Notes on continuity result for conformable diffusion equation on the sphere: the linear case

• Autorzy: Nguyen Van Tien

Identyfikatory

DOI

10.1515/dema-2022-0178

• Języki publikacji: EN

Abstrakty

EN

In this article, we are interested in the linear conformable diffusion equation on the sphere. Our main goal is to establish some results on the continuity problem with respect to fractional order. The main technique is based on several evaluations on the sphere using spherical basis functions. To overcome the difficulty, we also need to use some calculations to control the generalized integrals.

Słowa kluczowe

EN

parabolic equation; conformable derivative; sphere; spherical harmonics

PL

równanie paraboliczne; pochodna funkcji; kula; harmoniki sferyczne

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2022
• Tom: Vol. 55, nr 1
• Strony: 952--962
• Opis fizyczny: Bibliogr. 39 poz.

Twórcy

autor

Nguyen Van Tien

• Faculty of Maths, FPT University HCM, Saigon Hi-tech Park, Ho Chi Minh City, Vietnam

Bibliografia

• [1] Q. T. Le Gia, Approximation of parabolic PDEs on spheres using spherical basis functions, Adv. Comput. Math. 22 (2005), no. 4, 377–397, DOI: https://doi.org/10.1007/s10444-003-3960-9.
• [2] Q. T. Le Gia, Galerkin approximation of elliptic PDEs on spheres, J. Approx. Theory 130 (2004), no. 2, 125–149, DOI: https://doi.org/10.1016/j.jat.2004.07.008.
• [3] Q. T. Le Gia, I. H. Sloan, and T. Tran, Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere, Math. Comp. 78 (2009), no. 265, 79–101, DOI: https://doi.org/10.1016/S0025-5718(08)02150-9.
• [4] Z. Brzeźniak, B. Goldys, and Q. T. LeGia, Random attractors for the stochastic Navier-Stokes equations on the 2D unit sphere, J. Math. Fluid Mech. 20 (2018), no. 1, 227–253, DOI: https://doi.org/10.1007/s00021-017-0351-4.
• [5] M. Enelund, and P. Olsson, Damping described by fading memory-analysis and application to fractional derivative models, Int. J. Solids Struct. 36 (1999), no. 7, 939–970, DOI: https://doi.org/10.1016/S0020-7683(97)00339-9.
• [6] L. D. Long, H. D. Binh, D. Kumar, N. H. Luc, and N. H. Can, Stability of fractional order of time nonlinear fractional diffusion equation with Riemann-Liouville derivative, Math. Methods Appl. Sci. 45 (2022), 6194–6216, DOI: https://doi.org/10.1002/mma.8166.
• [7] H. Afshari, and E. Karapinar, A solution of the fractional differential equations in the setting of b-metric space, Carpathian Math. Publ. 13 (2021), no. 3, 764–774, DOI: https://doi.org/10.15330/cmp.13.3.764-774.
• [8] E. Karapinar, A. Fulga, N. Shahzad, and A. F. Roldan Lopez de Hierro, Solving integral equations by means of fixed point theory, J. Funct. Spaces 2022 (2022), 1–16, DOI: https://doi.org/10.1155/2022/7667499.
• [9] N. D. Phuong and N. H. Luc, Note on a nonlocal pseudo-parabolic equation on the unit sphere, Dynamic Syst. Appl. 30 (2021), no. 2, 295–304, DOI: https://doi.org/10.46719/dsa20213029.
• [10] T. T. Binh, Semilinear parabolic diffusion systems on the sphere with Caputo-Fabrizio operator, Adv. Theory Nonlinear Anal. Appl. 6 (2022), no. 2, 148–156, DOI: https://doi.org/10.31197/atnaa.1012869.
• [11] H. Wendland, A high-order approximation method for semilinear parabolic equations on spheres, Math. Comp. 82 (2013), no. 281, 227–245, DOI: https://doi.org/10.1090/S0025-5718-2012-02623-8.
• [12] R. S. Adiguzel, U. Aksoy, E. Karapinar, and I. M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation, Math. Methods Appl. Sci. (2020), 1–12, DOI: https://doi.org/10.1002/mma.6652.
• [13] H. Afshari, and E. Karapinar, A discussion on the existence of positive solutions of the boundary value problems via psi-Hilfer fractional derivative on b-metric spaces, Adv. Difference Equations 1 (2020), no. 616, 1–11, DOI: https://doi.org/10.1186/s13662-020-03076-z.
• [14] H. Afshari, S. Kalantari, and E. Karapinar, Solution of fractional differential equations via coupled fixed point, Electron. J. Differ. Equ. Conf. 2015 (2015), no. 286, 1–12, DOI: http://ejde.math.unt.eduftpejde.math.txstate.edu.
• [15] B. Alqahtani, H. Aydi, E. Karapinar, and V. Rakocevic, A solution for Volterra fractional integral equations by hybrid contractions, Mathematics 7 (2019), no. 8, 694, DOI: https://doi.org/10.3390/math7080694.
• [16] E. Karapinar, A. Fulga, M. Rashid, L. Shahid, and H. Aydi, Large contractions on quasi-metric spaces with an application to nonlinear fractional differential-equations, Mathematics 7 (2019), no. 5, 444, DOI: https://doi.org/10.3390/math7050444.
• [17] A. Salim, B. Benchohra, E. Karapinar, and J. E. Lazreg, Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations, Adv. Difference Equations 1 (2020), 1–21, DOI: https://doi.org/10.1186/s13662-020-03063-4.
• [18] E. Karapınar, T. Abdeljawad, and F. Jarad, Applying new fixed point theorems on fractional and ordinary differential equations, Adv. Difference Equations 2019 (2019), no. 421, 1–25, DOI: https://doi.org/10.1186/s13662-019-2354-3.
• [19] A. Abdeljawad, R. P. Agarwal, E. Karapinar, and P. S. Kumari, Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-metric space, Symmetry 11 (2019), no. 5, 686, DOI: https://doi.org/10.3390/sym11050686.
• [20] N. D. Phuong, Note on a Allen-Cahn equation with Caputo-Fabrizio derivative, Results Nonlinear Anal. 4 (2021), no. 3, 179–185, DOI: https://doi.org/10.53006/rna.962068.
• [21] N. H. Tuan, N. H. Can, R. Wang, and Y. Zhou, Initial value problem for fractional Volterra integro-differential equations with Caputo derivative, Discrete Contin. Dyn. Syst. Ser. B 26 (2021), no. 12, 6483–6510, DOI: https://doi.org/10.3934/dcdsb.2021030.
• [22] T. B. Ngoc, N. H. Tuan, T. Caraballo, and Y. Zhou, Existence and regularity results for terminal value problem for nonlinear fractional wave equations, Nonlinearity 34 (2021), no. 3, 1448–1502, DOI: https://doi.org/10.1088/1361-6544/abc4d9.
• [23] R. S. Adiguzel, U. Aksoy, E. Karapinar, and I. M. Erhan, Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions, Rev. R. Acad. Cienc. Exactas Fiiiis. Nat. Ser. A Mat. RACSAM 115 (2021), no. 115, 1–16, DOI: https://doi.org/10.1007/s13398-021-01095-3.
• [24] R. S. Adiguzel, U. Aksoy, E. Karapinar, and I. M. Erhan, On the solutions of fractional differential equations via Geraghty type hybrid contractions, Appl. Comput. Math. 20 (2021), no. 2, 313–333.
• [25] N. H. Tuan, and T. Caraballo, On initial and terminal value problems for fractional nonclassical diffusion equations, Proc. Amer. Math. Soc. 149 (2021), 143–161, DOI: https://doi.org/10.1090/proc/15131.
• [26] N. A. Tuan, N. H. Tuan, and C. Yang, On Cauchy problem for fractional parabolic-elliptic Keller-Segel model, Adv. Nonlinear Anal. 12 (2023), no. 1, 97–116, DOI: https://doi.org/10.1515/anona-2022-0256.
• [27] N. H. Tuan, V. V. Au, and N. A. Tuan, Mild solutions to a time-fractional Cauchy problem with nonlocal nonlinearity in Besov spaces, Arch. Math. 118 (2022), no. 3, 305–314, DOI: https://doi.org/10.1007/s00013-022-01702-8.
• [28] N. H. Tuan, M. Foondun, T. N. Thach, and R. Wang, On backward problems for stochastic fractional reaction equations with standard and fractional Brownian motion, Bull. Sci. Math. 179 (2022), no. 103158, 58 pp, DOI: https://doi.org/10.1016/j.bulsci.2022.103158.
• [29] J. Xu, Z. Zhang, and T. Caraballo, Mild solutions to time fractional stochastic 2D-stokes equations with bounded and unbounded delay, J. Dynam. Differential Equations 34 (2022), 583–603, DOI: https://doi.org/10.1007/s10884-019-09809-3.
• [30] N. H. Tuan, N. A. Tuan, and C. Yang, Global well-posedness for fractional Sobolev-Galpern type equations, Discrete Contin. Dyn. Syst. 42 (2022), no. 6, 2637–2665, DOI: https://doi.org/10.3934/dcds.2021206.
• [31] N. A. Tuan, T. Caraballo, and 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 152 (2022), 989–1031, DOI: https://doi.org/10.1017/prm.2021.44.
• [32] T. B Ngoc, T. Caraballo, N. H. Tuan, and Y. Zhou, Existence and regularity results for terminal value problem for nonlinear fractional wave equations, Nonlinearity 34 (2021), 1448–1503, DOI: https://doi.org/10.1088/1361-6544/abc4d9.
• [33] J. Xu, Z. Zhang, and T. Caraballo, Non-autonomous nonlocal partial differential equations with delay and memory, J. Differ. Equ. 270 (2021), 505–546, DOI: https://doi.org/10.1016/j.jde.2020.07.037.
• [34] R. Khalil, M. AlHorani, A. Yousef, and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65–70, DOI: https://doi.org/10.1016/j.cam.2014.01.002.
• [35] N. H. Tuan, T. B. Ngoc, D. Baleanu, and D. O’Regan, On well-posedness of the sub-diffusion equation with conformable derivative model, Commun. Nonlinear Sci. Numer. Simul. 89 (2020), 105332, DOI: https://doi.org/10.1016/j.cnsns.2020.105332.
• [36] D. T. Dang, E. Nane, D. M. Nguyen, and N. H. Tuan, Continuity of solutions of a class of fractional equations, Potential Anal. 49 (2018), no. 3, 423–478, DOI: https://doi.org/10.1007/s11118-017-9663-5.
• [37] N. H. Tuan, D. O’Regan, and T. B. Ngoc, Continuity with respect to fractional order of the time fractional diffusion-wave equation, Evol. Equ. Control Theory 9 (2020), no. 3, 773–793, DOI: https://doi.org/10.3934/eect.2020033.
• [38] E. Karapinar, H. D. Binh, N. H. Luc, and N. H. Can, On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems, Adv. Difference Equations 2021 (2021), no. 70, 1–24, DOI: https://doi.org/10.1186/s13662-021-03232-z.
• [39] A. Jaiswal and D. Bahuguna, Semilinear conformable fractional differential equations in Banach spaces, Differ. Equ. Dyn. Syst. 27 (2019), no. 17, 313–325, DOI: https://doi.org/10.1007/s12591-018-0426-6.

Uwagi

PL

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).