Skip to main content
Log in

Ordinary Differential Equations with Singular Coefficients: An Intrinsic Formulation with Applications to the Euler–Bernoulli Beam Equation

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript


We study a class of linear ordinary differential equations (ODEs) with distributional coefficients. These equations are defined using an intrinsic multiplicative product of Schwartz distributions which is an extension of the Hörmander product of distributions with non-intersecting singular supports (Hörmander in The analysis of linear partial differential operators I, Springer, Berlin, 1983). We provide a regularization procedure for these ODEs and prove an existence and uniqueness theorem for their solutions. We also determine the conditions for which the solutions are regular and distributional. These results are used to study the Euler–Bernoulli beam equation with discontinuous and singular coefficients. This problem was addressed in the past using intrinsic products (under some restrictive conditions) and the Colombeau formalism (in the general case). Here we present a new intrinsic formulation that is simpler and more general. As an application, the case of a non-uniform static beam displaying structural cracks is discussed in some detail.

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.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others


  1. Hilbert space methods are also very important, namely in the context of singular perturbations of Schrödinger operators [1, 2, 11, 16].


  1. Albeverio, S., Gesztesy, F., Högh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics, 2nd edn. AMS, Chelsea (2005)

    MATH  Google Scholar 

  2. Albeverio, S., Kurasov, P.: Singular Perturbations of Differential Operators and Solvable Schrödinger Type Operators. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  3. Atanackovic, T.M.: Stability Theory of Elastic Rods. World Scientific, Singapore (1997)

    Book  Google Scholar 

  4. Bagarello, F.: Multiplication of distribution in one dimension: possible approaches and applications to \(\delta \)-function and its derivatives. J. Math. Anal. Appl. 196, 885–901 (1995)

    Article  MathSciNet  Google Scholar 

  5. Bagarello, F.: Multiplication of distribution in one dimension and a first application to quantum field theory. J. Math. Anal. Appl. 266, 298–320 (2002)

    Article  MathSciNet  Google Scholar 

  6. Biondi, B., Caddemi, S.: Euler–Bernoulli beams with multiple singularities in the flexural stiffness. Eur. J. Mech. A Solids 46(5), 789–809 (2007)

    Article  MathSciNet  Google Scholar 

  7. Caddemi, S., Cali, I.: Exact solution of the multi-cracked Euler–Bernoulli column. Int. J. Solids Struct. 45(5), 1332–1351 (2008)

    Article  Google Scholar 

  8. Colombeau, J.F.: New Generalized Functions and Multiplication of Distributions. North-Holland Mathematical Studies, vol. 84. North-Holland, Amsterdam (1984)

    MATH  Google Scholar 

  9. Colombeau, J.F., Le Roux, A.Y., Noussair, A., Perrot, B.: Microscopic profiles of shock waves and ambiguities in multiplication of distributions. SIAM J. Numer. Anal. 26, 871–883 (1989)

    Article  MathSciNet  Google Scholar 

  10. Dias, N.C., Prata, J.N.: A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients. J. Math. Anal. Appl. 359, 216–228 (2009)

    Article  MathSciNet  Google Scholar 

  11. Dias, N.C., Posilicano, A., Prata, J.N.: Self-adjoint, globally defined Hamiltonian operators for systems with boundaries. Commun. Pure Appl. Anal. 10(6), 1687–1706 (2011)

    Article  MathSciNet  Google Scholar 

  12. Dias, N.C., Jorge, C., Prata, J.N.: One-dimensional Schrödinger operators with singular potentials: a Schwartz distributional formulation. J. Differ. Equ. 260(8), 6548–6580 (2016)

    Article  Google Scholar 

  13. Dias, N.C., Jorge, C, Prata, J.N.: An existence and uniqueness result about algebras of Schwartz distributions (submitted)

  14. Dias, N.C., Jorge, C., Prata, J.N.: Ordinary differential equations with point interactions: an inverse problem. J. Math. Anal. Appl. 471, 53–72 (2019)

    Article  MathSciNet  Google Scholar 

  15. Fleming, W.: Functions of Several Variables, 2nd edn. Springer, Berlin (1977)

    Book  Google Scholar 

  16. Golovaty, Y.D., Hryniv, R.O.: Norm resolvent convergence of sigularly scaled Schrödinger operators and \(\delta ^{\prime }\)-potentials. Proc. R. Soc. Edinb. A 143(4), 791–816 (2013)

    Article  Google Scholar 

  17. Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric Theory of Generalized Functions. Mathematics and Its Applications, vol. 537. Kluwer Academic Publishers, Dordrecht (2001)

    MATH  Google Scholar 

  18. Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer, Berlin (1983)

    MATH  Google Scholar 

  19. Hörmann, G., Oparnica, Lj: Distributional solution concepts for the Euler–Bernoulli beam equation with discontinuous coefficients. Appl. Anal. 86(11), 1347–1363 (2007)

    Article  MathSciNet  Google Scholar 

  20. Hörmann, G., Oparnica, Lj: Generalized solutions for the Euler–Bernoulli model with distributional forces. J. Math. Anal. Appl. 357(1), 142–153 (2009)

    Article  MathSciNet  Google Scholar 

  21. Hörmann, G., Konjik, S., Oparnica, L.J.: Generalized solutions for the Euler–Bernoulli model with Zener viscoelastic foundations and distributional forces. Anal. Appl. 11(2), 1350017 (2013)

    Article  MathSciNet  Google Scholar 

  22. Kanwal, R.P.: Generalized Functions: Theory and Technique, 2nd edn. Birkhäuser, Boston (1998)

    MATH  Google Scholar 

  23. Kurasov, P.: Distribution theory for discontinuous test functions and differential operators with generalized coefficients. J. Math. Anal. Appl. 201, 297–323 (1996)

    Article  MathSciNet  Google Scholar 

  24. Kurasov, P., Boman, J.: Finite rank singular perturbations and distributions with discontinuous test functions. Proc. Am. Math. Soc. 126, 1673–1683 (1998)

    Article  MathSciNet  Google Scholar 

  25. Oberguggenberger, M.: Multiplication of Distributions and Applications to Partial Differential Equations. Pitman Research Notes in Mathematics, vol. 259. Longman, Harlow (1992)

    MATH  Google Scholar 

  26. Rosinger, E.E.: Generalized Solutions of Nonlinear Partial Differential Equations. North Holland Mathematics Studies, vol. 146. North Holland, Amsterdam (1987)

    MATH  Google Scholar 

  27. Sarrico, C.: Collision of delta-waves in a turbulent model studied via a distribution product. Nonlinear Anal. 73(9), 2868–2875 (2010)

    Article  MathSciNet  Google Scholar 

  28. Schwartz, L.: Sur l’lmpossibilite de la multiplication des distributions. C. R. Acad. Sci. Paris Ser. I Math. 239, 847–848 (1954)

    MATH  Google Scholar 

  29. Teschl, G.: Ordinary Differential Equations and Dynamical Systems. Graduate Studies in Mathematics, vol. 140. American Mathematical Society, Providence (2012)

    Book  Google Scholar 

  30. Yavari, A., Sarkani, S., Moyer, E.T.: On applications of generalized functions to beam bending problems. Int. J. Solids Struct. 37(40), 5675–5705 (2000)

    Article  MathSciNet  Google Scholar 

  31. Yavari, A., Sarkani, S., Reddy, J.N.: On nonuniform Euler–Bernoulli and Timoshenko beams with jump discontinuities: application of distribution theory. Int. J. Solid. Struct. 38(46–47), 8389–8406 (2001)

    Article  MathSciNet  Google Scholar 

  32. Whitney, H.: Functions differentiable on the boundaries of regions. Ann. Math. 35, 485 (1934)

    MathSciNet  MATH  Google Scholar 

Download references


Cristina Jorge was supported by the Ph.D. Grant SFRH/BD/85839/2012 of the Portuguese Science Foundation. N.C. Dias and J.N. Prata were supported by the Portuguese Science Foundation (FCT) under the Grant PTDC/MAT-CAL/4334/2014.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Nuno Costa Dias.

Additional information

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

Dias, N.C., Jorge, C. & Prata, J.N. Ordinary Differential Equations with Singular Coefficients: An Intrinsic Formulation with Applications to the Euler–Bernoulli Beam Equation. J Dyn Diff Equat 33, 593–619 (2021).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification