Abstract
In the present paper, we successfully solve some linear fractional differential equations (FDE) analytically by solving an auxiliary linear differential equation with an integer order. The idea of the suggested method is based on transforming the given FDE into a linear differential equation with an integer order. This transformation removes certain terms of the solution of the considered FDE, resulting in the remaining terms being a solution to the auxiliary equation. To demonstrate the ability and efficacy of this idea, several examples have been provided.
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References
Abro KA, Atangana A (2020) Dual fractional modeling of rate type fluid through non-local differentiation. Numer Methods Partial Differ Equ 20:20
Ahmadova A, Mahmudov NI (2021) A class of Langevin time-delay differential equations with general fractional orders and their applications to vibration theory. J Comput Appl Math 388:113299
Ahmadova A, Huseynov IT, Fernandez A, Mahmudov NI (2021) Trivariate Mittag–Leffler functions used to solve multi-order systems of fractional differential equations. Commun Nonlinear Sci Numer Simul 97:105735
Alchikh R, Khuri S (2020) Numerical solution of a fractional differential equation arising in optics. Optik 208:163911
Arshad S, Baleanu D, Tang Y (2019) Fractional differential equations with bio-medical applications. In: Baleanu D, Lopes AM (eds) Applications in engineering, life and social sciences, Part A. De Gruyter, Berlin, pp 1–20
Bayrak MA, Demir A (2018) A new approach for space-time fractional partial differential equations by residual power series method. Appl Math Comput 336:215
Caputo M (1990) The splitting of the free oscillations of the Earth caused by the rheology. Rendiconti Lincei 1:119
Cesbron L, Mellet A, Trivisa K (2012) Anomalous transport of particles in plasma physics. App Math Lett 25:2344
Chouhan D, Mishra V, Srivastava H (2021) Bernoulli wavelet method for numerical solution of anomalous infiltration and diffusion modeling by nonlinear fractional differential equations of variable order. Results Appl Math 10:100146
Cresson J (2010) Inverse problem of fractional calculus of variations for partial differential equations. Commun Nonlinear Sci Numer Simul 15:987
Demirci E, Ozalp N (2012) A method for solving differential equations of fractional order. J Comput Appl Math 236:2754
Diethelm K (2010) The analysis of fractional differential equations. Springer, Berlin
Diethelm K, Ford NJ, Freed AD (2004) Detailed error analysis for a fractional Adams method. Numer Algorithms 36:31
Djeghali N, Djennoune S, Bettayeb M, Ghanes M, Barbot J-P (2016) Observation and sliding mode observer for nonlinear fractional-order system with unknown input. ISA Trans 63:1
Erturk VS, Momani S, Odibat Z (2008) Application of generalized differential transform method to multi-order fractional differential equations. Commun Nonlinear Sci Numer Simul 13:1642
Faghih A, Mokhtary P (2021) A new fractional collocation method for a system of multi-order fractional differential equations with variable coefficients. J Comput Appl Math 383:113139
He X, Rafiee M, Mareishi S, Liew K (2015) Large amplitude vibration of fractionally damped viscoelastic CNTs/fiber/polymer multiscale composite beams. Compos Struct 131:1111
Hilfer R (2000) Applications of fractional calculus in physics. World Scietific, Singapore
Huseynov IT, Mahmudov NI (2020) Particular solution of linear sequential fractional differential equation with constant coefficients by inverse fractional differential operators. Math Methods Appl Sci 20:20
Huseynov IT, Mahmudov NI (2021) A class of Langevin time-delay differential equations with general fractional orders and their applications to vibration theory. J King Saud Univ Sci 33:101596
Huseynov IT, Ahmadova A, Fernandez A, Mahmudov NI (2021) Explicit analytical solutions of incommensurate fractional differential equation systems. Appl Math Comput 390:125590
Jamil B, Anwar MS, Rasheed A, Irfan M (2020) MHD Maxwell flow modeled by fractional derivatives with chemical reaction and thermal radiation. Chin J Phys 67:512
Jiang Y-L, Ding X-L (2013) Waveform relaxation methods for fractional differential equations with the Caputo derivatives. J Comput Appl Math 238:51
Jong K, Choi H, Jang K, Pak S (2021) A new approach for solving one-dimensional fractional boundary value problems via Haar wavelet collocation method. Appl Numer Math 160:313
Khalaf SL, Khudair AR (2017) Particular solution of linear sequential fractional differential equation with constant coefficients by inverse fractional differential operators. Differ Equ Dyn Syst 25:373
Khudair AR (2013) On solving non-homogeneous fractional differential equations of Euler type. Comput Appl Math 32:577
Khudair AR, Haddad S, Khalaf SL (2017) Restricted fractional differential transform for solving irrational order fractional differential equations. Chaos Solitons Fractals 101:81
Kilbas AA (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam (ISBN 9780444518323)
Kilbas A, Rivero M, Rodríguez-Germá L, Trujillo J (2006) Caputo linear fractional differential equations. IFAC Proc Vol 39:52
Koca I (2015) A method for solving differential equations of q-fractional order. Appl Math Comput 266:1
Kumar V, Malik M, Debbouche A (2021) Stability and controllability analysis of fractional damped differential system with non-instantaneous impulses. Appl Math Comput 391:125633
Li C, Deng W (2007) Remarks on fractional derivatives. Appl Math Comput 187:777
Li C, Zhang F, Kurths J, Zeng F (2013) Equivalent system for a multiple-rational-order fractional differential system. Philos Trans R Soc A Math Phys Eng Sci 371:20120156
Li H-L, Jiang Y-L, Wang Z, Zhang L, Teng Z (2015) Global Mittag–Leffler stability of coupled system of fractional-order differential equations on network. Appl Math Comput 270:269
Martin O (2019) Stability approach to the fractional variational iteration method used for the dynamic analysis of viscoelastic beams. J Comput Appl Math 346:261
Mier J, Sánchez R, Newman D (2020) Tracer particle transport dynamics in the diffusive sandpile cellular automaton. Chaos Solitons Fractals 140:110117
Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New York
Mittag-Leffler GM (1903) Sur la nouvelle fonction E\(\alpha \) (x). CR Acad Sci Paris 137:554
Momani S, Odibat Z (2007) Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons Fractals 31:1248
Odibat ZM, Shawagfeh NT (2007) Generalized Taylor’s formula. Appl Math Comput 186:286
Oeser M, Pellinien T (2012) Computational framework for common visco-elastic models in engineering based on the theory of rheology. Comput Geotech 42:145
Oldham KB (2010) Fractional differential equations in electrochemistry. Adv Eng Softw 41:9
Oldham K, Spanier J (1974) The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier, New York
Ozalp N, Mizrak OO (2017) Fractional Laplace transform method in the framework of the CTIT transformation. J Comput Appl Math 317:90
Panda R, Dash M (2006) Fractional generalized splines and signal processing. Signal Process 86:2340
Peng X, Wang Y, Zuo Z (2021) Stabilization of non-smooth variable order switched nonlinear systems. ISA Trans 110:160
Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, New York
Podlubny I, Magin RL, Trymorush I (2017) Niels Henrik Abel and the birth of fractional calculus. Fract Calc Appl Anal 20:25
Radwan A, Moaddy K, Salama K, Momani S, Hashim I (2014) Control and switching synchronization of fractional order chaotic systems using active control technique. J Adv Res 5:125
Rauf A, Mahsud Y, Siddique I (2020) Multi-layer flows of immiscible fractional Maxwell fluids in a cylindrical domain. Chin J Phys 67:265
Ross B (1977) Fractional calculus. Math Mag 50:115
Samko SG, Kilbas AA, Marichev OI et al (1993) Fract Integrals Deriv, vol 1. Gordon and Breach Science Publishers, Yverdon Yverdon-les-Bains
Sweilam NH, El-Sayed AAE, Boulaaras S (2021) Fractional-order advection-dispersion problem solution via the spectral collocation method and the non-standard finite difference technique. Chaos Solitons Fractals 144:110736
Wittbold P, Wolejko P, Zacher R (2021) Bounded weak solutions of time-fractional porous medium type and more general nonlinear and degenerate evolutionary integro-differential equations. J Math Anal Appl 499:125007
Zhang T, Tong C (2018) A remark on the fractional order differential equations. J Comput Appl Math 340:375
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Communicated by Agnieszka Malinowska.
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Jalil, A.F.A., Khudair, A.R. Toward solving fractional differential equations via solving ordinary differential equations. Comp. Appl. Math. 41, 37 (2022). https://doi.org/10.1007/s40314-021-01744-8
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DOI: https://doi.org/10.1007/s40314-021-01744-8
Keywords
- Fractional differential equations
- Fractional calculus
- Caputo fractional derivatives
- Ordinary differential equation
- Laplace transform