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A matrix formulated algorithm for solving parabolic equations with nonlocal boundary conditions

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Abstract

We use a matrix formulated algorithm to approximate solutions of a class of nonlinear reaction-diffusion equations with nonlocal boundary conditions. Some theoretical results are presented to simplify application of operational matrix formulation and reduce the computational cost. Convergence analysis and error estimation of the method are also investigated. Finally, some numerical examples are given to demonstrate accuracy and efficiency of the proposed method.

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References

  1. Ang, W.T.: Numerical solution of non-classical parabolic problem: an integro-differential approach. Appl. Math. Comput. 175(2), 969–979 (2006)

    MathSciNet  MATH  Google Scholar 

  2. Ashyralyev, A., Yildirim, O.: A note on the second order of accuracy stable difference schemes for the non-local boundary value hyperbolic problem. Abstr. Appl. Anal. Article ID 846582, 1–29 (2012)

    MATH  Google Scholar 

  3. Avalishvili, G., Avalishvili, M., Gordeziani, D: On a non-local problem with integral boundary conditions for a multidimensional elliptic equation. Appl. Math. Lett. 24(4), 566–571 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Avalishvili, G., Avalishvili, M., Gordeziani, D: On integral non-local boundary value problems for some partial differential equations. Bull. Georegian Natl. Acad. Sci. 5(1), 31–37 (2011)

    MATH  Google Scholar 

  5. Berikelashvili, G., Khomeriki, N: On a numerical solution of one non-local boundary-value problem with mixed Dirichlet-Neumann conditions. Lith. Math. J. 53(4), 367–380 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bialecki, B., Fairweather, G., López-Marcos, J.C: The Crank-Nicolson Hermite cubic orthogonal spline collocation method for the heat equation with non-local boundary conditions. Adv. Appl. Math. Mech. 5, 442–460 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods Fundamentals in Single Domains. Springer-Verlag Berlin Heidelberg, Berlin (2006)

    MATH  Google Scholar 

  8. Debnath, L., Mikusinski, P.: Introduction to Hilbert Spaces with Applications. Academic Press, San Diego (1990)

    MATH  Google Scholar 

  9. Day, W.A.: Existence of a property of solutions of the heat equation subject to linear thermoelasticity and other theories. Quart. Appl. Math. 40, 319–330 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ekolin, G.: Finite difference methods for a non-local boundary value problem for the heat equation. BIT 31, 245–261 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Golbabi, A., Javidi, M.: A numerical solution for non-classical parabolic problem based on Chebyshev spectral collocation method. Appl. Math. Comput. 190, 179–185 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Gottlieb, D.: Orszag, S.A., Numerical Analysis of Spectral Methods: Theory and Application. SIAM, Philadelphia (1977)

    Book  Google Scholar 

  13. Hosseini, S.M., Shahmorad, S.: A matrix formulation of the Tau method for Fredholm and Volterra linear integro-differential equations. Korean J. Comput. Appl. Math. 9(2), 497–507 (2002)

    MathSciNet  MATH  Google Scholar 

  14. Hosseini, S.M., Shahmorad, S.: Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases. Appl. Math. Model. 27, 145–154 (2003)

    Article  MATH  Google Scholar 

  15. Hosseini, S.A., Shahmorad, S., Masoumi, H.: Extension of the operational Tau method for solving 1-D nonlinear transient heat conduction equation. J. King Saud Univ. 25, 283–288 (2013)

    Article  Google Scholar 

  16. Hosseini, S.A., Shahmorad, S., Talati, F.: A matrix based method for two dimensional nonlinear Volterra-Fredholm integral equations. Numer. Algor. 68, 511–529 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lánczos, C.: Trigonometric interpolation of empirical and analytical functions. J. Math. Phys. 17, 123–199 (1938)

    Article  MATH  Google Scholar 

  18. Lin, Y., Xu, S., Yin, H.: Finite difference approximations for a class of non-local parabolic equations. Internat. J. Maths. Mathl. Sci. 20, 147–163 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Martín-Vaquero, J.: Polynomial-based mean weighted residuals methods for elliptic problems with non-local boundary conditions in the rectangle. Nonlinear Anal. Modell. Control 19(3), 448–459 (2014)

    Article  MATH  Google Scholar 

  20. Martín-Vaquero, J., Queiruga-Dios, A., Encinas, A.H.: Numerical algorithms for diffusion-reaction problems with non-classical conditions. Appl. Math. Comp. 218, 5487–5495 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Martín-Vaquero, J.: Two-level fourth-order explicit schemes for diffusion equations subject to boundary integral specifications. Chaos, Solitons Fractals 42(4), 2364–2372 (2009)

    Article  MATH  Google Scholar 

  22. Martín-Vaquero, J., Vigo-Aguiar, J.: A note on efficient techniques for the second-order parabolic equation subject to non-local conditions. Appl. Numer. Math. 59(6), 1258–1264 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Martín-Vaquero, J., Wade, B.A: On efficient numerical methods for an initial-boundary value problem with non-local boundary conditions. Appl. Math. Model. 36(8), 3411–3418 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nie, C., Yu, H.: Some error estimates on the finite element approximation for two-dimensional elliptic problem with non-local boundary. Appl. Numer. Math. 68 (0), 31–38 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Ortiz, E.L., Samara, H.: Numerical solution of differential eigenvalue problems with an operational approach to the Tau method. Computing 31, 95–103 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ortiz, E.L., Samara, H.: Numerical solution of partial differential equations with variable coefficients with an operational approach to the Tau method. Comp. Math. Appl. 10, 5–13 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pao, C.V.: Asymptotic behavior of solutions of reaction-diffusion equations with non-local boundary conditions. J. Comput. Appl. Math. 88, 225–238 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pao, C.V.: Numerical solutions of reaction-diffusion equations with non-local boundary conditions. J. Comput. Appl. Math. 136, 227–243 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rudin, W.: Real and Complex Analysis, p 19. McGraw-Hill Book Co., Singapore (1987)

    MATH  Google Scholar 

  31. Saeedi, L., Tari, A., Masuleh, S H M: Numerical solution of some nonlinear Volterra integral equations of first kind. Appl. Appl. Math. 8, 214–226 (2013)

    MathSciNet  MATH  Google Scholar 

  32. Sajavic̆ius, S.: Radial basis function method for multidimensional elliptic equation with non-local conditions. Comput. Math. Appl. 67, 1407–1420 (2014)

    Article  MathSciNet  Google Scholar 

  33. Serbina, L: The solution of an initial-boundary value problem of the filtration theory with non-local boundary condition. Nonlinear Anal. Modell. Control 19(3), 488–502 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shahmorad, S., Borhanifar, A., Soltanalizadeh, B.: An approximation method based on matrix formulated algorithm for the heat equation with non-local boundary conditions. Comput. Method. Appl. Math. 12(1), 92–107 (2012)

    Article  MATH  Google Scholar 

  35. S̆tikonien, O., Sapagovas, M., C̆iupaila, R: On iterative methods for some elliptic equations with non-local conditions. Nonlinear Anal. Modell. Control 19(3), 517–535 (2014)

    Article  Google Scholar 

  36. Tari, A., Rahimi, M.Y., Shahmorad, S., Talati, F.: Development of the Tau method for the numerical solution of two-dimensional linear volterra integro-differential equations. Comput. Method. Appl. Math. 9(4), 421–435 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tari, A., Shahmorad, S.A.: Computational method for solving two-dimensional linear Volterra integral equations of the first kind. Scientia Iranica 19(3), 829–835 (2012)

    Article  MATH  Google Scholar 

  38. Volpert, V.: Elliptic Partial Differential Equations: Reaction-Diffusion Equations, vol. 2. Birkhäser Basel, France (2014)

  39. Yin, Y: On nonlinear parabolic equations with non-local boundary conditions. J. Math. Anal. Appl. 185, 161–174 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, Ardabil, Iran

    A. Borhanifar & E. Feizi

  2. Department of Applied Mathematics, University of Tabriz, Tabriz, Iran

    S. Shahmorad

Authors
  1. A. Borhanifar
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  2. S. Shahmorad
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  3. E. Feizi
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Correspondence to S. Shahmorad.

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Borhanifar, A., Shahmorad, S. & Feizi, E. A matrix formulated algorithm for solving parabolic equations with nonlocal boundary conditions. Numer Algor 74, 1203–1221 (2017). https://doi.org/10.1007/s11075-016-0192-x

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  • DOI: https://doi.org/10.1007/s11075-016-0192-x

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