Approximating a Class of Linear Third-Order Ordinary Differential Problems

  • Emilio Defez
  • Michael M. TungEmail author
  • J. Ibáñez
  • Jorge Sastre
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 30)


In this work, a procedure to approximate the solution of special linear third-order matrix differential problems of the type Y(3)(x) = A(x)Y (x) + B(x) with higher-order matrix splines is proposed. An illustrative example is given.



The authors wish to thank for financial support by the Spanish Ministerio de Economía y Competitividad under grant TIN2017-89314-P, and by the Universitat Politècnica de València under grant PAID-06-18/SP20180016.


  1. 1.
    Defez, E., Tung, M.M., Ibáñez, J., Sastre, J.: Approximating and computing nonlinear matrix differential models. Math. Comput. Model. 55(7), 2012–2022 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Duffy, B., Wilson, S.: A third-order differential equation arising in thin-film flows and relevant to Tanner’s law. Appl. Math. Lett. 10(3), 63–68 (1997)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Famelis, I.T., Tsitouras, C.: Symbolic derivation of Runge–Kutta–Nyström type order conditions and methods for solving y‴(x) = f(x, y). Appl. Math. Comput. 297, 50–60 (2017)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Golub, G.H., Loan, C.F.V.: Matrix Computations, 2nd edn. The Johns Hopkins University Press, Baltimore (1989)zbMATHGoogle Scholar
  5. 5.
    Gregus, M.: Third Order Linear Differential Equations, vol. 22. Springer, Berlin (2012)Google Scholar
  6. 6.
    Loscalzo, F.R., Talbot, T.D.: Spline function approximations for solutions of ordinary differential equations. SIAM J. Numer. Anal. 4(3), 433–445 (1967)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Momoniat, E.: Symmetries, first integrals and phase planes of a third-order ordinary differential equation from thin film flow. Math. Comput. Model. 49(1–2), 215–225 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Senu, N., Mechee, M., Ismail, F., Siri, Z.: Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations y‴(x) = f(x, y). Appl. Math. Comput. 240, 281–293 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Tod, K.: Einstein–Weyl spaces and third-order differential equations. J. Math. Phys. 41(8), 5572–5581 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Tuck, E., Schwartz, L.: A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows. SIAM Rev. 32(3), 453–469 (1990)MathSciNetCrossRefGoogle Scholar
  11. 11.
    You, X., Chen, Z.: Direct integrators of Runge–Kutta type for special third-order ordinary differential equations. Appl. Numer. Math. 74, 128–150 (2013)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Emilio Defez
    • 1
  • Michael M. Tung
    • 1
    Email author
  • J. Ibáñez
    • 2
  • Jorge Sastre
    • 3
  1. 1.Instituto de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValenciaSpain
  2. 2.Instituto de Instrumentación para Imagen MolecularUniversitat Politècnica de ValènciaValenciaSpain
  3. 3.Instituto de Telecomunicaciones y Aplicaciones MultimediaUniversitat Politècnica de ValènciaValenciaSpain

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