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Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems of diagonal form with large BV data
On general fractional abstract Cauchy problem
1. | School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China, China |
2. | Department of Basic Courses, Xi'an Technological University, North Institute of Information Engineering, Xi'an 710025, China |
References:
[1] |
W. Arendt, C. Batty, M. Hiever and F. Neubrander, "Vector-Valued Laplace Transforms and Cauchy Problems," Monogr. Math., vol. 96, Birh$\ddota$user, Basel, 2001. |
[2] |
E. Bazhlekova, "Fractional Evolution Equations in Banach Spaces," University Press Facilities, Eindhoven University of Technology, 2001. |
[3] |
M. Caputo, "Elasticita Dissipacione," Bologna: Zanichelli, 1969. |
[4] |
C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum, 80 (2010), 121-142.
doi: 10.1007/s00233-009-9184-7. |
[5] |
S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255.
doi: 10.1016/j.jde.2003.12.002. |
[6] |
A. Erdé, "Higher Transcendental Functions," vol. 3, McGraw-Hill, New Yourk, 1955. |
[7] |
R. Hilfer, Fractional diffusion based on Riemann-Liouville fractional derivatives, J. Phys. Chem. B, 104 (2000), 3914-3917.
doi: 10.1021/jp9936289. |
[8] |
R. Hilfer, Fractional time evolution, in "Applications of Fractional Calculus in Physics" (R. Hilfer ed.), World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000, pp. 87-130.
doi: 10.1142/9789812817747_0002. |
[9] |
R. Hilfer, Fractional calculus and regular variation in thermodynamics, In "Applications of Fractional Calculus in Physics" (R. Hilfer ed.), World Scientific, Singapore (2000). |
[10] |
R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chem. Phys., 284 (2002), 399-408.
doi: 10.1016/S0301-0104(02)00670-5. |
[11] |
R. Hilfer, Y. Luchko and Ž. Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal., 12 (2009), 299-318. |
[12] |
K. X. Li and J. G. Peng, Fractional resolvents and fractional evolution equations, Applied Mathematics Letters, 25 (2012), 808-812.
doi: 10.1016/j.aml.2011.10.023. |
[13] |
M. Li, C. Chen and F. B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal., 259 (2010), 2702-2726.
doi: 10.1016/j.jfa.2010.07.007. |
[14] |
M. M. Meerschaert, E. Nane and P. Vellaisamy, Fractional Cauchy Problems on bounded domains, Ann. Anal., 37 (2009), 979-1007.
doi: 10.1214/08-AOP426. |
[15] |
R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.
doi: 10.1016/S0370-1573(00)00070-3. |
[16] |
K. S. Miller and B. Ross, "An Introduction to the Fractional Differential Equations," New York: Wiley, 1993. |
[17] |
F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: a tutorial survey, Fract. Calc. Appl. Anal., 10 (2007), 269-308. |
[18] |
R. R. Nigmatullin, To the theoretical explanation of the "universal response", Phys. Sta. Sol. (b), 123 (1984), 739-745.
doi: 10.1002/pssb.2221230241. |
[19] |
K. B. Oldham and J. Spanier, "The Fractional Calculus," New York: Academic, 1974. |
[20] |
I. Podlubny, "Fractional Differential Equations," Academic Press, New Yourk, 1999. |
[21] |
J. Prüs, "Evolutionary Integral Equations and Applications," Birkh$\ddota$ser, Basel, Berlin, 1993. |
[22] |
T. Sandev, R. Metzler and Ž. Tomovski, Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative, J. Phys. A: Math. Theor., 44 (2011), 255203 (21pp).
doi: 10.1088/1751-8113/44/25/255203. |
[23] |
T. Sandev and Ž. Tomovski, The general time fractional Fokker-Planck equation with a constant external force, Proc. Symposium on Fractional Signals and Systems, Coimbra, Portugal, 4-5 November 2011, pages 27-39. |
[24] |
H. M. Srivastava and Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput., 211 (2009), 198-210.
doi: 10.1016/j.amc.2009.01.055. |
[25] |
Ž. Tomovski, R. Hilferb and H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms and Special Functions, 21 (2010), 797-814.
doi: 10.1080/10652461003675737. |
[26] |
G. M. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos, Phy. D., 76 (1994), 110-122.
doi: 10.1016/0167-2789(94)90254-2. |
show all references
References:
[1] |
W. Arendt, C. Batty, M. Hiever and F. Neubrander, "Vector-Valued Laplace Transforms and Cauchy Problems," Monogr. Math., vol. 96, Birh$\ddota$user, Basel, 2001. |
[2] |
E. Bazhlekova, "Fractional Evolution Equations in Banach Spaces," University Press Facilities, Eindhoven University of Technology, 2001. |
[3] |
M. Caputo, "Elasticita Dissipacione," Bologna: Zanichelli, 1969. |
[4] |
C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum, 80 (2010), 121-142.
doi: 10.1007/s00233-009-9184-7. |
[5] |
S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255.
doi: 10.1016/j.jde.2003.12.002. |
[6] |
A. Erdé, "Higher Transcendental Functions," vol. 3, McGraw-Hill, New Yourk, 1955. |
[7] |
R. Hilfer, Fractional diffusion based on Riemann-Liouville fractional derivatives, J. Phys. Chem. B, 104 (2000), 3914-3917.
doi: 10.1021/jp9936289. |
[8] |
R. Hilfer, Fractional time evolution, in "Applications of Fractional Calculus in Physics" (R. Hilfer ed.), World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000, pp. 87-130.
doi: 10.1142/9789812817747_0002. |
[9] |
R. Hilfer, Fractional calculus and regular variation in thermodynamics, In "Applications of Fractional Calculus in Physics" (R. Hilfer ed.), World Scientific, Singapore (2000). |
[10] |
R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chem. Phys., 284 (2002), 399-408.
doi: 10.1016/S0301-0104(02)00670-5. |
[11] |
R. Hilfer, Y. Luchko and Ž. Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal., 12 (2009), 299-318. |
[12] |
K. X. Li and J. G. Peng, Fractional resolvents and fractional evolution equations, Applied Mathematics Letters, 25 (2012), 808-812.
doi: 10.1016/j.aml.2011.10.023. |
[13] |
M. Li, C. Chen and F. B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal., 259 (2010), 2702-2726.
doi: 10.1016/j.jfa.2010.07.007. |
[14] |
M. M. Meerschaert, E. Nane and P. Vellaisamy, Fractional Cauchy Problems on bounded domains, Ann. Anal., 37 (2009), 979-1007.
doi: 10.1214/08-AOP426. |
[15] |
R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.
doi: 10.1016/S0370-1573(00)00070-3. |
[16] |
K. S. Miller and B. Ross, "An Introduction to the Fractional Differential Equations," New York: Wiley, 1993. |
[17] |
F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: a tutorial survey, Fract. Calc. Appl. Anal., 10 (2007), 269-308. |
[18] |
R. R. Nigmatullin, To the theoretical explanation of the "universal response", Phys. Sta. Sol. (b), 123 (1984), 739-745.
doi: 10.1002/pssb.2221230241. |
[19] |
K. B. Oldham and J. Spanier, "The Fractional Calculus," New York: Academic, 1974. |
[20] |
I. Podlubny, "Fractional Differential Equations," Academic Press, New Yourk, 1999. |
[21] |
J. Prüs, "Evolutionary Integral Equations and Applications," Birkh$\ddota$ser, Basel, Berlin, 1993. |
[22] |
T. Sandev, R. Metzler and Ž. Tomovski, Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative, J. Phys. A: Math. Theor., 44 (2011), 255203 (21pp).
doi: 10.1088/1751-8113/44/25/255203. |
[23] |
T. Sandev and Ž. Tomovski, The general time fractional Fokker-Planck equation with a constant external force, Proc. Symposium on Fractional Signals and Systems, Coimbra, Portugal, 4-5 November 2011, pages 27-39. |
[24] |
H. M. Srivastava and Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput., 211 (2009), 198-210.
doi: 10.1016/j.amc.2009.01.055. |
[25] |
Ž. Tomovski, R. Hilferb and H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms and Special Functions, 21 (2010), 797-814.
doi: 10.1080/10652461003675737. |
[26] |
G. M. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos, Phy. D., 76 (1994), 110-122.
doi: 10.1016/0167-2789(94)90254-2. |
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