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Explicit Formulae for Integro-Differential Operational Matrices

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Abstract

In this work we deduce explicit formulae for the elements of the matrices representing the action of integro-differential operators over the coefficients of generalized Fourier series. Our formulae are obtained by performing operations on the bases of orthogonal polynomials and result directly from the three-term recurrence relation satisfied by the polynomials. Moreover we give exact formulae for the coefficients for some families of orthogonal polynomials. Some tests are given to demonstrate the robustness of the formulas presented.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1974)

    MATH  Google Scholar 

  2. Chang, R.Y., Wang, M.L.: Shifted Legendre direct method for variational problems. J. Optim. Theory Appl. 39(2), 299–307 (1983)

    Article  MathSciNet  Google Scholar 

  3. Doha, E., Ahmed, H.: On the coefficients of integrated expansions of Bessel polynomials. J. Comput. Appl. Math. 187(1), 58–71 (2006)

    Article  MathSciNet  Google Scholar 

  4. Doha, E.H.: Explicit formulae for the coefficients of integrated expansions of Jacobi polynomials and their integrals. Integr. Transforms Spec. Funct. 14(1), 69–86 (2003)

    Article  MathSciNet  Google Scholar 

  5. Dongarra, J., Straughan, B., Walker, D.: Chebyshev tau-qz algorithm methods for calculating spectra of hydrodynamic stability problems. Appl. Numer. Math. 22(4), 399–434 (1996)

    Article  MathSciNet  Google Scholar 

  6. Greengard, L.: Spectral integration and two-point boundary value problems. SIAM J. Numer. Anal. 28(4), 1071–1080 (1991)

    Article  MathSciNet  Google Scholar 

  7. Koepf, W.: Identities for families of orthogonal polynomials and special functions. Integr. Transforms Spec. Funct. 5(1–2), 69–102 (1997)

    Article  MathSciNet  Google Scholar 

  8. Koepf, W., Schmersau, D.: Representations of orthogonal polynomials. J. Comput. Appl. Math. 90(1), 57–94 (1998)

    Article  MathSciNet  Google Scholar 

  9. Kong, W., Wu, X.: Chebyshev tau matrix method for Poisson-type equations in irregular domain. J. Comput. Appl. Math. 228(1), 158–167 (2009)

    Article  MathSciNet  Google Scholar 

  10. Krall, H.L., Frink, O.: A new class of orthogonal polynomials: the Bessel polynomials. Trans. Am. Math. Soc. 65(1), 100–115 (1949)

    Article  MathSciNet  Google Scholar 

  11. Matos, J.C., Matos, J.M.A., Rodrigues, M.J.: Solving differential and integral equations with Tau method. Math. Comput. Sci. 12(2), 197–205 (2018)

    Article  MathSciNet  Google Scholar 

  12. Matos, J.M.A., Rodrigues, M.J.: Almost exact computation of eigenvalues in approximate differential problems. Math. Comput. Appl. 24(4), 96 (2019)

    MathSciNet  Google Scholar 

  13. Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V. (eds.) NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/. Release 1.0.17 of 2017-12-22, 2010

  14. Ortiz, E.L., Samara, H.: An operational approach to the Tau method for the numerical solution of non-linear differential equations. Computing 27(1), 15–25 (1981)

    Article  MathSciNet  Google Scholar 

  15. Phillips, T.N.: On the Legendre coefficients of a general-order derivative of an infinitely differentiable function. IMA J. Numer. Anal. 8(4), 455–459 (1988)

    Article  MathSciNet  Google Scholar 

  16. Shaban, M., Kazem, S., Rad, J.: A modification of the homotopy analysis method based on Chebyshev operational matrices. Math. Comput. Modell. 57(5), 1227–1239 (2013)

    Article  MathSciNet  Google Scholar 

  17. Singh, S., Patel, V.K., Singh, V.K.: Operational matrix approach for the solution of partial integro-differential equation. Appl. Math. Comput. 283, 195–207 (2016)

    MathSciNet  MATH  Google Scholar 

  18. Singh, S., Patel, V.K., Singh, V.K.: Convergence rate of collocation method based on wavelet for nonlinear weakly singular partial integro-differential equation arising from viscoelasticity. Numer. Methods Part. Differ. Equ. 34(5), 1781–1798 (2018)

    Article  MathSciNet  Google Scholar 

  19. Singh, S., Patel, V.K., Singh, V.K., Tohidi, E.: Numerical solution of nonlinear weakly singular partial integro-differential equation via operational matrices. Appl. Math. Comput. 298, 310–321 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Singh, V.K., Postnikov, E.B.: Operational matrix approach for solution of integro-differential equations arising in theory of anomalous relaxation processes in vicinity of singular point. Appl. Math. Model. 37(10), 6609–6616 (2013)

    Article  MathSciNet  Google Scholar 

  21. Vasconcelos, P.B., Matos, J., Trindade, M.S.: Spectral Lanczos’ Tau Method for Systems of Nonlinear Integro-Differential Equations, pp. 305–314. Springer, Cham (2017)

    MATH  Google Scholar 

  22. Vasconcelos, P.B., Matos, J., Trindade, M.S.: Dealing with non-polynomial coefficients within Tau method. Math. Comput. Sci. 12(2), 183–195 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was partially supported by CMUP (UID/ MAT/ 00144/ 2019), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020.

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Authors and Affiliations

  1. Laboratório de Engenharia Matemática, Instituto Superior de Engenharia do Porto, Porto, Portugal

    José M. A. Matos & João Carrilho de Matos

  2. Centro de Matemática da, Universidade do Porto, Porto, Portugal

    José M. A. Matos & Maria João Rodrigues

  3. Faculdade de Ciências da, Universidade do Porto, Porto, Portugal

    Maria João Rodrigues

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  1. José M. A. Matos
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  2. Maria João Rodrigues
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  3. João Carrilho de Matos
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Correspondence to Maria João Rodrigues.

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Matos, J.M.A., Rodrigues, M.J. & de Matos, J.C. Explicit Formulae for Integro-Differential Operational Matrices. Math.Comput.Sci. 15, 45–61 (2021). https://doi.org/10.1007/s11786-020-00465-1

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