Skip to main content
SpringerLink
Log in

Application of Euler Matrix Method for Solving Linear and a Class of Nonlinear Fredholm Integro-Differential Equations

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

The main aim of this paper is to apply a novel matrix method to compute analytic approximate solutions for the Fredholm integro-differential equations under mixed conditions. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution in terms of Euler polynomials. The introduced approach is applied to already worked problems in the literature by means of different numerical methods. Comparisons clearly show that our scheme is better and even more superior as compared to the existing ones.

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

Access this article

Log in via an institution

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

  1. Abdou M.A., Badr A.A.: On a method for solving an integral equation in the displacement contact problem. J. Appl. Math. Comput. 127, 65–78 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables. National bureau of standards, Washington, DC (1964)

  3. Arfken, G.: Mathematical Methods for Physicists, 3rd ed. Academic Press, San Diego (1985)

  4. Asady B., Tavassoli Kajani M., Vencheh A.H., Heydari A.: Solving second kind integral equations with hybrid of Fourier and block-pulse functions. Appl. Math. Comput. 160, 517–522 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bhrawy A.H., Tohidi E., Soleymani F.: A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals. Appl. Math. Comput. 219, 482–497 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Caraus I., Al Faquid F.M.: Convergence of the collocation method and the mechanical quadrature method for systems of singular integro-differential equations in Lebesgue spaces. J. Comput. Appl. Math. 236, 3796–3804 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cheon G.-S.: A note on the Bernoulli and Euler polynomials. Appl. Math. Lett. 16, 365–368 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Darania P., Ebadian A.: A method for the numerical solution of the integro-differential equations. Appl. Math. Comput. 188, 657–668 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dehghan M., Salehi R.: The numerical solution of the non-linear integro-differential equations based on the meshless method. J. Comput. Appl. Math. 236, 2367–2377 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. El-Sayed S.M., Abdel-Aziz M.R.: A comparison of Adomians decomposition method and wavelet-galerkin method for solving integro-differential equations. Appl. Math. Comput. 136, 151–159 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 1. McGraw-Hill, New York (1953)

  12. Farnoosh R., Ebrahimi M.: Monte Carlo method for solving Fredholm integral equations. Appl. Math. Comput. 195, 309–315 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Grimmer R., Liu J.H.: Singular perturbations in viscoelasticity. Rocky Mountain J. Math. 24, 61–75 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hansen, E.R.: A Table of Series and Products. Prentice hall, Englewood cliffs (1975)

  15. Jackiewicz Z., Rahman M., Welfert B.D.: Numerical solution of a Fredholm integro-differential equation modelling neural networks. Appl. Numer. Math. 56, 423–432 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Jangveladze T., Kiguradze Z., Neta B.: Finite difference approximation of a nonlinear integro-differential system. Appl. Math. Comput. 215, 615–628 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Jerri, A.J.: Introduction to Integral Equations with Applications. Wiely, New York (1985)

  18. Kajani M.T., Ghasemi M., Babolian E.: Comparision between the homotopy perturbation method and the sine-cosine wavelet method for solving linear integro- differential equations. Appl. Math. Comput. 54, 1162–1168 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Keller J.B., Olmstead W.E.: Temperature of nonlinearly radiating semi-infinite solid. Q. Appl. Math. Comput. 29, 61–75 (1972)

    Google Scholar 

  20. Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics, Third enlarged edition. Springer, New York (1966)

  21. Maleknejad K., Arzhang A.: Numerical solution of the Fredholm singular integro-differential equation with Cauchy kernel by using Taylor-series expansion and Galerkin method. Appl. Math. Comput. 182, 888–897 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Maleknejad K., Basirat B., Hashemizadeh E.: A Bernstein operational matrix approach for solving a system of high order linear Volterra–Fredholm integro-differential equations. Math. Comput. Model. 55, 1363–1372 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  23. Maleknejad K., Mirzaee F.: Numerical solution of integro-differential equations by using rationalized Haar functions method. Kybernetes 35, 1735–1744 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Maleknejad K., Nedaiasl K.: Application of Sinc-collocation method for solving a class of nonlinear Fredholm integral equations. Comput. Math. Appl. 62, 3292–3303 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Maleknejad K., Shahrezaee M., Khatami H.: Numerical solution of integral equations system of the second kind by Block-Pulse functions. Appl. Math. Comput. 166, 15–24 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  26. Mirzaee F., Bimesl S.: A new approach to numerical solution of second-order linear hyperbolic partial differential equations arising from physics and engineering. Resu. Phys. 3, 241–247 (2013)

    Article  Google Scholar 

  27. Mirzaee, F., Bimesl, S.: A new Euler matrix method for solving systems of linear Volterra integral equations with variable coefficients, J. Egypt. Math. Soci. (2014) (in press)

  28. Neta B., Igwe J.O.: Finite differences versus finite elements for solving nonlinear integro-differential equations. J. Math. Anal. Appl. 112, 607–618 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  29. Olmstead W.E., Handelsman R.A.: Diffusion in semi-infinite region with non-linear surface dissipation. SIAM Rev. 18, 275–291 (1996)

    Article  MathSciNet  Google Scholar 

  30. Ordokhani Y.: An application of Walsh functions for Fredholm Hammerstein integro-differential equations. Int. J. Contemp. Math. Sci. 5, 1055–1063 (2010)

    MATH  MathSciNet  Google Scholar 

  31. Rashidinia, J., Zarebnia, M.: The numerical solution of integro-differential equation by means of the Sinc method. Appl. Math. Comput. 188, 1124–1130 (2007)

    Google Scholar 

  32. Reihani M.H., Abadi Z.: Rationalized Haar functions method for solving Fredholm and Volterra integral equations. J. Comput. Appl. Math. 200, 12–20 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  33. Roman, S.: The Umbral Calculus. Academic Press, New York (1984)

  34. Shahmorad S.: Numerical solution of the general form linear Fredholm Volterra integro-differential equations by the Tau method with an error estimation. Appl. Math. Comput. 167, 1418–1429 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  35. Srivastava, H.M., Choi, J.: Series Associated with the Zeta and Related Functions. Kluwer Academic, Dordreeht (2001)

  36. Yalcinbas S., Sezar M.: The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials. Appl. Math. Comput. 112, 291–308 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  37. Young P.T.: Congruences for Bernoulli, Euler and Stirling numbers. J. Number Theory 78, 204–227 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  38. Yuzbasi S., Shahin N., Sezer M.: Numerical solutions of systems of linear Fredholm integro-differential equations with Bessel polynomial bases. Comput. Math. Appl. 61, 3079–3096 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Malayer University, Malayer, Iran

    Farshid Mirzaee & Saeed Bimesl

Authors
  1. Farshid Mirzaee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saeed Bimesl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Farshid Mirzaee.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mirzaee, F., Bimesl, S. Application of Euler Matrix Method for Solving Linear and a Class of Nonlinear Fredholm Integro-Differential Equations. Mediterr. J. Math. 11, 999–1018 (2014). https://doi.org/10.1007/s00009-014-0391-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00009-014-0391-4

Mathematics Subject Classification

Keywords

Search

Navigation