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September  2020, 9(3): 773-793. doi: 10.3934/eect.2020033

## Continuity with respect to fractional order of the time fractional diffusion-wave equation

 1 Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam 2 Department of Mathematics, University of Science, Vietnam National University, 227 Nguyen Van Cu Street, District 5, Ho Chi Minh City, Viet Nam 3 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland 4 Faculty of Natural Sciences, Thu Dau Mot University, Thu Dau Mot City 820000, Binh Duong Province, Vietnam

* Corresponding author: tranbaongoc@tdmu.edu.vn (Tran Bao Ngoc)

Received  August 2019 Revised  October 2019 Published  December 2019

This paper studies a time-fractional diffusion-wave equation with a linear source function. First, some stability results on parameters of the Mittag-Leffler functions are established. Then, we focus on studying the continuity of the solution of both the initial problem and the inverse initial value problems corresponding to the fractional-order in our main results. One of the difficulties encounteblack comes from estimating all constants independently of the fractional orders. Finally, we present some numerical results to confirm the effectiveness of our methods.

Citation: Nguyen Huy Tuan, Donal O'Regan, Tran Bao Ngoc. Continuity with respect to fractional order of the time fractional diffusion-wave equation. Evolution Equations & Control Theory, 2020, 9 (3) : 773-793. doi: 10.3934/eect.2020033
##### References:
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Holm, Physical interpretation of fractional diffusion-wave equation via lossy media obeying frequency power law, Mathematical Physics, (2003), https://arXiv.org/abs/math-ph/0303040. Google Scholar [6] B. Cuahutenango-Barro, M. A. Taneco-Hernández and J. F. Gómez-Aguilar, On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel, Chaos Solitons Fractals, 115 (2018), 283-299.  doi: 10.1016/j.chaos.2018.09.002.  Google Scholar [7] 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), 423-478.  doi: 10.1007/s11118-017-9663-5.  Google Scholar [8] A. Deiveegan, J. J. Nieto and P. Prakash, The revised generalized Tikhonov method for the backward time-fractional diffusion equation, J. Appl. Anal. Comput., 9 (2019), 45-56.   Google Scholar [9] X. L. Ding and J. J. Nieto, Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms, Fractional Calculus and Applied Analysis, 21 (2018), 312-335.  doi: 10.1515/fca-2018-0019.  Google Scholar [10] H. Dong and D. Kim, $L_p$-estimates for time fractional parabolic equations with coefficients measurable in time, Adv. Math., 345 (2019), 289-345.  doi: 10.1016/j.aim.2019.01.016.  Google Scholar [11] W. Fan, F. Liu, X. Jiang and I. Turner, A novel unstructublack mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain, Fractional Calculus and Applied Analysis, 20 (2017), 352-383.  doi: 10.1515/fca-2017-0019.  Google Scholar [12] J. F. Gómez-Aguilar, L. Torres, H. Yépez-Martínez, D. Baleanu, J. M. Reyes and I. O. Sosa, Fractional Liénard type model of a pipeline within the fractional derivative without singular kernel, Adv. Difference Equ., 173 (2016), 13 pp. doi: 10.1186/s13662-016-0908-1.  Google Scholar [13] J. F. Gómez-Aguilar, H. Yépez-Martínez, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, J. M. Reyes and I. O. Sosa, Series solution for the time-fractional coupled mKdV equation using the homotopy analysis method, Math. Probl. Eng., 2016 (2016), Art. ID 7047126, 8 pp.  Google Scholar [14] J. F. Gómez-Aguilar and A. Atangana, Fractional Hunter-Saxton equation involving partial operators with bi-order in Riemann-Liouville and Liouville-Caputo sense, A. Eur. Phys. J. Plus, 132 (2017), 15 pp. Google Scholar [15] J. F. Gómez-Aguilar, M. Miranda-Hernández, M. G. López-López, V. M. Alvarado-Martínez and D. Baleanu, Modeling and simulation of the fractional space-time diffusion equation, Commun. Nonlinear Sci. Numer. Simul., 30 (2016), 115-127.  doi: 10.1016/j.cnsns.2015.06.014.  Google Scholar [16] J. F. Gómez-Aguilar, Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel, Phys. A, 465 (2017), 562-572.  doi: 10.1016/j.physa.2016.08.072.  Google Scholar [17] S. Guo, L. Mei and Y. Li, An efficient Galerkin spectral method for two-dimensional fractional nonlinear reaction-diffusion-wave equation, Computers and Mathematics with Applications, 74 (2017), 2449-2465.  doi: 10.1016/j.camwa.2017.07.022.  Google Scholar [18] L. N. Huynh, Y. Zhou, D. O'Regan and N. H. Tuan, Fractional Landweber method for an initial inverse problem for time-fractional wave equations, Applicable Analysis, 2019. doi: 10.1080/00036811.2019.1622682.  Google Scholar [19] J. Janno and N. Kinash, Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements, Inverse Problems, 34 (2018), 025007, 19 pp. doi: 10.1088/1361-6420/aaa0f0.  Google Scholar [20] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag Berlin Heidelberg, 1995.  Google Scholar [21] Y. Kian and M. Yamamoto, On existence and uniqueness of solutions for semilinear fractional wave equations, Fractional Calculus and Applied Analysis, 20 (2017), 117-138.  doi: 10.1515/fca-2017-0006.  Google Scholar [22] D. Kumar, J. Singh and D. Baleanu, A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves, Mathematical Methods in the Applied Sciences, 40 (2017), 5642-5653.  doi: 10.1002/mma.4414.  Google Scholar [23] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.  doi: 10.1142/9781848163300.  Google Scholar [24] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9 (1996), 23-28.  doi: 10.1016/0893-9659(96)00089-4.  Google Scholar [25] F. Mainardi and P. Paradisi, Fractional diffusive waves, Journal of Computational Acoustics, 9 (2001), 1417-1436.  doi: 10.1142/S0218396X01000826.  Google Scholar [26] V. F. Morales-Delgado, M. A. Taneco-Hernández and J. F. Gómez-Aguilar, On the solutions of fractional order of evolution equations, J. F. Eur. Phys. J. Plus, 132 (2017), 14 pp. doi: 10.1016/j.physa.2017.02.016.  Google Scholar [27] R. H. Nochetto, E. Otárola and A. J. Salgado, A PDE Approach to Space-Time Fractional Parabolic Problems, SIAM J. Numer. Anal., 54 (2016), 848-873.  doi: 10.1137/14096308X.  Google Scholar [28] I. Podlubny, Fractional Differential Equations, Academic Press, London, 1999.   Google Scholar [29] K. M. Saad and J. F. Gómez-Aguilar, Analysis of reaction-diffusion system via a new fractional derivative with non-singular kernel, Phys. A, 509 (2018), 703-716.  doi: 10.1016/j.physa.2018.05.137.  Google Scholar [30] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar [31] D. D. Trong, D. N. D. Hai and N. M. Dien, On a time–space fractional backward diffusion problem with inexact orders, Computers and Mathematics with Applications, 78 (2019), 1572-1593.  doi: 10.1016/j.camwa.2019.03.014.  Google Scholar [32] N. H. Tuan, A. Debbouche and T. B. Ngoc, Existence and regularity of final value problems for time fractional wave equations, Computers and Mathematics with Applications, 78 (2019), 1396-1414.  doi: 10.1016/j.camwa.2018.11.036.  Google Scholar [33] N. H. Tuan, L. N. Huynh, T. B. Ngoc and Y. Zhou, On a backward problem for nonlinear fractional diffusion equations, Appl. Math. Lett., 92 (2019), 76-84.  doi: 10.1016/j.aml.2018.11.015.  Google Scholar [34] T. Wei and Y. Zhang, The backward problem for a time-fractional diffusion-wave equation in a bounded domain, Computers and Mathematics with Applications, 75 (2018), 3632-3648.  doi: 10.1016/j.camwa.2018.02.022.  Google Scholar [35] J. Xian and T. Wei, Determination of the initial data in a time-fractional diffusion-wave problem by a final time data, Computers and Mathematics with Applications, 78 (2019), 2525-2540.  doi: 10.1016/j.camwa.2019.03.056.  Google Scholar [36] X. J. Yang, D. Baleanu and H. M. Srivastava, Local Fractional Integral Transforms and Their Applications,, Elsevier/Academic Press, Amsterdam, 2016.   Google Scholar [37] Y. Zhou, Basic Theory of Fractional Differential Equations, , World Scientific, Singapore, 2014. doi: 10.1142/9069.  Google Scholar [38] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier/Academic Press, London, 2016.   Google Scholar [39] G. Zou, A. Atangana and Y. Zhou, Error estimates of a semidiscrete finite element method for fractional stochastic diffusion-wave equations, Numer. Methods Partial Differential Equations, 34 (2018), 1834-1848.  doi: 10.1002/num.22252.  Google Scholar

show all references

##### References:
 [1] P. Agarwal, J. J. Nieto and M. J. Luo, Extended Riemann-Liouville type fractional derivative operator with applications, Open Math., 15 (2017), 1667-1681.  doi: 10.1515/math-2017-0137.  Google Scholar [2] E. Alvarez, C. G. Gal, V. Keyantuo and M. Warma, Well-posedness results for a class of semi-linear super-diffusive equations, Nonlinear Anal., 181 (2019), 24-61.  doi: 10.1016/j.na.2018.10.016.  Google Scholar [3] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.  Google Scholar [4] D. Baleanu, J. A. T. Machado and Z. B. Guvenc (Eds.), New Trends in Nanotechnology and Fractional Calculus Applications, Springer, Netherlands, 2010. doi: 10.1007/978-90-481-3293-5.  Google Scholar [5] W. Chen and S. Holm, Physical interpretation of fractional diffusion-wave equation via lossy media obeying frequency power law, Mathematical Physics, (2003), https://arXiv.org/abs/math-ph/0303040. Google Scholar [6] B. Cuahutenango-Barro, M. A. Taneco-Hernández and J. F. Gómez-Aguilar, On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel, Chaos Solitons Fractals, 115 (2018), 283-299.  doi: 10.1016/j.chaos.2018.09.002.  Google Scholar [7] 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), 423-478.  doi: 10.1007/s11118-017-9663-5.  Google Scholar [8] A. Deiveegan, J. J. Nieto and P. Prakash, The revised generalized Tikhonov method for the backward time-fractional diffusion equation, J. Appl. Anal. Comput., 9 (2019), 45-56.   Google Scholar [9] X. L. Ding and J. J. Nieto, Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms, Fractional Calculus and Applied Analysis, 21 (2018), 312-335.  doi: 10.1515/fca-2018-0019.  Google Scholar [10] H. Dong and D. Kim, $L_p$-estimates for time fractional parabolic equations with coefficients measurable in time, Adv. Math., 345 (2019), 289-345.  doi: 10.1016/j.aim.2019.01.016.  Google Scholar [11] W. Fan, F. Liu, X. Jiang and I. Turner, A novel unstructublack mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain, Fractional Calculus and Applied Analysis, 20 (2017), 352-383.  doi: 10.1515/fca-2017-0019.  Google Scholar [12] J. F. Gómez-Aguilar, L. Torres, H. Yépez-Martínez, D. Baleanu, J. M. Reyes and I. O. Sosa, Fractional Liénard type model of a pipeline within the fractional derivative without singular kernel, Adv. Difference Equ., 173 (2016), 13 pp. doi: 10.1186/s13662-016-0908-1.  Google Scholar [13] J. F. Gómez-Aguilar, H. Yépez-Martínez, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, J. M. Reyes and I. O. Sosa, Series solution for the time-fractional coupled mKdV equation using the homotopy analysis method, Math. Probl. Eng., 2016 (2016), Art. ID 7047126, 8 pp.  Google Scholar [14] J. F. Gómez-Aguilar and A. Atangana, Fractional Hunter-Saxton equation involving partial operators with bi-order in Riemann-Liouville and Liouville-Caputo sense, A. Eur. Phys. J. Plus, 132 (2017), 15 pp. Google Scholar [15] J. F. Gómez-Aguilar, M. Miranda-Hernández, M. G. López-López, V. M. Alvarado-Martínez and D. Baleanu, Modeling and simulation of the fractional space-time diffusion equation, Commun. Nonlinear Sci. Numer. Simul., 30 (2016), 115-127.  doi: 10.1016/j.cnsns.2015.06.014.  Google Scholar [16] J. F. Gómez-Aguilar, Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel, Phys. A, 465 (2017), 562-572.  doi: 10.1016/j.physa.2016.08.072.  Google Scholar [17] S. Guo, L. Mei and Y. Li, An efficient Galerkin spectral method for two-dimensional fractional nonlinear reaction-diffusion-wave equation, Computers and Mathematics with Applications, 74 (2017), 2449-2465.  doi: 10.1016/j.camwa.2017.07.022.  Google Scholar [18] L. N. Huynh, Y. Zhou, D. O'Regan and N. H. Tuan, Fractional Landweber method for an initial inverse problem for time-fractional wave equations, Applicable Analysis, 2019. doi: 10.1080/00036811.2019.1622682.  Google Scholar [19] J. Janno and N. Kinash, Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements, Inverse Problems, 34 (2018), 025007, 19 pp. doi: 10.1088/1361-6420/aaa0f0.  Google Scholar [20] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag Berlin Heidelberg, 1995.  Google Scholar [21] Y. Kian and M. Yamamoto, On existence and uniqueness of solutions for semilinear fractional wave equations, Fractional Calculus and Applied Analysis, 20 (2017), 117-138.  doi: 10.1515/fca-2017-0006.  Google Scholar [22] D. Kumar, J. Singh and D. Baleanu, A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves, Mathematical Methods in the Applied Sciences, 40 (2017), 5642-5653.  doi: 10.1002/mma.4414.  Google Scholar [23] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.  doi: 10.1142/9781848163300.  Google Scholar [24] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9 (1996), 23-28.  doi: 10.1016/0893-9659(96)00089-4.  Google Scholar [25] F. Mainardi and P. Paradisi, Fractional diffusive waves, Journal of Computational Acoustics, 9 (2001), 1417-1436.  doi: 10.1142/S0218396X01000826.  Google Scholar [26] V. F. Morales-Delgado, M. A. Taneco-Hernández and J. F. Gómez-Aguilar, On the solutions of fractional order of evolution equations, J. F. Eur. Phys. J. Plus, 132 (2017), 14 pp. doi: 10.1016/j.physa.2017.02.016.  Google Scholar [27] R. H. Nochetto, E. Otárola and A. J. Salgado, A PDE Approach to Space-Time Fractional Parabolic Problems, SIAM J. Numer. Anal., 54 (2016), 848-873.  doi: 10.1137/14096308X.  Google Scholar [28] I. Podlubny, Fractional Differential Equations, Academic Press, London, 1999.   Google Scholar [29] K. M. Saad and J. F. Gómez-Aguilar, Analysis of reaction-diffusion system via a new fractional derivative with non-singular kernel, Phys. A, 509 (2018), 703-716.  doi: 10.1016/j.physa.2018.05.137.  Google Scholar [30] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar [31] D. D. Trong, D. N. D. Hai and N. M. Dien, On a time–space fractional backward diffusion problem with inexact orders, Computers and Mathematics with Applications, 78 (2019), 1572-1593.  doi: 10.1016/j.camwa.2019.03.014.  Google Scholar [32] N. H. Tuan, A. Debbouche and T. B. Ngoc, Existence and regularity of final value problems for time fractional wave equations, Computers and Mathematics with Applications, 78 (2019), 1396-1414.  doi: 10.1016/j.camwa.2018.11.036.  Google Scholar [33] N. H. Tuan, L. N. Huynh, T. B. Ngoc and Y. Zhou, On a backward problem for nonlinear fractional diffusion equations, Appl. Math. Lett., 92 (2019), 76-84.  doi: 10.1016/j.aml.2018.11.015.  Google Scholar [34] T. Wei and Y. Zhang, The backward problem for a time-fractional diffusion-wave equation in a bounded domain, Computers and Mathematics with Applications, 75 (2018), 3632-3648.  doi: 10.1016/j.camwa.2018.02.022.  Google Scholar [35] J. Xian and T. Wei, Determination of the initial data in a time-fractional diffusion-wave problem by a final time data, Computers and Mathematics with Applications, 78 (2019), 2525-2540.  doi: 10.1016/j.camwa.2019.03.056.  Google Scholar [36] X. J. Yang, D. Baleanu and H. M. Srivastava, Local Fractional Integral Transforms and Their Applications,, Elsevier/Academic Press, Amsterdam, 2016.   Google Scholar [37] Y. Zhou, Basic Theory of Fractional Differential Equations, , World Scientific, Singapore, 2014. doi: 10.1142/9069.  Google Scholar [38] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier/Academic Press, London, 2016.   Google Scholar [39] G. Zou, A. Atangana and Y. Zhou, Error estimates of a semidiscrete finite element method for fractional stochastic diffusion-wave equations, Numer. Methods Partial Differential Equations, 34 (2018), 1834-1848.  doi: 10.1002/num.22252.  Google Scholar
A comparison between $u_\alpha$, $u_{\alpha^*}$ for $\alpha\in\{1.500\}$, $\alpha^* \in \{1.505;1.510; 1.515; 1.520\}$ at $t = 0.1$, $x \in (0,\pi)$. Here $N_x = N_t = 40$
A comparison between $u_\alpha$, $u_{\alpha^*}$ for $\alpha\in 1.500$, $\alpha^* \in 1.505,1.510, 1.515, 1.520$ at $t=0.5$, $x \in (0,\pi)$. Here $N_x = N_t = 40$
A comparison between $u_\alpha$, $u_{\alpha^*}$ for $\alpha\in \{1.500\}$, $\alpha^* \in \{1.505,1.510, 1.515, 1.520\}$ at $t = 0.9$, $x \in (0,\pi)$. Here $N_x = N_t = 40$
The solution $u_\alpha$ for $\alpha \in \{1.1; 1.2; 1.3; 1.4; 1.5; 1.6; 1.7; 1.8; 1.9\}$, $N_x = N_t = 40$
The output errors for $t\in\{0.1; 0.5; 0.9\}$, $x \in (0,\pi)$
 $\{\alpha, \alpha^*\}$ $\mathbf{N_x} = 40, \mathbf{N_t} = 40$ $|\alpha - \alpha^*|$ $\mathrm{Error(t=0.1)}$ $\mathrm{Error(t=0.5)}$ $\mathrm{Error(t=0.9)}$ $\{1.500,\,1.505\}$ 0.005 0.000932294832753 0.020776926030918 0.043799703445885 $\{1.500,\,1.510\}$ 0.010 0.001846284758162 0.041394223517507 0.087636643120345 $\{1.500,\,1.515\}$ 0.015 0.002742293511105 0.061850625744690 0.131503482942371 $\{1.500,\,1.520\}$ 0.020 0.003620640443306 0.082144942461500 0.175392929962381
 $\{\alpha, \alpha^*\}$ $\mathbf{N_x} = 40, \mathbf{N_t} = 40$ $|\alpha - \alpha^*|$ $\mathrm{Error(t=0.1)}$ $\mathrm{Error(t=0.5)}$ $\mathrm{Error(t=0.9)}$ $\{1.500,\,1.505\}$ 0.005 0.000932294832753 0.020776926030918 0.043799703445885 $\{1.500,\,1.510\}$ 0.010 0.001846284758162 0.041394223517507 0.087636643120345 $\{1.500,\,1.515\}$ 0.015 0.002742293511105 0.061850625744690 0.131503482942371 $\{1.500,\,1.520\}$ 0.020 0.003620640443306 0.082144942461500 0.175392929962381
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