The extension of analytic solutions to FDEs to the negative half-line

Inga Timofejeva; Zenonas Navickas; Tadas Telksnys; Romas Marcinkevičius; Xiao-Jun Yang; Minvydas Ragulskis; Inga Timofejeva; Zenonas Navickas; Tadas Telksnys; Romas Marcinkevičius; Xiao-Jun Yang; Minvydas Ragulskis

doi:10.3934/math.2021195

AIMS Mathematics

2021, Volume 6, Issue 4: 3257-3271. doi: 10.3934/math.2021195

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The extension of analytic solutions to FDEs to the negative half-line

1.
Center for Nonlinear Systems, Kaunas University of Technology, Studentu 50-147, Kaunas LT-51368, Lithuania
2.
Department of Software Engineering, Kaunas University of Technology, Studentu 50-415, Kaunas LT-51368, Lithuania
3.
School of Mathematics and State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, No.1, Daxue Road, Xuzhou 221116, Jiangsu, P. R. China

Received: 14 October 2020 Accepted: 31 December 2020 Published: 15 January 2021
MSC : 34A08, 34A25, 26A33

An analytical framework for the extension of solutions to fractional differential equations (FDEs) to the negative half-line is presented in this paper. The proposed technique is based on the construction of a special characteristic equation corresponding to the original FDE (when the characteristic equation does exist). This characteristic equation enables the construction analytic solutions to FDEs are expressed in the form of infinite fractional power series. Necessary and sufficient conditions for the existence of such an extension are discussed in detail. It is demonstrated that the extension of solutions to FDEs to the negative half-line is not a single-valued operation. Computational experiments are used to illustrate the efficacy of the proposed scheme.
- fractional differential equation,
- operator calculus,
- negative half-line,
- solitary wave,
- inverse balancing
Citation: Inga Timofejeva, Zenonas Navickas, Tadas Telksnys, Romas Marcinkevičius, Xiao-Jun Yang, Minvydas Ragulskis. The extension of analytic solutions to FDEs to the negative half-line[J]. AIMS Mathematics, 2021, 6(4): 3257-3271. doi: 10.3934/math.2021195

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Abstract

An analytical framework for the extension of solutions to fractional differential equations (FDEs) to the negative half-line is presented in this paper. The proposed technique is based on the construction of a special characteristic equation corresponding to the original FDE (when the characteristic equation does exist). This characteristic equation enables the construction analytic solutions to FDEs are expressed in the form of infinite fractional power series. Necessary and sufficient conditions for the existence of such an extension are discussed in detail. It is demonstrated that the extension of solutions to FDEs to the negative half-line is not a single-valued operation. Computational experiments are used to illustrate the efficacy of the proposed scheme.

References

[1]	B. Ross, The development of fractional calculus 1695–1900, Hist. Math., 4 (1977), 75–89.
[2]	A. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Vol. 204, Elsevier Science Limited, 2006.
[3]	R. L. Bagley, P. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983), 201–210.
[4]	K. Adolfsson, M. Enelund, P. Olsson, On the fractional order model of viscoelasticity, Mech. Time-depend. Mat., 9 (2005), 15–34.
[5]	S. Pal, Mechanical properties of biological materials, Design of Artificial Human Joints & Organs, (2014), 23–40. Springer.
[6]	N. Grahovac, M. Žigić, Modelling of the hamstring muscle group by use of fractional derivatives, Comput. Math. Appl., 59 (2010), 1695–1700.
[7]	C. Ionescu, A. Lopes, D. Copot, J. T. Machado, J. Bates, The role of fractional calculus in modeling biological phenomena: A review, Commun. Nonlinear Sci., 51 (2017), 141–159.
[8]	M. P. Aghababa, M. Borjkhani, Chaotic fractional-order model for muscular blood vessel and its control via fractional control scheme, Complexity, 20 (2014), 37–46.
[9]	A. Ortega, J. Rosales, J. Cruz-Duarte, M. Guia, Fractional model of the dielectric dispersion, Optik, 180 (2019), 754–759.
[10]	V. E. Tarasov, Fractional integro-differential equations for electromagnetic waves in dielectric media, Theor. Math. Phys., 158 (2009), 355–359.
[11]	B. Ducharne, B. Newell, G. Sebald, A unique fractional derivative operator to simulate all dynamic piezoceramic dielectric manifestations: From aging to frequency-dependent hysteresis, IEEE T. Ultrason. Ferr., 67 (2019), 197–206.
[12]	R. Zhang, Z. Han, Y. Shao, Z. Wang, Y. Wang, The numerical study for the ground and excited states of fractional Bose–Einstein condensates, Comput. Math. Appl., 78 (2019), 1548–1561.
[13]	V. E. Tarasov, V. V. Tarasova, Macroeconomic models with long dynamic memory: Fractional calculus approach, Appl. Math. Comput., 338 (2018), 466–486.
[14]	V. E. Tarasov, V. V. Tarasova, Time-dependent fractional dynamics with memory in quantum and economic physics, Ann. Phys., 383 (2017), 579–599.
[15]	A. A. Arafa, A. M. S. Hagag, A new analytic solution of fractional coupled Ramani equation, Chinese J. Phys., 60 (2019), 388–406.
[16]	E. Demirci, N. Ozalp, A method for solving differential equations of fractional order, J. Comput. Appl. Math., 236 (2012), 2754–2762.
[17]	K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3–22.
[18]	M. Kumar, V. Daftardar-Gejji, A new family of predictor-corrector methods for solving fractional differential equations, Appl. Math. Comput., 363 (2019), 124633.
[19]	A. Saadatmandi, M. Dehghan, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl., 59 (2010), 1326–1336.
[20]	M. Pourbabaee, A. Saadatmandi, A novel Legendre operational matrix for distributed order fractional differential equations, Appl. Math. Comput., 361 (2019), 215–231.
[21]	M. Dehghan, J. Manafian, A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numerical Methods for Partial Differential Equations: An International Journal, 26 (2010), 448–479.
[22]	L. Aceto, D. Bertaccini, F. Durastante, P. Novati, Rational Krylov methods for functions of matrices with applications to fractional partial differential equations, J. Comput. Phys., 396 (2019), 470–482.
[23]	Y.-L. Wang, L.-n. Jia, H.-l. Zhang, Numerical solution for a class of space-time fractional equation by the piecewise reproducing kernel method, Int. J. Comput. Math., 96 (2019), 2100–2111.
[24]	A. R. Khudair, S. Haddad, Restricted fractional differential transform for solving irrational order fractional differential equations, Chaos, Solitons & Fractals, 101 (2017), 81–85.
[25]	C. Zúñiga-Aguilar, H. Romero-Ugalde, J. Gómez-Aguilar, R. Escobar-Jiménez, M. Valtierra-Rodriguez, Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks, Chaos, Solitons & Fractals, 103 (2017), 382–403.
[26]	G. E. Andrews, R. Askey, R. Roy, Special functions, Cambridge University Press, Cambridge, 2001.
[27]	Z. Navickas, T. Telksnys, R. Marcinkevicius, M. Ragulskis, Operator-based approach for the construction of analytical soliton solutions to nonlinear fractional-order differential equations, Chaos Solitons and Fractals, 104 (2017), 625–634.
[28]	L. Bikulčienė, Z. Navickas, Expressions of solutions of linear partial differential equations using algebraic operators and algebraic convolution, International Conference on Numerical Analysis and Its Applications, (2008), 208–215. Springer.
[29]	Z. Navickas, T. Telksnys, I. Timofejeva, R. Marcinkevicius, M. Ragulskis, An operator-based approach for the construction of closed-form solutions to fractional differential equations, Math. Model. Anal., 23 (2018), 665–685.
[30]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Netherlands, 2006.
[31]	J. W. Brown, R. V. Churchill, Complex variables and applications eighth edition, Boston: McGraw-Hill Higher Education, 2009.
[32]	Z. Navickas, L. Bikulciene, Expressions of solutions of ordinary differential equations by standard functions, Math. Model. Anal., 11 (2006), 399–412.
[33]	Z. Navickas, M. Ragulskis, L. Bikulciene, Special solutions of Huxley differential equation, Math. Model. Anal., 16 (2011), 248–259.
[34]	Z. Navickas, L. Bikulciene, M. Rahula, M. Ragulskis, Algebraic operator method for the construction of solitary solutions to nonlinear differential equations, Commun. Nonlinear Sci., 18 (2013), 1374–1389.
[35]	F. Ndairou, I. Area, J. J. Nieto, D. F. Torres, Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan, Chaos, Solitons & Fractals, (2020), 109846.
[36]	G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo, et al., Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nat. Med., (2020), 1–6.

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