Skip to main content
Log in

Toward solving fractional differential equations via solving ordinary differential equations

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abro KA, Atangana A (2020) Dual fractional modeling of rate type fluid through non-local differentiation. Numer Methods Partial Differ Equ 20:20

    Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Alchikh R, Khuri S (2020) Numerical solution of a fractional differential equation arising in optics. Optik 208:163911

    Article  Google Scholar 

  • 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

    Google Scholar 

  • 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

    MathSciNet  MATH  Google Scholar 

  • Caputo M (1990) The splitting of the free oscillations of the Earth caused by the rheology. Rendiconti Lincei 1:119

    Article  Google Scholar 

  • Cesbron L, Mellet A, Trivisa K (2012) Anomalous transport of particles in plasma physics. App Math Lett 25:2344

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Cresson J (2010) Inverse problem of fractional calculus of variations for partial differential equations. Commun Nonlinear Sci Numer Simul 15:987

    Article  MathSciNet  MATH  Google Scholar 

  • Demirci E, Ozalp N (2012) A method for solving differential equations of fractional order. J Comput Appl Math 236:2754

    Article  MathSciNet  MATH  Google Scholar 

  • Diethelm K (2010) The analysis of fractional differential equations. Springer, Berlin

    Book  MATH  Google Scholar 

  • Diethelm K, Ford NJ, Freed AD (2004) Detailed error analysis for a fractional Adams method. Numer Algorithms 36:31

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • Hilfer R (2000) Applications of fractional calculus in physics. World Scietific, Singapore

    Book  MATH  Google Scholar 

  • 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

    Google Scholar 

  • 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

    Article  Google Scholar 

  • Huseynov IT, Ahmadova A, Fernandez A, Mahmudov NI (2021) Explicit analytical solutions of incommensurate fractional differential equation systems. Appl Math Comput 390:125590

    MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Jiang Y-L, Ding X-L (2013) Waveform relaxation methods for fractional differential equations with the Caputo derivatives. J Comput Appl Math 238:51

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Khudair AR (2013) On solving non-homogeneous fractional differential equations of Euler type. Comput Appl Math 32:577

    Article  MathSciNet  MATH  Google Scholar 

  • Khudair AR, Haddad S, Khalaf SL (2017) Restricted fractional differential transform for solving irrational order fractional differential equations. Chaos Solitons Fractals 101:81

    Article  MathSciNet  MATH  Google Scholar 

  • Kilbas AA (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam (ISBN 9780444518323)

    MATH  Google Scholar 

  • Kilbas A, Rivero M, Rodríguez-Germá L, Trujillo J (2006) Caputo linear fractional differential equations. IFAC Proc Vol 39:52

    Article  Google Scholar 

  • Koca I (2015) A method for solving differential equations of q-fractional order. Appl Math Comput 266:1

    MathSciNet  MATH  Google Scholar 

  • 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

    MathSciNet  MATH  Google Scholar 

  • Li C, Deng W (2007) Remarks on fractional derivatives. Appl Math Comput 187:777

    MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Mier J, Sánchez R, Newman D (2020) Tracer particle transport dynamics in the diffusive sandpile cellular automaton. Chaos Solitons Fractals 140:110117

    Article  MathSciNet  Google Scholar 

  • Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New York

    MATH  Google Scholar 

  • Mittag-Leffler GM (1903) Sur la nouvelle fonction E\(\alpha \) (x). CR Acad Sci Paris 137:554

    MATH  Google Scholar 

  • Momani S, Odibat Z (2007) Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons Fractals 31:1248

    Article  MathSciNet  MATH  Google Scholar 

  • Odibat ZM, Shawagfeh NT (2007) Generalized Taylor’s formula. Appl Math Comput 186:286

    MathSciNet  MATH  Google Scholar 

  • Oeser M, Pellinien T (2012) Computational framework for common visco-elastic models in engineering based on the theory of rheology. Comput Geotech 42:145

    Article  Google Scholar 

  • Oldham KB (2010) Fractional differential equations in electrochemistry. Adv Eng Softw 41:9

    Article  MATH  Google Scholar 

  • Oldham K, Spanier J (1974) The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier, New York

    MATH  Google Scholar 

  • Ozalp N, Mizrak OO (2017) Fractional Laplace transform method in the framework of the CTIT transformation. J Comput Appl Math 317:90

    Article  MathSciNet  MATH  Google Scholar 

  • Panda R, Dash M (2006) Fractional generalized splines and signal processing. Signal Process 86:2340

    Article  MATH  Google Scholar 

  • Peng X, Wang Y, Zuo Z (2021) Stabilization of non-smooth variable order switched nonlinear systems. ISA Trans 110:160

    Article  Google Scholar 

  • 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

    MATH  Google Scholar 

  • Podlubny I, Magin RL, Trymorush I (2017) Niels Henrik Abel and the birth of fractional calculus. Fract Calc Appl Anal 20:25

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • Rauf A, Mahsud Y, Siddique I (2020) Multi-layer flows of immiscible fractional Maxwell fluids in a cylindrical domain. Chin J Phys 67:265

    Article  MathSciNet  Google Scholar 

  • Ross B (1977) Fractional calculus. Math Mag 50:115

    Article  MathSciNet  MATH  Google Scholar 

  • Samko SG, Kilbas AA, Marichev OI et al (1993) Fract Integrals Deriv, vol 1. Gordon and Breach Science Publishers, Yverdon Yverdon-les-Bains

    Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang T, Tong C (2018) A remark on the fractional order differential equations. J Comput Appl Math 340:375

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by Agnieszka Malinowska.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-021-01744-8

Keywords

Mathematics Subject Classification

Navigation