Abstract.
This study considers the space-time fractional partial differential equations with multi delays under a unique formulation, proposing a numerical method involving advanced matrix system. This matrix system is made up of the matching polynomial of complete graph together with fractional Caputo and Jumarie derivative types. Also, the derivative types are scrutinized to determine which of them is more proper for the method. Convergence analysis of the method is established via an average value of residual function using double integrals. The obtained solutions are improved with the aid of a residual error estimation. A general computer program module, which contains few steps, is developed. Tables and figures prove the efficiency and simplicity of the method. Eventually, an algorithm is given to illustrate the basis of the method.
Similar content being viewed by others
References
H.G. Sun, W. Chen, C. Li, Y.Q. Chen, Physica A 389, 2719 (2010)
A.V. Chechkin, R. Gorenflo, I.M. Sokolov, J. Phys. A 38, 679 (2005)
J. Sabatier, O.P. Agrawal, Advances in Fractional Calculus, in Theoretical Developments and Applications in Physics and Engineering (Springer, 2007)
V.A. Vyawahare, P.S.V. Nataraj, in Proceedings of the 2013 European Control Conference (ECC) (Zürich, Switzerland, 2013)
X.-N. Cao, J.-L. Fu, H. Huang, Adv. Appl. Math. Mech. 4, 848 (2012)
M.S. Tavazoei, M. Haeri, Phys. Lett. A 372, 798 (2008)
H. Thabet, S. Kendre, Chaos, Solitons Fractals 109, 238 (2018)
K. Moaddy, S. Momani, I. Hashim, Comput. Math. Appl. 61, 1209 (2011)
K.S. Tasneem, D.J. Higham, Appl. Numer. Math. 24, 425 (1997)
I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999)
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, San Diego 2006)
L. Zhu, Y. Wang, Nonlinear Dyn. 89, 1915 (2017)
F.A. Rihan, Numer. Methods Partial Differ. Equ. 26, 1556 (2009)
S.Yu. Reutskiy, Appl. Math. Model. 45, 238 (2017)
H. Zhang, F. Liu, V. Anh, Appl. Math. Comput. 217, 2534 (2010)
D.B. Dhaigude, G.A. Birajdar, Adv. Appl. Math. Mech. 6, 107 (2014)
N. Kurt, M. Sezer, J. Franklin Inst. 345, 839 (2008)
N. Baykuş, M. Sezer, Numer. Methods Partial Differ. Equ. 27, 1327 (2011)
N. Baykuş Savaşaneril, M. Sezer, Appl. Math. Inf. Sci. 11, 1795 (2017)
B. Bülbül, M. Sezer, Int. J. Comput. Math. 88, 533 (2011)
S, Yüzbaşi, N. Sahin, M. Sezer, Math. Methods Appl. Sci. 35, 1126 (2012)
Ö.K. Kürkçü, E. Aslan, M. Sezer, Appl. Math. Comput. 276, 324 (2016)
Ö.K. Kürkçü, E. Aslan, M. Sezer, Sains Malays. 46, 335 (2017)
Ö.K. Kürkçü, E. Aslan, M. Sezer, Appl. Numer. Math. 121, 134 (2017)
Ö.K. Kürkçü, E. Aslan, M. Sezer, Ö. I, Int. J. Comput. Meth. 15, 1850039 (2018)
Ö.K. Kürkçü, E. Aslan, M. Sezer, Turk. J. Math. 43, 373 (2019)
M. Caputo, Elasticità e Dissipazione (Zanichelli, Bologna, 1969)
G. Jumarie, Comput. Math. Appl. 51, 1367 (2006)
U. Ghosh, S. Sengupta, S. Sarkar, S. Das, Eur. J. Acad. Essays 2, 70 (2015)
E.J. Farrell, J. Combin. Theory Ser. B 27, 75 (1979)
I. Gutman, Publ. Inst. Math. Beograd 22, 63 (1977)
I. Gutman, Match Commun. Math. Co. 6, 75 (1979)
J. Aihara, Bull. Chem. Soc. Jpn. 52, 1529 (1979)
F. Harary, Graph Theory (Addison-Wesley, New York, 1994)
J.L. Gross, J. Yellen, P.F. Zhang, Handbook of Graph Theory (CRC Press, New York, 2013)
E. Aslan, Bull. Int. Math. Virtual Inst. 4, 53 (2014)
G. Bacak-Turan, A. Kirlangiç, Ars Combin. 102, 333 (2011)
G. Gunther, B.L. Hartnell, Proceedings of the Eighth Manitoba Conference on Numerical Mathematics and Computing, Vol. 285 (1978)
O.J. Heilmann, E.H. Lieb, Commun. Math. Phys. 25, 190 (1972)
E.W. Weisstein, Matching Polynomial, in MathWorld: A Wolfram Web Resource, http://mathworld.wolfram.com/MatchingPolynomial.html
H. Hosoya, Discr. Appl. Math. 19, 239 (1988)
W. Yan, Y.-N. Yeh, Discr. Appl. Math. 157, 195 (2009)
K. Diethelm, The Analysis of Fractional Differential Equations (Springer-Verlag, Berlin Heidelberg, 2010)
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, with Formulas, graphs, and Mathematical Tables, in Appl. Math. Ser., Vol. 55 (National Bureau of Standards, 1964)
M. Du, X. Qiao, B. Wang, Y. Wang, B. Gao, Appl. Math. Comput. 347, 15 (2019)
X.Q. Du, S. Gao, Acta Oceanol. Sin. 34, 121 (2012)
N. Qiu, J. Shenyang Univ. 28, 170 (2016)
Y.L. Wang, L.J. Su, Comput. Math. Appl. 61, 421 (2011)
V.R. Hosseini, W. Chen, Z. Avazzadeh, Eng. Anal. Bound. Elem. 38, 31 (2014)
M. Zheng, F. Liu, V. Anh, I. Turner, Appl. Math. Model. 40, 4970 (2016)
M. Dehghan, M. Safarpoor, M. Abbaszadeh, J. Comput. Appl. Math. 290, 174 (2015)
C. Li, W. Deng, Adv. Appl. Math. Mech. 9, 282 (2017)
A. Mohebbi, M. Abbaszadeh, M. Dehghan, J. Comput. Phys. 240, 36 (2013)
F.H. Zeng, C.P. Li, F.W. Liu, I. Turner, SIAM J. Sci. Comput. 37, A55 (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kıvanç Kürkçü, Ö., Aslan, E. & Sezer, M. An advanced method with convergence analysis for solving space-time fractional partial differential equations with multi delays. Eur. Phys. J. Plus 134, 393 (2019). https://doi.org/10.1140/epjp/i2019-12761-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2019-12761-4