Research Article
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Stability of a nonlinear fractional pseudo-parabolic equation system regarding fractional order of the time

Year 2022, Volume: 6 Issue: 3, 405 - 419, 30.09.2022
https://doi.org/10.31197/atnaa.961417

Abstract

In this work, we investigate an issue of fractional order continuity for a system of pseudo-parabolic equations. Specifically, we focus on investigating the stability of the derivative index, the solution $w_{a}$ is continuously with respect to fractional order $a$ in the appropriate sense.

Supporting Institution

Thu Dau Mot University

References

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  • [10] K.A. Abro, A. Atangana, Mathematical analysis of memristor through fractal -fractional differential operators: A numerical study, Mathematical Methods in the Applied Sciences., 43(10) (2020), pp. 6378–6395.
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  • [49] K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), pp. 426–447.
  • [50] M. Kh. Beshtokov, To boundary-value problems for degenerating pseudo-parabolic equations with Gerasimov–Caputo fractional derivative, Izv. Vyssh. Uchebn. Zaved. Mat., 10 (2018), pp. 3–16.
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  • [52] M.Kh. Beshtokov, Boundary value problems for a pseudoparabolic equation with the Caputo fractional derivative, Differ. Equ., 55(7) (2019), pp. 884–893.
  • [53] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives, Theory and Applications, Gordon and Breach Science, Naukai Tekhnika, Minsk, 1987.
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  • [55] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag Berlin Heidelberg, 1995.
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Year 2022, Volume: 6 Issue: 3, 405 - 419, 30.09.2022
https://doi.org/10.31197/atnaa.961417

Abstract

References

  • [1] N.H. Sweilam, S.M. Al-Mekhlafi, T. Assiri, A. Atangana, Optimal control for cancer treatment mathematical model using Atangana−Baleanu−Caputo fractional derivative, Advances in Difference Equations., 2020 (1), pp. 1–21.
  • [2] S. Kumar, A. Atangana, A numerical study of the nonlinear fractional mathematical model of tumor cells in presence of chemotherapeutic treatment, International Journal of Biomathematics., 13 (03), pp. 205002.1.
  • [3] A. Atangana, A. Akgül, K.M. Owolabi, Analysis of fractal fractional differential equations, Alexandria Engineering Journal., 59.3 (2020), pp. 1117–1134.
  • [4] A. Atangana, Z. Hammouch, Fractional calculus with power law: The cradle of our ancestors, The European Physical Journal Plus., 134 (9), pp. 429.
  • [5] A. Atangana, E. Bonyah, Fractional stochastic modeling: New approach to capture more heterogeneity, Chaos: An Inter- disciplinary Journal of Nonlinear Science., 29 (1), pp. 013118.
  • [6] A. Atangana A, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm Sci 2016. OnLine-First (00). 18. 10.2298/TSCI160111018A.
  • [7] A. Atangana, E.F.D. Goufo, Cauchy problems with fractal-fractional operators and applications to groundwater dynamics, Fractals., 2020, doi:10.1142/s0218348x20400435.
  • [8] H.G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y.Q. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer.Simul., 64 (2018), pp. 213–231.
  • [9] I. Podlubny, Fractional Differential Equations, Academic press., California, (1999).
  • [10] K.A. Abro, A. Atangana, Mathematical analysis of memristor through fractal -fractional differential operators: A numerical study, Mathematical Methods in the Applied Sciences., 43(10) (2020), pp. 6378–6395.
  • [11] T.B. Ngoc, D. Baleanu, L.M. Duc, N.H. Tuan, Regularity results for fractional diffusion equations involving fractional derivative with Mittag-Leffler kernel, Mathematical Methods in the Applied Sciences., 43(12) (2020), pp. 7208–7226.
  • [12] V. Kiryakova, Generalized Fractional Calculus and Applications., CRC press, 1993.
  • [13] N.H. Tuan, D. Baleanu, T.N. Thach, D. O’Regan, N.H. Can, Final value problem for nonlinear time fractional reaction- diffusion equation with discrete data, J. Comput. Appl. Math., 376 (2020), 25 pages.
  • [14] N.H. Luc, L.N. Huynh, D. Baleanu, N.H. Can, Identifying the space source term problem for a generalization of the fractional diffusion equation with hyper-Bessel operator, Adv. Difference Equ., (261) (2020), 23 pages.
  • [15] Binh, H. D., Hoang, L. N., Baleanu, D., Van, H. T. K. (2021). Continuity Result on the Order of a Nonlinear Fractional Pseudo-Parabolic Equation with Caputo Derivative, Fractal and Fractional, 5(2), 41.
  • [16] L. Li, G.J. Liu, A generalized definition of Caputo derivatives and its application to fractional ODEs, SIAM J. Math. Anal., 50(3) (2018), pp. 2867–2900.
  • [17] T.B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc., 272(1220 ) (1972), PP. 47-78.
  • [18] W.T. Ting, Certain non-steady flows of second-order fluids, Arch. Ration. Mech. Anal., 14 (1963), PP. 1–26.
  • [19] V. Padron, Effect of aggregation on population recovery modeled by a forward-backward pseudo-parabolic equation, Trans. Am. Math. Soc., 356 (2004), PP. 2739–2756.
  • [20] D. Huafei, S. Yadong, Z. Xiaoxiao, Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms, Disc. Contin. Dyn. Syst, Ser. B., 21(3) (2016), pp. 781–801.
  • [21] F. Sun, L. Liu and Y. Wu, Global existence and finite time blow-up of solutions for the semi-linear pseudo-parabolic equation with a memory term, Applicable Analysis., 98(4) (2019), 22 pages.
  • [22] H. Ding, J. Zhou, Global existence and blow-up for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity, J. Math. Anal. Appl., 478 (2019), pp. 393–420.
  • [23] L. Jin, L. Li, S. Fang, The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation, Computers and Mathematics with Applications., 73(10) (2017), pp. 2221–2232.
  • [24] X. Zhu, F. Li, Y. Li, Global solutions and blow up solutions to a class of pseudo-parabolic equations with nonlocal term, Applied Mathematics and Computation., 329 (2018), pp. 38–51.
  • [25] Yang Cao, Jingxue Yin, Chunpeng Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differential Equations., 246 (2009), pp. 4568–4590.
  • [26] Y. Cao, C. Liu, Initial boundary value problem for a mixed pseudo- parabolic p-Laplacian type equation with logarithmic nonlinearity, Electronic Journal of Differential Equations., 2018(116) (2018), pp. 1–19.
  • [27] Y. He, H. Gao , H. Wang, Blow-up and decay for a class of pseudo-parabolic p-Laplacian equation with logarithmic nonlinearity, Computers and Mathematics with Applications., 75(2) (2018), pp. 459–469.
  • [28] Y. Lu and L. Fei, Bounds for blow-up time in a semilinear pseudo-parabolic equation with nonlocal source, Journal of Inequalities and Applications., (2016), pp. 229.
  • [29] 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.
  • [30] N. D. Phuong, N. H. Luc, L. D. Long, Modified Quasi Boundary Value method for inverse source problem of the bi-parabolic equation, Advances in the Theory of Nonlinear Analysis and its Applications 4 (2020) No. 3, 132–142.
  • [31] B. D. Nghia, N. H. Luc, H. D. Binh, L. D. Long, Regularization method for the problem of determining the source function using integral conditions, Advances in the Theory of Nonlinear Analysis and its Applications 5 (2021) No. 3, 351–362.
  • [32] N. N. Hung, H. D. Binh, N. H. Luc, N. T. K. An, L. D. Long, Stochastic sub-diffusion equation with conformable derivative driven by standard Brownian motion , Advances in the Theory of Nonlinear Analysis and its Applications 5 (2021) No. 3, 287–299.
  • [33] Z. Baitiche, C. Derbazi, M. Benchohra,ψ-Caputo fractional differential equations with multi-point boundary conditions by Topological Degree Theory, Results in Nonlinear Analysis, 2020 V 3I4,167 − 178.
  • [34] A. Ardjounia , A. Djoudi, Existence and uniqueness of solutions for nonlinear hybrid implicit Caputo-Hadamard fractional differential equations, Results in Nonlinear Analysis, 2(3),136 − 142
  • [35] S. Redhwan, S. Shaikh, M. Abdo, Some properties of Sadik transform and its applications of fractional-order dynamical systems in control theory, Advances in the Theory of Nonlinear Analysis and its Application, 4 (1) , 51-66.
  • [36] T.B. Ngoc, V.V. Tri, Z. Hammouch, N.H. Can, Stability of a class of problems for timespace fractional pseudo-parabolic equation with datum measured at terminal time, Applied Numerical Mathematics, 167,308 − 329
  • [37] E. Karapinar, H.D. Binh, N.H. Luc, N.H. Can, On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems, Adv. Difference Equ., 70,26pp
  • [38] J. E. Lazreg, S. Abbas, M. Benchohra, and E. Karapinar, Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces , Open Mathematics 2021; 19: 363-372,
  • [39] J. Patila, A. Chaudharib, M. S. Abdoc, B. Hardan, Upper and Lower Solution method for Positive solution of generalized Caputo fractional differential equations, Advances in the Theory of Nonlinear Analysis and its Application, 4 (2020), 279-291.
  • [40] R. S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation , Mathematical Methods in the Applied Sciences. https://doi.org/10.1002/mma.6652
  • [41] R. S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions , RACSAM (2021) 115:155.
  • [42] R. S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On The Solutions Of Fractional Differential Equations Via Geraghty Type Hybrid Contractions, Appl. Comput. Math., 20, N.2, 2021,313-333.
  • [43] A.T. Nguyen, T. Caraballo, N.H. Tuan, On the initial value problem for a class of nonlinear biharmonic equation with time-fractional derivative, Proc. Roy. Soc. Edinburgh Sect. A, (2021) 1–43.
  • [44] T. Wei, Y. Zhang, The backward problem for a time-fractional diffusion-wave equation in a bounded domain, Computers and Mathematics with Applications., 75(10) (2018) pp. 3632–3648.
  • [45] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam, 2006.
  • [46] C.V. J. Sousa, C.E. de Oliveira, Fractional order pseudo-parabolic partial differential equation: Ulam−Hyers stability, Bull. Braz. Math. Soc., 50(2) (2019), pp. 481–496.
  • [47] D.T. Dang, E. Nane, D.M. Nguyen, N.H. Tuan, Continuity of Solutions of a Class of Fractional Equations, Potential Anal., 49 (2018), pp. 423–478.
  • [48] K. Diethelm, The analysis of fractional differential equations, Springer, Berlin, 2010.
  • [49] K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), pp. 426–447.
  • [50] M. Kh. Beshtokov, To boundary-value problems for degenerating pseudo-parabolic equations with Gerasimov–Caputo fractional derivative, Izv. Vyssh. Uchebn. Zaved. Mat., 10 (2018), pp. 3–16.
  • [51] M.Kh. Beshtokov, Boundary-value problems for loaded pseudo-parabolic equations of fractional order and difference methods of their solving, Russian Mathematics., 63(2) (2019), pp. 1–10.
  • [52] M.Kh. Beshtokov, Boundary value problems for a pseudoparabolic equation with the Caputo fractional derivative, Differ. Equ., 55(7) (2019), pp. 884–893.
  • [53] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives, Theory and Applications, Gordon and Breach Science, Naukai Tekhnika, Minsk, 1987.
  • [54] D.T. Dang, E. Nane, D.M. Nguyen, N.H. Tuan, Continuity of Solutions of a Class of Fractional Equations, Potential Anal., 49 (2018), pp. 423–478.
  • [55] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag Berlin Heidelberg, 1995.
  • [56] 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.
  • [57] 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.
  • [58] Z. Baitichea, C. Derbazia, M. Benchohrab, ψ–Caputo Fractional Differential Equations with Multi-point Boundary Conditions by Topological Degree Theory, Results in Nonlinear Analysis 3 (2020) No. 4, 167-–178
  • [59] M.Kh. Beshtokov, Boundary value problems for a pseudoparabolic equation with the Caputo fractional derivative, Differ. Equ., 55(7) (2019), pp. 884–893.
  • [60] N. H. Tuan, D. O’Regan, T. B. Ngoc, Continuity with respect to fractional order of the time fractional diffusion-wave equation, Evolution Equations and Control Theory., 9(3) (2020), pp.773.
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Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Nguyen Duc Phuong 0000-0003-3779-197X

Le Dinh Long 0000-0001-8805-4588

Tuan Nguyen Anh 0000-0002-8757-9742

Ho Binh 0000-0003-1925-4601

Publication Date September 30, 2022
Published in Issue Year 2022 Volume: 6 Issue: 3

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