• Previous Article
    Dirac-concentrations in an integro-pde model from evolutionary game theory
  • DCDS-B Home
  • This Issue
  • Next Article
    Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments
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  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 & 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. ErvinN. 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. LiuS. 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. YangI. TurnerF. 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. ZengD. BaleanuY. 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. ErvinN. 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. LiuS. 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. YangI. TurnerF. 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. ZengD. BaleanuY. 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

[1]

Jiří Neustupa. On $L^2$-Boundedness of a $C_0$-Semigroup generated by the perturbed oseen-type operator arising from flow around a rotating body. Conference Publications, 2007, 2007 (Special) : 758-767. doi: 10.3934/proc.2007.2007.758

[2]

Yu-Xia Liang, Ze-Hua Zhou. Supercyclic translation $C_0$-semigroup on complex sectors. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 361-370. doi: 10.3934/dcds.2016.36.361

[3]

Dieter Bothe, Petra Wittbold. Abstract reaction-diffusion systems with $m$-completely accretive diffusion operators and measurable reaction rates. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2239-2260. doi: 10.3934/cpaa.2012.11.2239

[4]

Xin Yu, Guojie Zheng, Chao Xu. The $C$-regularized semigroup method for partial differential equations with delays. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5163-5181. doi: 10.3934/dcds.2016024

[5]

András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks & Heterogeneous Media, 2012, 7 (1) : 43-58. doi: 10.3934/nhm.2012.7.43

[6]

Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022

[7]

Antoine Hochart. An accretive operator approach to ergodic zero-sum stochastic games. Journal of Dynamics & Games, 2019, 6 (1) : 27-51. doi: 10.3934/jdg.2019003

[8]

Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311

[9]

Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007

[10]

Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032

[11]

Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032

[12]

Hao Yang, Fuke Wu, Peter E. Kloeden. Existence and approximation of strong solutions of SDEs with fractional diffusion coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5553-5567. doi: 10.3934/dcdsb.2019071

[13]

Claude Bardos, François Golse, Ivan Moyano. Linear Boltzmann equation and fractional diffusion. Kinetic & Related Models, 2018, 11 (4) : 1011-1036. doi: 10.3934/krm.2018039

[14]

T. Diogo, P. Lima, M. Rebelo. Numerical solution of a nonlinear Abel type Volterra integral equation. Communications on Pure & Applied Analysis, 2006, 5 (2) : 277-288. doi: 10.3934/cpaa.2006.5.277

[15]

Philip M. J. Trevelyan. Approximating the large time asymptotic reaction zone solution for fractional order kinetics $A^n B^m$. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 219-234. doi: 10.3934/dcdss.2012.5.219

[16]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250

[17]

Jacek Banasiak, Marcin Moszyński. Hypercyclicity and chaoticity spaces of $C_0$ semigroups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 577-587. doi: 10.3934/dcds.2008.20.577

[18]

Piotr Kościelniak, Marcin Mazur. On $C^0$ genericity of various shadowing properties. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 523-530. doi: 10.3934/dcds.2005.12.523

[19]

Kingshook Biswas. Maximal abelian torsion subgroups of Diff( C,0). Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 839-844. doi: 10.3934/dcds.2011.29.839

[20]

Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (84)
  • HTML views (428)
  • Cited by (0)

Other articles
by authors

[Back to Top]