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February  2019, 24(2): 719-735. doi: 10.3934/dcdsb.2018204

## The Rothe method for multi-term time fractional integral diffusion equations

 1 College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, 610225, Sichuan Province, China 2 Jagiellonian University in Krakow, Chair of Optimization and Control, ul. Lojasiewicza 6, 30348 Krakow, Poland 3 Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30348 Krakow, Poland

* Corresponding author: Shengda Zeng

Dedicated to Professor Zhenhai Liu on the occasion of his 60th birthday.

Received  July 2017 Revised  February 2018 Published  February 2019 Early access  June 2018

Fund Project: Project supported by the National Science Center of Poland under Maestro Project No. UMO-2012/06/A/ST1/00262, National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611, and the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No. 3792/GGPJ/H2020/2017/0.

In this paper we study a class of multi-term time fractional integral diffusion equations. Results on existence, uniqueness and regularity of a strong solution are provided through the Rothe method. Several examples are given to illustrate the applicability of main results.

Citation: Stanisław Migórski, Shengda Zeng. The Rothe method for multi-term time fractional integral diffusion equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 719-735. doi: 10.3934/dcdsb.2018204
##### References:
 [1] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Models and Numerical Methods, World Scientific, Boston, 2012. [2] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Math. Sci. Engrg., 190, London, 1993. [3] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. doi: 10.1007/978-1-4419-9158-4. [4] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. [5] S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differ. Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002. [6] V. J. Ervin, N. Heuer and J. P. Roop, Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Numer. Anal., 45 (2007), 572-591.  doi: 10.1137/050642757. [7] W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics 30, Americal Mathematical Society, Providence, RI, International Press, Somerville, MA, 2002. [8] R. Herrmann, Fractional Calculus: An Introduction for Physicists, Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. doi: 10.1142/8934. [9] J. Kačur, Application of Rothe's method to perturbed linear hyperbolic equations and variational inequalities, Czechoslovak Mathematical Journal, 34 (1984), 92-106. [10] J. Kačur, Method of Rothe in Evolution Equations, Teubner-Texte zur Mathematik 80, B. G. Teubner, Leipzig, 1985. [11] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [12] X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), 2108-2131.  doi: 10.1137/080718942. [13] Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001. [14] Z. H. Liu, S. D. Zeng and Y. R. Bai, Maximum principles for multi-term space-time variable order fractional diffusion equations and their applications, Fract. Calc. Appl. Anal., 19 (2016), 188-211.  doi: 10.1515/fca-2016-0011. [15] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, London, 2010. doi: 10.1142/9781848163300. [16] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5. [17] A. Pazy, Semigroup of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [18] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [19] A. Raheem and D. Bahuguna, Rothe's method for solving some fractional integral diffusion equation, Appl. Math. Comput., 236 (2014), 161-168.  doi: 10.1016/j.amc.2014.03.025. [20] S. Reich, Product formulas, nonlinear semigroups, and accretive operators, J. Funct. Anal., 36 (1980), 147-168.  doi: 10.1016/0022-1236(80)90097-X. [21] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser Verlag, Basel, Boston, Berlin, 2005. [22] M. Sofonea, W. Han and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage, Chapman & Hall/CRC, Boca Raton, FL, 2006. [23] Y. B. Xiao and N. J. Huang, Generalized quasi-variational-like hemivariational inequalities, Nonlinear Anal. Theory Methods and Appl., 69 (2008), 637-646.  doi: 10.1016/j.na.2007.06.011. [24] Y. B. Xiao and N. J. Huang, Sub-super-solution method for a class of higher order evolution hemivariational inequalities, Nonlinear Anal. Theory Methods and Appl., 71 (2009), 558-570.  doi: 10.1016/j.na.2008.10.093. [25] Q. Yang, I. Turner, F. Liu and M. Ilić, Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions, SIAM J. Sci. Comput., 33 (2011), 1159-1180.  doi: 10.1137/100800634. [26] E. Zeidler, Nonlinear Functional Analysis and Applications Ⅱ A/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0. [27] S. D. Zeng, D. Baleanu, Y. R. Bai and G. C. Wu, Fractional differential equations of Caputo-Katugampola type and numerical solutions, Appl. Math. Comput., 315 (2017), 549-554.  doi: 10.1016/j.amc.2017.07.003. [28] S. D. Zeng and S. Migórski, Noncoercive hyperbolic variational inequalities with applications to contact mechanics, J. Math. Anal. Appl., 455 (2017), 619-637.  doi: 10.1016/j.jmaa.2017.05.072. [29] S. D. Zeng, Z. H. Liu and S. Migórski, A class of fractional differential hemivariational inequalities with application to contact problem, Z. Angew. Math. Phys., 69: 36 (2018), 1-23, in press. https://doi.org/10.1007/s00033-018-0929-6. doi: 10.1007/s00033-018-0929-6. [30] S. D. Zeng and S. Migórski, A class of time-fractional hemivariational inequalities with application to frictional contact problem, Communications in Nonlinear Science and Numerical Simulation, 56 (2018), 34-48.  doi: 10.1016/j.cnsns.2017.07.016. [31] Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Problems, 27 (2011), 035010, 12 pp. doi: 10.1088/0266-5611/27/3/035010.

show all references

Dedicated to Professor Zhenhai Liu on the occasion of his 60th birthday.

##### References:
 [1] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Models and Numerical Methods, World Scientific, Boston, 2012. [2] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Math. Sci. Engrg., 190, London, 1993. [3] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. doi: 10.1007/978-1-4419-9158-4. [4] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. [5] S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differ. Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002. [6] V. J. Ervin, N. Heuer and J. P. Roop, Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Numer. Anal., 45 (2007), 572-591.  doi: 10.1137/050642757. [7] W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics 30, Americal Mathematical Society, Providence, RI, International Press, Somerville, MA, 2002. [8] R. Herrmann, Fractional Calculus: An Introduction for Physicists, Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. doi: 10.1142/8934. [9] J. Kačur, Application of Rothe's method to perturbed linear hyperbolic equations and variational inequalities, Czechoslovak Mathematical Journal, 34 (1984), 92-106. [10] J. Kačur, Method of Rothe in Evolution Equations, Teubner-Texte zur Mathematik 80, B. G. Teubner, Leipzig, 1985. [11] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [12] X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), 2108-2131.  doi: 10.1137/080718942. [13] Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001. [14] Z. H. Liu, S. D. Zeng and Y. R. Bai, Maximum principles for multi-term space-time variable order fractional diffusion equations and their applications, Fract. Calc. Appl. Anal., 19 (2016), 188-211.  doi: 10.1515/fca-2016-0011. [15] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, London, 2010. doi: 10.1142/9781848163300. [16] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5. [17] A. Pazy, Semigroup of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [18] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [19] A. Raheem and D. Bahuguna, Rothe's method for solving some fractional integral diffusion equation, Appl. Math. Comput., 236 (2014), 161-168.  doi: 10.1016/j.amc.2014.03.025. [20] S. Reich, Product formulas, nonlinear semigroups, and accretive operators, J. Funct. Anal., 36 (1980), 147-168.  doi: 10.1016/0022-1236(80)90097-X. [21] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser Verlag, Basel, Boston, Berlin, 2005. [22] M. Sofonea, W. Han and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage, Chapman & Hall/CRC, Boca Raton, FL, 2006. [23] Y. B. Xiao and N. J. Huang, Generalized quasi-variational-like hemivariational inequalities, Nonlinear Anal. Theory Methods and Appl., 69 (2008), 637-646.  doi: 10.1016/j.na.2007.06.011. [24] Y. B. Xiao and N. J. Huang, Sub-super-solution method for a class of higher order evolution hemivariational inequalities, Nonlinear Anal. Theory Methods and Appl., 71 (2009), 558-570.  doi: 10.1016/j.na.2008.10.093. [25] Q. Yang, I. Turner, F. Liu and M. Ilić, Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions, SIAM J. Sci. Comput., 33 (2011), 1159-1180.  doi: 10.1137/100800634. [26] E. Zeidler, Nonlinear Functional Analysis and Applications Ⅱ A/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0. [27] S. D. Zeng, D. Baleanu, Y. R. Bai and G. C. Wu, Fractional differential equations of Caputo-Katugampola type and numerical solutions, Appl. Math. Comput., 315 (2017), 549-554.  doi: 10.1016/j.amc.2017.07.003. [28] S. D. Zeng and S. Migórski, Noncoercive hyperbolic variational inequalities with applications to contact mechanics, J. Math. Anal. Appl., 455 (2017), 619-637.  doi: 10.1016/j.jmaa.2017.05.072. [29] S. D. Zeng, Z. H. Liu and S. Migórski, A class of fractional differential hemivariational inequalities with application to contact problem, Z. Angew. Math. Phys., 69: 36 (2018), 1-23, in press. https://doi.org/10.1007/s00033-018-0929-6. doi: 10.1007/s00033-018-0929-6. [30] S. D. Zeng and S. Migórski, A class of time-fractional hemivariational inequalities with application to frictional contact problem, Communications in Nonlinear Science and Numerical Simulation, 56 (2018), 34-48.  doi: 10.1016/j.cnsns.2017.07.016. [31] Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Problems, 27 (2011), 035010, 12 pp. doi: 10.1088/0266-5611/27/3/035010.
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