, 44:246 | Cite as

On the numerical solution of fractional differential equations with cubic nonlinearity via matching polynomial of complete graph

  • Ömür KIVANÇ KürkçüEmail author
  • Ersİn Aslan
  • Mehmet Sezer


This study deals with a generalized form of fractional differential equations with cubic nonlinearity, employing a matrix-collocation method dependent on the matching polynomial of complete graph. The method presents a simple and efficient algorithmic infrastructure, which contains a unified matrix expansion of fractional-order derivatives and a general matrix relation for cubic nonlinearity. The method also performs a sustainable approximation for high value of computation limit, thanks to the inclusion of the matching polynomial in matrix system. Using the residual function, the convergence and error estimation are investigated via the second mean value theorem having a weight function. In comparison with the existing results, highly accurate results are obtained. Moreover, the oscillatory solutions of some model problems arising in several applied sciences are simulated. It is verified that the proposed method is reliable, efficient and productive.


Fractional differential equations matrix-collocation method convergence analysis Laplace–Padé method 



The authors would like to thank the reviewers for their constructive and valuable comments to improve the paper.


  1. 1.
    Caputo M 1969 Elasticità e dissipazione. Bologna: ZanichelliGoogle Scholar
  2. 2.
    Diethelm K 2010 The analysis of fractional differential equations. Berlin–Heidelberg: Springer-VerlagzbMATHGoogle Scholar
  3. 3.
    Engheta N 1996 On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans. Antennas Propag. 44: 554–566MathSciNetzbMATHGoogle Scholar
  4. 4.
    Gülsu M, Öztürk Y and Anapalı A 2013 A collocation method for solving fractional Riccati differential equation. Adv. Appl. Math. Mech. 5: 872–884MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hilfer R 2000 Applications of fractional calculus in physics. Singapore: World ScientificzbMATHGoogle Scholar
  6. 6.
    Kulish V V and Lage J L 2002 Application of fractional calculus to fluid mechanics. J. Fluids Eng. 124: 803–806Google Scholar
  7. 7.
    Ortigueira M D, Ionescu C M, Machado J T and Trujillo J J 2015 Fractional signal processing and applications. Signal Process. 107: 197Google Scholar
  8. 8.
    Podlubny I 1999 Fractional differential equations. New York: Academic PresszbMATHGoogle Scholar
  9. 9.
    Verotta D 2010 Fractional dynamics pharmacokinetics–pharmacodynamic models. J. Pharmacokinet. Pharmacodyn. 37: 257–276Google Scholar
  10. 10.
    West B J 2015 Exact solution to fractional logistic equation. Physica A 429: 103–108MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kürkçü Ö K, Aslan E and Sezer M 2019 A novel graph-operational matrix method for solving multidelay fractional differential equations with variable coefficients and a numerical comparative survey of fractional derivative types. Turk. J. Math. 43: 373–392MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kürkçü Ö K, Aslan E and Sezer M 2019 An integrated numerical method with error analysis for solving fractional differential equations of quintic nonlinear type arising in applied sciences. Math. Methods Appl. Sci. Google Scholar
  13. 13.
    Yüzbaşı Ş 2013 Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials. Appl. Math. Comput. 219: 6328–6343MathSciNetzbMATHGoogle Scholar
  14. 14.
    Saeed U 2017 CAS Picard method for fractional nonlinear differential equation. Appl. Math. Comput. 307: 102–112MathSciNetzbMATHGoogle Scholar
  15. 15.
    Yarmohammadi M, Javadi S and Babolian E 2018 Spectral iterative method and convergence analysis for solving nonlinear fractional differential equation. J. Comput. Phys. 359: 436–450MathSciNetzbMATHGoogle Scholar
  16. 16.
    Rehman M and Khan R A 2011 The Legendre wavelet method for solving fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 16: 4163–4173MathSciNetzbMATHGoogle Scholar
  17. 17.
    Baykuş Savaşaneril N and Sezer M 2017 Hybrid Taylor–Lucas collocation method for numerical solution of high-order pantograph type delay differential equations with variables delays. Appl. Math. Inf. Sci. 11: 1795–1801MathSciNetGoogle Scholar
  18. 18.
    Baykuş Savaşaneril N and Sezer M 2011 Solution of high-order linear Fredholm integro-differential equations with piecewise intervals. Numer. Methods Partial Differ. Equ. 27: 1327–1339MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kurt N and Sezer M 2008 Polynomial solution of high-order linear Fredholm integro differential equations with constant coefficients. J. Franklin Inst. 345: 839–850MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kürkçü Ö K, Aslan E and Sezer M 2017 A numerical method for solving some model problems arising in science and convergence analysis based on residual function. Appl. Numer. Math. 121: 134–148MathSciNetzbMATHGoogle Scholar
  21. 21.
    Baleanu D, Mousalou A and Rezapour S 2017 A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo–Fabrizio derivative. Adv. Differ. Equ. 2017Google Scholar
  22. 22.
    Caputo M and Fabrizio M 2015 A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1: 1–13Google Scholar
  23. 23.
    Losada J and Nieto J J 2015 Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1: 87–92Google Scholar
  24. 24.
    Atangana A and Baleanu D 2016 New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20: 763–769Google Scholar
  25. 25.
    Doungmo Goufo E F 2016 Application of the Caputo–Fabrizio fractional derivative without singular kernel to Korteweg-de Vries–Bergers equation. Math. Model. Anal. 21: 188–198MathSciNetGoogle Scholar
  26. 26.
    Doungmo Goufo E F 2016 Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: basic theory and applications. Chaos Interdiscip. J. Nonlinear Sci. 26: 084305MathSciNetzbMATHGoogle Scholar
  27. 27.
    Heilmann O J and Lieb E H 1972 Theory of monomer-dimer systems. Commun. Math. Phys. 25: 190–232.MathSciNetzbMATHGoogle Scholar
  28. 28.
    Farrell E J 1979 An introduction to matching polynomials. J. Comb. Theory Ser. B 27: 75–86MathSciNetzbMATHGoogle Scholar
  29. 29.
    Gutman I 1977 The acyclic polynomial of a graph. Publ. Inst. Math. Beograd 22: 63–69.MathSciNetzbMATHGoogle Scholar
  30. 30.
    Aihara J 1979 Matrix representation of an olefinic reference structure for monocyclic conjugated compounds. Bull. Chem. Soc. Japan 52: 1529–1530Google Scholar
  31. 31.
    Harary F 1969 Graph theory. New York: Addison-WesleyzbMATHGoogle Scholar
  32. 32.
    Gutman I 1979 The matching polynomial. MATCH Commun. Math. Comput. Chem. 6: 75–91MathSciNetzbMATHGoogle Scholar
  33. 33.
    Godsil C D and Gutman I 1981 On the theory of the matching polynomial. J. Graph Theor. 5: 137–145MathSciNetzbMATHGoogle Scholar
  34. 34.
    Weisstein E W Matching polynomial. In: Math World: A Wolfram Web Resource.
  35. 35.
    Hosoya H 1988 On some counting polynomials in chemistry. Discrete Appl. Math. 19: 239–257MathSciNetzbMATHGoogle Scholar
  36. 36.
    Ghosh T, Mondal S and Mandal B 2018 Matching polynomial coefficients and the Hosoya indices of poly(p-phenylene) graphs. Mol. Phys. 116: 361–377Google Scholar
  37. 37.
    Araujo O, Estrada M, Morales D A and Rada J 2005 The higher-order matching polynomial of a graph. Int. J. Math. Math. Sci. 10: 1565–1576MathSciNetzbMATHGoogle Scholar
  38. 38.
    Aslan E 2014 A measure of graphs vulnerability: edge scattering number. Bull. Int. Math. Virtual Inst. 4: 53–60MathSciNetzbMATHGoogle Scholar
  39. 39.
    Bacak-Turan G and Kırlangıç A 2011 Neighbor rupture degree and the relations between other parameters. Ars Comb. 102: 333–352MathSciNetzbMATHGoogle Scholar
  40. 40.
    Momani S and Ertürk V S 2008 Solutions of non-linear oscillators by the modified differential transform method. Comput. Math. Appl. 55: 833–842MathSciNetzbMATHGoogle Scholar
  41. 41.
    Sweilam N H and Khader M M 2009 Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method. Comput. Math. Appl. 58: 2134–2141MathSciNetzbMATHGoogle Scholar
  42. 42.
    Hobson E W 1909 On the second mean value theorem of the integral calculus. Proc. London Math. Soc. 2–7: 14–23 doi: MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Dixon A 1929 The second mean value theorem in the integral calculus. Math. Proc. Cambridge 25: 282–284zbMATHGoogle Scholar
  44. 44.
    Nourazar S and Mirzabeigy A 2013 Approximate solution for nonlinear Duffing oscillator with damping effect using the modified differential transform method. Sci. Iran. B 20: 364–368Google Scholar
  45. 45.
    Akyüz-Daşcıoğlu A and Çerdik-Yaslan H 2011 The solution of high-order nonlinear ordinary differential equations by Chebyshev series. Appl. Math. Comput. 217: 5658–5666MathSciNetzbMATHGoogle Scholar
  46. 46.
    Kaur H, Mittal R C and Mishra V 2014 Haar wavelet solutions of nonlinear oscillator equations. Appl. Math. Model. 38: 4958–4971MathSciNetzbMATHGoogle Scholar
  47. 47.
    Lev B I, Tymchyshyn V B and Zagorodny A G 2017 On certain properties of nonlinear oscillator. Phys. Lett. A 381: 3417–3423MathSciNetzbMATHGoogle Scholar
  48. 48.
    Wu Y and He J H 2018 Homotopy perturbation method for nonlinear oscillators with coordinate-dependent mass. Results Phys. 10: 270–271Google Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  • Ömür KIVANÇ Kürkçü
    • 1
    Email author
  • Ersİn Aslan
    • 2
  • Mehmet Sezer
    • 3
  1. 1.Department of Mathematicsİzmir University of EconomicsIzmirTurkey
  2. 2.Department of Software EngineeringManisa Celal Bayar UniversityManisaTurkey
  3. 3.Department of MathematicsManisa Celal Bayar UniversityManisaTurkey

Personalised recommendations