Existence of an initial value problem for time-fractional Oldroyd-B fluid equation using Banach fixed point theorem

Year 2021, Volume: 5 Issue: 4, 523 - 530, 30.12.2021

Vo Viet Trı

https://doi.org/10.31197/atnaa.943242

Cited By: 1

Abstract

In this paper, we study the initial boundary value problem for time-fractional Oldroyd-B fluid equation. Our model contains two Riemann-Liouville fractional derivatives which have many applications, for example, in viscoelastic flows. For the linear case, we obtain regularity results under some different assumptions of the initial data and the source function. For the non-linear case, we obtain the existence of a unique solution using Banach's fixed point theorem.

Keywords

Time-fractional Oldroyd-B fluid problem, Riemman-Liouville, regularity, Banach fixed point theory

References

• [1] E. Bazhlekova, I. Bazhlekov, Viscoelastic flows with fractional derivative models: computational approach by convolutional calculus of Dimovski, Fract. Calc. Appl. Anal. 17 (2014), no. 4, 954-976.
• [2] J. Manimaran, L. Shangerganesh, A. Debbouche, Finite element error analysis of a time-fractional nonlocal diffusion equation with the Dirichlet energy, J. Comput. Appl. Math. 382 (2021), 113066, 11 pp
• [3] J. Manimaran, L. Shangerganesh, A. Debbouche, A time-fractional competition ecological model with cross-diffusion, Math. Methods Appl. Sci. 43 (2020), no. 8, 5197-5211
• [4] N.H. Tuan, A. Debbouche, T.B. Ngoc, Existence and regularity of final value problems for time fractional wave equations, Comput. Math. Appl. 78 (2019), no. 5, 1396-1414.
• [5] M. Al-Maskari, S. Karaa, Galerkin FEM for a time-fractional Oldroyd-B fluid problem, Adv. Comput. Math. 45 (2019), no. 2, 1005-1029.
• [6] E. Karapinar, H.D. Binh, N.H. Luc, & N.H. Can, N. H. (2021), On continuity of the fractional derivative of the time- fractional semilinear pseudo-parabolic systems, Advances in Difference Equations, 2021(1), 1-24.
• [7] H.T. Qi, M.Y. Xu, Stokes' first problem for a viscoelastic fluid with the generalized Oldroyd-B model, Acta. Mech. Sin. 23, 463-469 (2007).
• [8] Y. Zhang, J. Jiang, Y. Bai, MHD flow and heat transfer analysis of fractional Oldroyd-B nanofluid between two coaxial cylinders, Comput. Math. Appl. 78, 3408-3421 (2019).
• [9] L. Feng, F. Liu, I. Turner, P. Zhuang, Numerical methods and analysis for simulating the flow of a generalized Oldroyd-B fluid between two infinite parallel rigid plates, Int. J. Heat Mass Transf. 115, 1309-1320 (2017).
• [10] J. Zhang, F. Liu, V. Anh, Analytical and numerical solutions of a two-dimensional multi-term time fractional Oldroyd-B model, Numer. Methods Part. Differ. Equ. 35, 875-893 (2019).
• [11] N.H. Tuan, Y. Zhou, T.N. Thach, N.H. Can, Initial inverse problem for the nonlinear fractional Rayleigh-Stokes equation with random discrete data, Commun. Nonlinear Sci. Numer. Simul. 78 (2019), 104873, 18 pp.
• [12] T.B. Ngoc, N.H. Luc, V.V. Au, N.H. Tuan, Z. Yong, Existence and regularity of inverse problem for the nonlinear fractional Rayleigh-Stokes equations, Mathematical Methods in the Applied Sciences, 1-27.
• [13] Y. Zhou, J. N. Wang, The nonlinear Rayleigh-Stokes problem with Riemann-Liouville fractional derivative, Mathematical Methods in the Applied Sciences, https://doi.org/10.1002/mma.5926.
• [14] M. Abdullah, A.R. Butt, N. Raza and E.U. Haque, Semi-analytical technique for th solution of fractional Maxwell fluid, Can. J. Phys., 94 (2017), 472-478.
• [15] E. Bazhlekova and I. Bazhlekov, Peristaltic transport of viscoelastic bio-fluids with fractional derivative models, Biomath, 5 (2016) 1605151.
• [16] M. Jamil, A. Rauf, A.A. Zafar and N.A. Khan, New exact analytical solutions for Stokes first problem of Maxwell fluid with fractional derivative approach, Comput. Math. Appl., 62 (2011), 1013-1023.
• [17] H. Afshari, E, Karapinar, A discussion on the existence of positive solutions of the boundary value problems via-Hilfer fractional derivative on b-metric spaces, Advances in Difference Equations volume 2020, Article number: 616 (2020).
• [18] H. Afshari, S. Kalantari, E. Karapinar, Solution of fractional differential equations via coupled fixed point, Electronic Journal of Differential Equations,Vol. 2015 (2015), No. 286, pp. 1-12.
• [19] B.Alqahtani, H. Aydi, E. Karapinar, V. Rakocevic, A Solution for Volterra Fractional Integral Equations by Hybrid Contractions, Mathematics 2019, 7, 694.
• [20] E. Karapinar, A.Fulga, M. Rashid, L.Shahid, H. Aydi, Large Contractions on Quasi-Metric Spaces with an Application to Nonlinear Fractional Differential-Equations, Mathematics 2019, 7, 444.
• [21] I.S. Kim, Semilinear problems involving nonlinear operators of monotone type, Results in Nonlinear Analysis, 2(1), 25-35.
• [22] F.S. Bachir, S. Abbas, M. Benbachir, M. Benchohra, Hilfer-Hadamard Fractional Differential Equations, Existence and Attractivity, Advances in the Theory of Nonlinear Analysis and its Application, 2021, Vol 5, Issue 1, Pages 49-57.
• [23] A. Salim, M. Benchohra, J. Lazreg, J. Henderson, Nonlinear Implicit Generalized Hilfer-Type Fractional Differential Equations with Non-Instantaneous Impulses in Banach Spaces, Advances in the Theory of Nonlinear Analysis and its Application, Vol 4, Issue 4, Pages 332-348, 2020.
• [24] Z. Baitichea, C. Derbazia, M. Benchohrab, ψ-Caputo Fractional Differential Equations with Multi-point Boundary Condi- tions by Topological Degree Theory, Results in Nonlinear Analysis 3 (2020) No. 4, 167-178.
• [25] T.N. Thach, N.H. Can, V.V. Tri, Identifying the initial state for a parabolic diffusion from their time averages with fractional derivative, Mathematical Methods in the Applied Sciences, (2021), pp. 1-16.
• [26] S. Muthaiah, Murugesan, N. Thangaraj, Existence of Solutions for Nonlocal Boundary Value Problem of Hadamard Fractional Differential Equations, Advances in the Theory of Nonlinear Analysis and its Application, 3 (3) , 162-173.
• [27] A. Ardjouni, A. Djoudi, Existence and uniqueness of solutions for nonlinear hybrid implicit Caputo-Hadamard fractional differential equations, Results in Nonlinear Analysis, 2 (3), 136-142.
• [28] J.E. Lazreg, S. Abbas, M. Benchohra, & E. Karapinar, Impulsive Caputo-Fabrizio fractional di?erential equations in b-metric spaces, Open Mathematics, 19(1), 363-372.
• [29] S. Muthaiah, M. Murugesan, N. Thangaraj, Existence of Solutions for Nonlocal Boundary Value Problem of Hadamard Fractional Differential Equations, Advances in the Theory of Nonlinear Analysis and its Application, 3 (3), 162-173.
• [30] N.D. Phuong, L.V.C. Hoan, E. Karapinar, J. Singh, H.D. Binh, & N.H. Can, Fractional order continuity of a time semi- linear fractional diffusion-wave system, Alexandria Engineering Journal, 59(6), 4959-4968.