anharmonic Trap. Chaos, Solitons and Fractals, 143, Article ID: 110580. https://doi.org/10.1016/j.chaos.2020.110580
[18]
Zhang, J. and Wei, Z.H. (2011) A Class of Fractional-Order Multi-Scale Variational Models and Alternating Projection Algorithm for Image Denoising. Applied Mathematical Modelling, 35, 2516-2528. https://doi.org/10.1016/j.apm.2010.11.049
[19]
Zhang, J., Wei, Z.H. and Xiao, L. (2012) Adaptive Fractional-Order Multi-Scale Method for Image Denoising. Journal of Mathematical Imaging and Vision, 43, 39-49. https://doi.org/10.1007/s10851-011-0285-z
[20]
Zhang, Y., Pu, Y.F., Hu, J.R. and Zhou, J.L. (2012) A Class of Fractional-Order Variational Image Inpainting Models. Applied Mathematics and Information Sciences, 6, Article 17.
[21]
Magin, R., Ortigueira, M.D., Podlubny, I. and Trujillo, J. (2011) On the Fractional Signals and Systems. Signal Processing, 91, 350-371. https://doi.org/10.1016/j.sigpro.2010.08.003
[22]
West, B.J. (2014) Colloquium: Fractional Calculus View of Complexity: A Tutorial. Reviews of Modern Physics, 86, 1169-1186. https://doi.org/10.1103/RevModPhys.86.1169
[23]
Yusuf, A., Inc, M., Aliyu, A.I. and Baleanu, D. (2019) Optical Solitons Possessing β Derivative of the Chen-Lee-Liu Equation in Optical Fibers. Frontiers in Physics, 7, Article 34. https://doi.org/10.3389/fphy.2019.00034
[24]
Caputo, M. and Mainardi, F. (1971) A New Dissipation Model Based on Memory Mechanism. Pure and Applied Geophysics, 91, 134-147. https://doi.org/10.1007/BF00879562
[25]
Caputo, M. and Fabricio, M. (2015) A New Definition of Fractional Derivative without Singular Kernel. Progress in Fractional Differentiation and Applications, 1, 73-85.
[26]
Atangana, A. and Baleanu, D. (2016) New Fractional Derivatives with Non-Local and Non-Singular Kernel Theory and Application to Heat Transfer Model. Thermal Science, 20, 763-769.
[27]
Atangana, A. and Secer, A. (2013) A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions. Abstract Applied Analysis, 2013, Article ID: 279681. https://doi.org/10.1155/2013/279681
[28]
Khalil, R., Al Horani, M., Yousef, A. and Sababheh, M. (2014) A New Definition of Fractional Derivative. Journal of Computation and Applied Mathematics, 264, 65-70. https://doi.org/10.1016/j.cam.2014.01.002
[29]
Atangana, A., Baleanu, D. and Alsaedi, A. (2016) Analysis of Time-Fractional Hunter-Saxton Equation: A Model of Neumatic Liquid Crystal. Open Physics, 14, 145-149. https://doi.org/10.1515/phys-2016-0010
[30]
Abdelhakim, A.A. (2019) The Flaw in the Conformable Calculus: It Is Conformable Because It Is Not Fractional. Fractional Calculus and Applied Analysis, 22, 242-254. https://doi.org/10.1515/fca-2019-0016
[31]
Ortigueira, M.D. and Tenreiro Machado, J.A. (2015) What Is a Fractional Derivative? Journal of Computational Physics, 293, 4-13. https://doi.org/10.1016/j.jcp.2014.07.019
[32]
Wang, C.H. (1980) Depolarized Rayleigh-Brillouin Scattering of Shear Waves and Molecular Reorientation in a Viscoelastic Liquid. Molecular Physics, 41, 541-565. https://doi.org/10.1080/00268978000102981
[33]
Abdeljawad, T. (2015) On Conformable Fractional Calculus. Journal of Computational and Applied Mathematics, 279, 57-66. https://doi.org/10.1016/j.cam.2014.10.016
[34]
Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.
[35]
Atanacković, T.M., Pilipović, S., Stanković, B. and Zorica, D. (2014) Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes. Wiley, Hoboken. https://doi.org/10.1002/9781118577530
[36]
Ortigueira, M.D. (2022) A New Look at the Initial Condition Problem. Mathematics, 10, Article 1771. https://doi.org/10.3390/math10101771
[37]
Lorenzo, C.F. and Hartley, T.T. (2008) Initialization of Fractional-Order Operators and Fractional Differential Equations. Journal of Computational and Nonlinear Dynamics, 3, Article ID: 021101. https://doi.org/10.1115/1.2833585
[38]
Lorenzo, C.F. and Hartley, T.T. (2007) Initialization, Conceptualization, and Application in the Generalized (Fractional) Calculus. Critical Reviews in Biomedical Engineering, 35, 447-553.
[39]
Lorenzo, C.F. and Hartley, T.T. (2000) Initialized Fractional Calculus. International Journal of Applied Mathematics and Computer Science, 3, 249-266.
[40]
Abramowitz, M. and Stegun, I.A. (1972) Handbook of Mathematical Functions. Dover, New York.
[41]
Lorenzo, C.F. and Hartley, T.T. (1999) Generalized Functions for the Fractional Calculus. NASA/TP—1999-209424/REV1, The NASA STI Program Office, Glenn Research Center.
[42]
Fa, K.S. (2008) Study of a Generalized Langevin Equation with Nonlocal Dissipative Force, Harmonic Potential and a Constant Load Force. The European Physical Journal B, 65, 265-270. https://doi.org/10.1140/epjb/e2008-00344-1
[43]
Duarte-Ortigueira, M. (2008) Fractional Central Differences and Derivatives. Journal of Vibration and Control, 14, 1255-1266. https://doi.org/10.1177/1077546307087453
[44]
Yepez-Martínez, H., Khater, M.M.A., Rezazadeh, H. and Inc, M. (2021) Analytical Novel Solutions to the Fractional Optical Dynamics in a Medium with Polynomial Law Nonlinearity and Higher Order Dispersion with a New Local Fractional Derivative. Physics Letters A, 420, Article ID: 127744. https://doi.org/10.1016/j.physleta.2021.127744
[45]
He, S., Sun, K., Mei, X., Yan, B. and Xu, S. (2017) Numerical Analysis of a Fractional-Order Chaotic System Based on Conformable Fractional-Order Derivative. The European Physical Journal Plus, 132, Article No. 36. https://doi.org/10.1140/epjp/i2017-11306-3
[46]
Chung, W.S. (2015) Fractional Newton Mechanics with Conformable Fractional Derivative. Journal of Computational and Applied Mathematics, 290, 150-158. https://doi.org/10.1016/j.cam.2015.04.049
[47]
Cenesiz, Y. and Kurt, A. (2015) The New Solution of Time Fractional Wave Equation with Conformable Fractional Derivative Definition. Journal of New Theory, 7, 79-85.
[48]
Cenesiz, Y., Baleanu, D., Kurt, A. and Tasbozan, O. (2017) New Exact Solutions of Burger’s Type Equations with Conformable Derivative. Waves in Random and Complex Media, 27, 103-116. https://doi.org/10.1080/17455030.2016.1205237
[49]
Zainab, I. and Akram, G. (2023) Effect of β-Derivative on Time Fractional Jaulent-Miodek System under Modified Auxiliary Equation Method and Exp (-g(Ω)) -Expansion Method. Chaos, Solitons and Fractals, 168, Article ID: 113147. https://doi.org/10.1016/j.chaos.2023.113147
[50]
Yusuf, A., Inc, M. and Baleanu, D. (2019) Optical Solitons with M-Truncated and Beta Derivatives in Nonlinear Optics. Frontiers in Physics, 7, Article 126. https://doi.org/10.3389/fphy.2019.00126
[51]
Wang, X., Ehsan, H., Abbas, M., Akram, G., Sadaf, M. and Abdeljawad, T. (2023) Analytical Solitary Wave Solutions of a Time-Fractional Thin-Film Ferroelectric Material Equation Involving β-Derivative Using Modified Auxiliary Equation Method. Results in Physics, 48, Article ID: 106411. https://doi.org/10.1016/j.rinp.2023.106411
[52]
Atangana, A. and Alqahtani, R.T. (2016) Modelling the Spread of River Blindness Disease via the Caputo Fractional Derivative and the β-Derivative. Entropy, 18, Article 40. https://doi.org/10.3390/e18020040
[53]
Akram, G., Arshed, S. and Sadaf, M. (2023) Soliton Solutions of Generalized Time-Fractional Boussinesq-Like Equation via Three Techniques. Chaos, Solitons and Fractals, 173, Article ID: 113653. https://doi.org/10.1016/j.chaos.2023.113653
