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OPERATIONAL MATRICES TO SOLVE NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS OF MULTI-ARBITRARY ORDER

Year 2016, Volume: 29 Issue: 4, 895 - 907, 19.12.2016

Abstract

Fractional calculus has been used for modelling many of physical and engineering processes, that many of them are described by linear and nonlinear Volterra- Fredholm integro- differential equations of multi-arbitrary order. Therefore, an efficient and suitable method for the solution of them is very important. In this paper, the generalized fractional order of the Chebyshev functions (GFCFs) based on the classical Chebyshev polynomials of the first kind used to obtain the solution of the linear and nonlinear multi-order Volterra-Fredholm integro-differential equations. Also the operational matrices of the fractional derivative, the product, and the fractional integration to transform the equations to a system of algebraic equations are introduced. Some examples are included to demonstrate the validity and applicability of the technique.

References

  • Kazem, S., Abbasbandy, S. and Kumar S., "Fractional-order Legendre functions for solving fractional-order differential equations", Appl. Math. Modell., 37: 5498-5510, (2013).
  • Delkhosh, M., "Introduction of Derivatives and Integrals of Fractional order and Its Applications", Appl. Math. and Phys., 1(4): 103-119, (2013).
  • Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, (2006).
  • Craven, B.D., "Stone’s Theorem and Completeness of Orthogonal Systems", J. Australian Math. Soc., 12(2): 211-223, (1971).
  • Szego, G., orthogonal polynomials, American Mathematical Society Providence, Rhode Island, (1975).
  • Boyd, J.P., Chebyshev and Fourier Spectral Methods, Second Edition, Dover Publications, Mineola, New York, (2000).
  • Bhrawy, A.H. and Alofi, A.S., "The operational matrix of fractional integration for shifted Chebyshev polynomials", Appl. Math. Letters, 26: 25-31 (2013).
  • Parand, K., Abbasbandy, S., Kazem, S. and Rezaei, A.R., "An improved numerical method for a class of astrophysics problems based on radial basis functions", Phys. Scr., 83(1): 015011, 11 pages, (2011).
  • Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S., "A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order", Comput. Math. Appl., 62: 2364-2373, (2011).
  • Saadatmandi, A. and Dehghan, M., "Numerical solution of hyperbolic telegraph equation using the Chebyshev Tau method", Num. Meth. Partial Diff. Equ. 26(1): 239-252, (2010).
  • Parand, K., Taghavi, A. and Shahini, M., "Comparison between rational Chebyshev and modified generalized Laguerre functions pseudospectral methods for solving Lane-Emden and unsteady gas equations", Acta Phys. Polo. B, 40(12): 1749-1763, (2009).
  • Parand, K., Rezaei, A.R. and Taghavi, A., "Numerical approximations for population growth model by rational Chebyshev and Hermite functions collocation approach: a comparison", Math. Method. Appl. Sci., 33(17): 2076-2086, (2010).
  • Parand, K. and Khaleqi, S., "The rational Chebyshev of Second Kind Collocation Method for Solving a Class of Astrophysics Problems", Euro. Phys. J. Plus, 131: 1-24, (2016).
  • Parand, K., Dehghan, M. and Taghavi, A., "Modified generalized Laguerre function Tau method for solving laminar viscous flow: The Blasius equation", Int. J. Numer. Method. H., 20(7): 728-743, (2010).
  • Parand, K. and Delkhosh, M., "Solving Volterra’s population growth model of arbitrary order using the generalized fractional order of the Chebyshev functions", Ricerche Mat., 65(1): 307-328, (2016).
  • Darani, M.A. and Nasiri, M., "A fractional type of the Chebyshev polynomials for approximation of solution of linear fractional differential equations", Comp. Meth. Diff. Equ., 1: 96-107, (2013).
  • Butcher, E.A., Ma, H., Bueler, E., Averina, V. and Szabo, Z., "Stability of linear time-periodic delay-differential equations via Chebyshev polynomials", Int. J. Numer. Meth. Eng., 59: 895-922, (2004).
  • Mason, J.C. and Handscomb, D.C., Chebyshev polynomials, CRC Press Company, ISBN 0-8493-0355-9, (2003).
  • Turkyilmazoglu, M., "High-order nonlinear Volterra-Fredholm-Hammerstein integro-differential equations and their effective computation", Appl. Math. Comput., 247: 410-416, (2014).
  • Ordokhani, Y. and Razzaghi, M., "Solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via a collocation method and rationalized Haar functions", Appl. Math. Lett., 21: 4-9, (2008).
Year 2016, Volume: 29 Issue: 4, 895 - 907, 19.12.2016

Abstract

References

  • Kazem, S., Abbasbandy, S. and Kumar S., "Fractional-order Legendre functions for solving fractional-order differential equations", Appl. Math. Modell., 37: 5498-5510, (2013).
  • Delkhosh, M., "Introduction of Derivatives and Integrals of Fractional order and Its Applications", Appl. Math. and Phys., 1(4): 103-119, (2013).
  • Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, (2006).
  • Craven, B.D., "Stone’s Theorem and Completeness of Orthogonal Systems", J. Australian Math. Soc., 12(2): 211-223, (1971).
  • Szego, G., orthogonal polynomials, American Mathematical Society Providence, Rhode Island, (1975).
  • Boyd, J.P., Chebyshev and Fourier Spectral Methods, Second Edition, Dover Publications, Mineola, New York, (2000).
  • Bhrawy, A.H. and Alofi, A.S., "The operational matrix of fractional integration for shifted Chebyshev polynomials", Appl. Math. Letters, 26: 25-31 (2013).
  • Parand, K., Abbasbandy, S., Kazem, S. and Rezaei, A.R., "An improved numerical method for a class of astrophysics problems based on radial basis functions", Phys. Scr., 83(1): 015011, 11 pages, (2011).
  • Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S., "A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order", Comput. Math. Appl., 62: 2364-2373, (2011).
  • Saadatmandi, A. and Dehghan, M., "Numerical solution of hyperbolic telegraph equation using the Chebyshev Tau method", Num. Meth. Partial Diff. Equ. 26(1): 239-252, (2010).
  • Parand, K., Taghavi, A. and Shahini, M., "Comparison between rational Chebyshev and modified generalized Laguerre functions pseudospectral methods for solving Lane-Emden and unsteady gas equations", Acta Phys. Polo. B, 40(12): 1749-1763, (2009).
  • Parand, K., Rezaei, A.R. and Taghavi, A., "Numerical approximations for population growth model by rational Chebyshev and Hermite functions collocation approach: a comparison", Math. Method. Appl. Sci., 33(17): 2076-2086, (2010).
  • Parand, K. and Khaleqi, S., "The rational Chebyshev of Second Kind Collocation Method for Solving a Class of Astrophysics Problems", Euro. Phys. J. Plus, 131: 1-24, (2016).
  • Parand, K., Dehghan, M. and Taghavi, A., "Modified generalized Laguerre function Tau method for solving laminar viscous flow: The Blasius equation", Int. J. Numer. Method. H., 20(7): 728-743, (2010).
  • Parand, K. and Delkhosh, M., "Solving Volterra’s population growth model of arbitrary order using the generalized fractional order of the Chebyshev functions", Ricerche Mat., 65(1): 307-328, (2016).
  • Darani, M.A. and Nasiri, M., "A fractional type of the Chebyshev polynomials for approximation of solution of linear fractional differential equations", Comp. Meth. Diff. Equ., 1: 96-107, (2013).
  • Butcher, E.A., Ma, H., Bueler, E., Averina, V. and Szabo, Z., "Stability of linear time-periodic delay-differential equations via Chebyshev polynomials", Int. J. Numer. Meth. Eng., 59: 895-922, (2004).
  • Mason, J.C. and Handscomb, D.C., Chebyshev polynomials, CRC Press Company, ISBN 0-8493-0355-9, (2003).
  • Turkyilmazoglu, M., "High-order nonlinear Volterra-Fredholm-Hammerstein integro-differential equations and their effective computation", Appl. Math. Comput., 247: 410-416, (2014).
  • Ordokhani, Y. and Razzaghi, M., "Solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via a collocation method and rationalized Haar functions", Appl. Math. Lett., 21: 4-9, (2008).
There are 20 citations in total.

Details

Journal Section Mathematics
Authors

Kourosh Parand This is me

Mehdi Delkhosh This is me

Publication Date December 19, 2016
Published in Issue Year 2016 Volume: 29 Issue: 4

Cite

APA Parand, K., & Delkhosh, M. (2016). OPERATIONAL MATRICES TO SOLVE NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS OF MULTI-ARBITRARY ORDER. Gazi University Journal of Science, 29(4), 895-907.
AMA Parand K, Delkhosh M. OPERATIONAL MATRICES TO SOLVE NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS OF MULTI-ARBITRARY ORDER. Gazi University Journal of Science. December 2016;29(4):895-907.
Chicago Parand, Kourosh, and Mehdi Delkhosh. “OPERATIONAL MATRICES TO SOLVE NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS OF MULTI-ARBITRARY ORDER”. Gazi University Journal of Science 29, no. 4 (December 2016): 895-907.
EndNote Parand K, Delkhosh M (December 1, 2016) OPERATIONAL MATRICES TO SOLVE NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS OF MULTI-ARBITRARY ORDER. Gazi University Journal of Science 29 4 895–907.
IEEE K. Parand and M. Delkhosh, “OPERATIONAL MATRICES TO SOLVE NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS OF MULTI-ARBITRARY ORDER”, Gazi University Journal of Science, vol. 29, no. 4, pp. 895–907, 2016.
ISNAD Parand, Kourosh - Delkhosh, Mehdi. “OPERATIONAL MATRICES TO SOLVE NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS OF MULTI-ARBITRARY ORDER”. Gazi University Journal of Science 29/4 (December 2016), 895-907.
JAMA Parand K, Delkhosh M. OPERATIONAL MATRICES TO SOLVE NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS OF MULTI-ARBITRARY ORDER. Gazi University Journal of Science. 2016;29:895–907.
MLA Parand, Kourosh and Mehdi Delkhosh. “OPERATIONAL MATRICES TO SOLVE NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS OF MULTI-ARBITRARY ORDER”. Gazi University Journal of Science, vol. 29, no. 4, 2016, pp. 895-07.
Vancouver Parand K, Delkhosh M. OPERATIONAL MATRICES TO SOLVE NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS OF MULTI-ARBITRARY ORDER. Gazi University Journal of Science. 2016;29(4):895-907.