Singular fractional evolution differential equations

• Authors: Mirjana Stojanovic
• Languages of publication: EN

Abstracts

EN

We give an existence-uniqueness result for linear and nonlinear time fractional evolution equations with singularities in corresponding norm in extended Colombeau algebra of generalized functions using fractional analog for Duhamel principle. Paper deals with some nonlinear models with singularities appearing in viscoelasticity and in anomalous processes which have met great interest among researchers who consider them as a challenge in recent years.

Keywords

EN

linear and nonlinear time fractional evolution equations; existence-uniqueness result; Duhamel principle; extended Colombeau algebra

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2013
• Volume: 11
• Issue: 10
• Pages: 1337-1349

Dates

published

1 - 10 - 2013

online

19 - 12 - 2013

Contributors

author

Mirjana Stojanovic

• Department of Mathematics and Informatics, University of Novi Sad, Trg D. Obradovica 4, 21 000, Novi Sad, Serbia

References

• [1] R. Gorenflo, F. Mainardi, Fractional Calculus; Integral and Differential Equations of fractional order, CISM Lecture notes, In: A. Carpinteri, F. Mainardi (Ed.), Fractals and Fractional Calculus in Continuum mechanics (Springer Verlag, Wien and New York, 1997) 233
• [2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, In: J. van Mill (Ed.), North-Holland Mathematics studies 204 (Elsevier, Amsterdam, Netherlands, 2006)
• [3] I. Podlubny, Fractional Differential Equations, (Academic Press, San Diego, 1999)
• [4] A. V. Chechkin, V. Yu. Gonchar, R. Gorenflo, N. Korabel, I. M. Sokolov, Phys. Rev. E 78, 021111 (2008) http://dx.doi.org/10.1103/PhysRevE.78.021111[Crossref]
• [5] A. V. Chechkin, R. Gorenflo, I. M. Sokolov, V. Yu. Gonchar, Fract. Calc. Appl. Anal. 6, 259 (2003)
• [6] A. V. Chechkin, J. Klafter, I. M. Sokolov, Phys. Rev. E 66, 046129 (2002) http://dx.doi.org/10.1103/PhysRevE.66.046129[Crossref]
• [7] A. V. Chechkin, R. Gorenflo, I. M. Sokolov, Phys. Rev. E 66, 046129 (2002) http://dx.doi.org/10.1103/PhysRevE.66.046129[Crossref]
• [8] A. V. Chechkin, R. Gorenflo, I. M. Sokolov, J. Phys. A: Math. Gen. 38, L679 (2005) http://dx.doi.org/10.1088/0305-4470/38/42/L03[Crossref]
• [9] I. M. Sokolov, A. V. Chechkin, J. Klafter, Acta Phys. Polon. 35, 1323 (2004)
• [10] V. D. Djordjevic, T. M. Atanackovic, J. Comput. Appl. Math. 222, 701 (2008) http://dx.doi.org/10.1016/j.cam.2007.12.013[Crossref]
• [11] W. Chen, Chaos 16, 023126 (2006) http://dx.doi.org/10.1063/1.2208452[Crossref]
• [12] W. Chen, J. Vib. Control 14, 1651 (2008) http://dx.doi.org/10.1177/1077546307087398[Crossref]
• [13] M. Stojanovic, Nonlinear Anal. 71, 5458 (2009) http://dx.doi.org/10.1016/j.na.2009.04.034[Crossref]
• [14] D. Rajter-Ciric, M. Stojanovic, Integr. Transfs. Spec. Funct. 22, 319 (2011) http://dx.doi.org/10.1080/10652469.2010.541047[Crossref]
• [15] S. Umarov, R. Gorenflo, J. Anal. Appl. 24, 449 (2005)
• [16] M. Stojanovic, Anal. Appl. (Singap.) 10, 439 (2012) http://dx.doi.org/10.1142/S0219530512500212[Crossref]
• [17] S. Umarov, E. Saydamatov, Fract. Calc. Appl. Anal. 9, 57 (2006)
• [18] M. Oberguggenberger, Integr. Transfs. Spec. Funct. 17, 101 (2006) http://dx.doi.org/10.1080/10652460500436601[Crossref]
• [19] S. Pilipovic, M. Stojanovic, Nonlinear Anal. 37, 319 (1999) http://dx.doi.org/10.1016/S0362-546X(98)00050-9[Crossref]
• [20] H. A. Biagioni, L. Caddedu, T. Gramchev, Diff. Integr. Equat. 12N5, 613 (1999)
• [21] M. Nedeljkov, S. Pilipovic, D. Rajter, Proc. Roy. Soc. Edingburgh 135 A, 863 (2005) http://dx.doi.org/10.1017/S0308210500004169[Crossref]
• [22] J. F. Colombeau, Multiplications of Distributions: A tool in mathematics, numerical engineering and theoretical physics (Springer-Verlag, Berlin and New York, 1992)
• [23] S. Pilipovic, Colombeau Generalized Functions and Pseudo-differential Operators, (Lecture in Mathematical Science, The University of Tokyo, 1994)
• [24] M. Grosser, M. Kunzinger, M. Oberguggenberger, R. Steinbauer, Geometric Generalized Functions with Applications to General Relativity, (Mathematics and its Applications, Kluwer, 2001) 537 http://dx.doi.org/10.1007/978-94-015-9845-3[Crossref]
• [25] H. A. Biagioni, M. Oberguggenberger, SIAM J. Math. Anal. 23, 923 (1992) http://dx.doi.org/10.1137/0523049[Crossref]
• [26] M. Oberguggenberger, arXiv:math/0612445v1 [math.AP]
• [27] Yu. A. Dubinskij, Russian Math. Surveys 37, 109 (1982) http://dx.doi.org/10.1070/RM1982v037n05ABEH004012[Crossref]
• [28] R. Gorenflo, Yu. F. Luchko, S. R. Umarov, Fract. Calc. Appl. Anal. 3, 249 (2000)
• [29] M. Stojanovic, Fract. Calc. Appl. Anal. 13, 21 (2010)

Identifiers

DOI

10.2478/s11534-013-0260-y