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

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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

Stanisław Migórski ^1,2, and Shengda Zeng ^3,,

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.

Keywords: Fractional integral diffusion equation, m-accretive operator, Rothe method, strong solution, C₀-semigroup.

Mathematics Subject Classification: Primary: 35A01, 35K57; Secondary: 39Axx, 46Txx, 47D03.

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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