An operator-based approach for the construction of closed-form solutions to fractional differential equations
Abstract
An operator-based approach for the construction of closed-form solutions to fractional differential equations is presented in this paper. The technique is based on the analysis of Caputo and Riemann-Liouville algebras of fractional power series. Explicit solutions to a class of linear fractional differential equations are obtained in terms of Mittag-Leffler and fractionally-integrated exponential functions in order to demonstrate the viability of the proposed technique.
Keyword : fractional differential equation, operator calculus, analytical solution, closed-form solution
How to Cite
Navickas, Z., Telksnys, T., Timofejeva, I., Marcinkevičius, R., & Ragulskis, M. (2018). An operator-based approach for the construction of closed-form solutions to fractional differential equations. Mathematical Modelling and Analysis, 23(4), 665-685. https://doi.org/10.3846/mma.2018.040
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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K.B. Oldham and J. Spanier(Eds.).F The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press,Cambridge, 1974.
F.W.J. Olver, D.M. Lozier, R.F. Boisvert and C.W. Clark. NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge, 2010.
S.D. Purohit and S.L. Kalla. On fractional partial differential equations related to quantum mechanics .J. Phys. A,44(4):045202, 2010.
M. Rivero, L. Rodriguez-Germa and J.J. Trujillo. Linear fractional differentia lequations with variable coefficients. Appl. Math. Lett.,21(5):892–897, 2008
M. Al-Refai, M. Ali Hajji and M.I. Syam. An efficient series solution for fractional differential equations. Abstr. Appl. Anal., ID891837, 2014. https://doi.org/10.1155/2014/891837
M.K. Al-Srihin and M. Al-Refai. An efficient series solution for nonlinear multiterm fractional differential equations. Discrete Dyn. Nat. Soc., ID5234151, 2017. https://doi.org/10.1155/2017/5234151
O.A. Arqub, A. El-Ajou and S. Momani. Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations. J. Comp. Phys., 293:385–399, 2015.https://doi.org/10.1016/j.jcp.2014.09.034
O.A. Arqub, A. El-Ajou, Z. Al Zhour and S. Momani. Multiple solutions ofnonlinear boundary value problems of fractional order: a new analytic iterativetechnique.Entropy,16(1):471–493, 2014. https://doi.org/10.3390/e16010471
D. Baleanu and J.J. Trujillo. Exact solutions of a class of fractional Hamiltonian equations involving Caputo derivatives. Phys Scr, 80(5):055101, 2009. https://doi.org/10.1088/0031-8949/80/05/055101
E. Bazhlekova and I. Dimovski. Exact solution of two-term time-fractional Thornley’s problem by operational method. Integr. Transf. Spec. F.,25(1):61–74, 2014. https://doi.org/10.1080/10652469.2013.815184
S. Bouzidi, H. Bechir and F. Bremand. Phenomenological isotropic viscohyperelasticity: a differential model based on fractional derivatives. J. Engrg. Math., 99(1):1–28, 2016.https://doi.org/10.1007/s10665-015-9818-6
S. Das. Functional Fractional Calculus. Springer, Berlin-Heidelberg, 2011. https://doi.org/10.1007/978-3-642-20545-3
M. Edelman. Fractional standard map: Riemann-Liouville vs. Ca-puto.Commun. Nonlin. Sci. Numer. Simulat., 16(12):4573–4580, 2011. https://doi.org/10.1016/j.cnsns.2011.02.007
A. El-Ajou, O. A. Arqub, Z. Al Zhour and S. Momani. New results on frac-tional power series: Theories and applications. Entropy, 15(12):5305–5323, 2013. https://doi.org/10.3390/e15125305
A. El-Ajou, O.A. Arqub, S. Momani, D. Baleanu and A. Alsaedi. Anovel expansion iterative method for solving linear partial differentialequations of fractional order. Appl. Math. Comp., 257:119–133, 2015. https://doi.org/10.1016/j.amc.2014.12.121
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward. Recurrence equences. American Mathematical Society, Providence, RI, 2003. https://doi.org/10.1090/surv/104
H. Fallahgoul, S. Focardi and F. Fabozzi. Fractional Calculus and Fractional Processes with Applications to Financial Economics: Theory and Application. Academic Press, 2016.
E.F. Doungmo Goufo and J.J. Nieto. Attractors for fractional differential problems of transition to turbulent flows. J. Comput. Appl. Math., 2017.
G. H. Hardy(Ed.). Divergent Series. Clarendon Press, Oxford, 1949.
R. Hilfer. Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000. https://doi.org/10.1142/3779
R. Kopka. Estimation of supercapacitor energy storage based on fractional differential equations. Nanoscale Res. Lett., 12(1):636, 2017. https://doi.org/10.1186/s11671-017-2396-y
S.-D. Lin, C.-H. Lu and S.-M. Su. Particular solutions of a certain class of associated Cauchy-Euler fractional partial differential equations via fractionalcalculus. Bound. Value Probl., 2013(1):126, 2013. https://doi.org/10.1186/1687-2770-2013-126
B.N. Lundstrom, M.H. Higgs, W.J. Spain and A.L. Fairhall. Fractional differentiation by neocortical pyramidal neurons.Nat. Neurosci., 11(11):1335–1342, 2008. https://doi.org/10.1038/nn.2212
R.L. Magin. Fractional Calculus in Bioengineering. Begell House Redding, 2006.
K.S. Miller and B. Ross(Eds.).An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York, 1993.
G.M. Mittag-Leffler. Sur la nouvelle fonction Eα(z).C. R. Acad. Sci.,137:554–558, 1903.
C.A. Monje, Y.-Q. Chen, B.M. Vinagre, D. Xue and V. Feliu. Fractional-order Systems and Controls. Springer, London, 2010. https://doi.org/10.1007/978-1-84996-335-0
Z. Navickas and L. Bikulciene. Expressions of solutions of ordinary differential equations by standard functions.Math. Model. Anal.,11:399–412, 2006.
Z. Navickas, T. Telksnys, R. Marcinkevicius and M. Ragulskis .Operator-based approach for the construction of analytical soliton solutions to nonlinear fractional-order differential equations.Chaos Solitons and Fractals,104:625–634,2017. https://doi.org/10.1016/j.chaos.2017.09.026
K.B. Oldham and J. Spanier(Eds.).F The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press,Cambridge, 1974.
F.W.J. Olver, D.M. Lozier, R.F. Boisvert and C.W. Clark. NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge, 2010.
S.D. Purohit and S.L. Kalla. On fractional partial differential equations related to quantum mechanics .J. Phys. A,44(4):045202, 2010.
M. Rivero, L. Rodriguez-Germa and J.J. Trujillo. Linear fractional differentia lequations with variable coefficients. Appl. Math. Lett.,21(5):892–897, 2008