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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 |
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.
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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