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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  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 & 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. Google Scholar

[2]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Math. Sci. Engrg., 190, London, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

R. Herrmann, Fractional Calculus: An Introduction for Physicists, Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. doi: 10.1142/8934.  Google Scholar

[9]

J. Kačur, Application of Rothe's method to perturbed linear hyperbolic equations and variational inequalities, Czechoslovak Mathematical Journal, 34 (1984), 92-106.   Google Scholar

[10]

J. Kačur, Method of Rothe in Evolution Equations, Teubner-Texte zur Mathematik 80, B. G. Teubner, Leipzig, 1985.  Google Scholar

[11]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, London, 2010. doi: 10.1142/9781848163300.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser Verlag, Basel, Boston, Berlin, 2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

E. Zeidler, Nonlinear Functional Analysis and Applications Ⅱ A/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Models and Numerical Methods, World Scientific, Boston, 2012. Google Scholar

[2]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Math. Sci. Engrg., 190, London, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

R. Herrmann, Fractional Calculus: An Introduction for Physicists, Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. doi: 10.1142/8934.  Google Scholar

[9]

J. Kačur, Application of Rothe's method to perturbed linear hyperbolic equations and variational inequalities, Czechoslovak Mathematical Journal, 34 (1984), 92-106.   Google Scholar

[10]

J. Kačur, Method of Rothe in Evolution Equations, Teubner-Texte zur Mathematik 80, B. G. Teubner, Leipzig, 1985.  Google Scholar

[11]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, London, 2010. doi: 10.1142/9781848163300.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser Verlag, Basel, Boston, Berlin, 2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

E. Zeidler, Nonlinear Functional Analysis and Applications Ⅱ A/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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