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Nonlinear second order systems of Fredholm integro-differential equations

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Abstract

A large class of physical phenomena in biophysics, chemical engineering, and physical sciences are modeled as systems of Fredhold integro-differential equations. In its simplest form, such systems are linear and analytic solutions might be obtained in some cases while numerical methods can be also used to solve such systems when analytic solutions are not possible. For more realistic and accurate study of underlying physical behavior, including nonlinear actions is useful. In this paper, we use the Chebyshev pseudo-spectral method to solve the pattern nonlinear second order systems of Fredholm integro-differential equations. The method reduces the operators to a nonlinear system of equations that can be solved alliteratively. The method is tested against the reproducing kernel Hilbert space (RKHS) method and shows good performance. The present method is easy to implement and yields very good accuracy for using a relatively small number of collocation points.

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References

  1. Al-Smadi, M., Abu Arqub, O., Shawagfeh, N., Momani, S.: Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method. Appl. Math. Model. 291, 137–148 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Abu Arqub, O., Abo-Hammour, Z.: Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm. Appl. Math. Inf. Sci. 279, 396–415 (2014)

    MathSciNet  MATH  Google Scholar 

  3. Akyüz-Daşcioǧlu, A.: A Chebyshev polynomial approach for linear Fredholm-Volterra integro-differential equations in the most general form. Appl. Math. Comput. 181, 103–112 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Akyüz-Daşcioǧlu, A., çerdik-Yaslan, H.: The solution of high-order nonlinear ordinary differential equations by Chebyshev Series. Appl. Math. Comput. 12, 5658–5666 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Azodi, H.: Euler polynomials approach to the system of nonlinear fractional differential equations. J. Math. 51, 71–87 (2019)

    MathSciNet  Google Scholar 

  6. Baccouch, M., Kaddeche, S.: Efficient Chebyshev pseudospectral methods for Viscous Burgers equations in one and two space dimensions. Int. J. Appl. Comput. Math. 2019, 9 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Bermejo, R., Sastre, P.: An implicit-explicit Runge-Kutta-Chebyshev finite element method for the nonlinear Lithium-ion battery equations. Appl. Math. Comput. 361, 398–420 (2019)

    MathSciNet  MATH  Google Scholar 

  8. Bai, A.Z., Gu, A.Y., Fan, C.: A direct Chebyshev collocation method for the numerical solutions of three-dimensional Helmholtz-type equations. Eng. Anal. Bound. Elem. 104, 26–33 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bakodah, H., Al-Mazmumy, M., Almuhalbedi, S.: Solving system of integro differential equations using discrete adomain decomposition method. J. Taibah Univer. Sci. 13, 805–812 (2019)

    Article  Google Scholar 

  10. Babolian, E., Mordad, M.: A numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions. Comput. Math. Appl. 62, 187–198 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Didgar, M., Vahidi, A., Biazar, J.: An approximate approach for system of fractional intgro-differential equations based on Taylor expansion. Kragujevac J. Math. 44, 379–392 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dehghan, M., Saadatmandi, A.: The numerical solution of a nonlinear system of second-order boundary value problems using the sinc-collocation method. Math. Comput. Model. 46, 1434–1441 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dehghan, M., Nikpour, A.: Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method. Appl. Math. Model. 37, 8578–8599 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. El-Gamel, M.: Sinc-collocation method for solving linear and nonlinear system of second-order boundary value problems. Appl. Math. 3, 1627–1633 (2012)

    Article  Google Scholar 

  15. El-Sayeda, A., Ahmed, R.: Solvability of a coupled system of functional integro-differential equations with infinite point and Riemann-Stieltjes integral conditions. Appl. Math. Comput. 370, 5 (2020)

    MathSciNet  Google Scholar 

  16. El-Gamel, M., Sameeh, M.: An efficient technique for finding the eigenvalues of fourth-order Sturm-Liouville problems. Appl. Math. 3, 920–925 (2012)

    Article  Google Scholar 

  17. El-Gamel, M., Sameeh, M.: A Chebychev collocation method for solving Troesch’s problem. Int. J. Math. Comput. Appl. Res. 3, 23–32 (2013)

    Google Scholar 

  18. El-Gamel, M.: Numerical comparison of sinc-collocation and Chebychev-collocation methods for determining the eigenvalues of Sturm-Liouville problems with parameter-dependent boundary conditions. SeMA J. 66, 29–42 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. El-Gamel, M., Mohamed, O., El-Shamy, N.: A robust and effective method for solving two-point BVP in modelling Viscoelastic flows. AM J. 11, 23–34 (2020)

    Article  Google Scholar 

  20. El-Gamel, M., Sameeh, M.: Numerical solution of singular two-point boundary value problems by the collocation method with the Chebyshev bases. SeMA J. 74, 627–641 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. El-Gamel, M.: Chebychev polynomial solutions of twelfth-order boundary-value problems. Br. J. Math. Comput. Sci. 6, 13–23 (2015)

    Article  Google Scholar 

  22. Ghimire, B., Li, X., Chen, C., Lamichhane, A.: Hybrid Chebyshev polynomial scheme for solving elliptic partial differential equations. Iran. J. Sci. Technol. 364, 9 (2020)

    MathSciNet  MATH  Google Scholar 

  23. Gu, Z.: Chebyshev spectral collocation method for system of nonlinear Volterra integral equations. Numer. Algor. 2019, 8 (2019)

    Google Scholar 

  24. Geng, F., Cui, M.: Homotopy perturbation-reproducing kernel method for nonlinear systems of second order boundary value problems. Appl. Math. Comput. 235, 2405–2411 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Geng, F., Cui, M.: Solving a nonlinear system of second order boundary value problems. J. Math. Anal. Appl. 279, 396–415 (2014)

    Google Scholar 

  26. Hassani, H., Machado, J., Naraghirad, E.: Generalized shifted Chebyshev polynomials for fractional optimal control problems. Commun. Nonlin. Sci. Num. Sim. 75, 50–61 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. Heydari, M.: Chebyshev cardinal functions for a new class of nonlinear optimal control problems generated by Atangana-Baleanu-Caputo variable-order fractional derivative. Chaos. Soliton Fract. 130, 5 (2020)

    Article  MathSciNet  Google Scholar 

  28. Hesameddini, E., Riahi, M.: Bernoulli Galerkin matrix method and its convergence analysis for solving system of Volterra-Fredholm integro-differential equations. Iran J. Sci. Technol. Trans. Sci. 43, 1203–1214 (2019)

    Article  MathSciNet  Google Scholar 

  29. Khan, S., Ali, I.: Convergence and error analysis of a spectral collocation method for solving system of nonlinear Fredholm integral equations of second kind. Comp. Appl. Math. 38, 125–139 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. Karunakar, P., Chakraverty, S.: Shifted Chebyshev polynomials based solution of partial differential equations. SN Appl. Sci. 2019, 8 (2019)

    Google Scholar 

  31. Lu, J.: Variational iteration method for solving a nonlinear system of second-order boundary value problems. Comput. Math. Appl. 54, 1133–1138 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Öztürk, Y.: An efficient numerical algorithm for solving system of Lane-Emden type equations arising in engineering. Nonlinear Eng. 8, 429–437 (2019)

    Article  Google Scholar 

  33. Öztürk, Y.: Solution for the system of Lane-Emden type equations using Chebyshev polynomials. Math. 2020, 5 (2020)

    Google Scholar 

  34. Rahimkhani, P., Ordokhani, Y.: Approximate solution of nonlinear fractional integro-differential equations using fractional alternative Legendre functions. J. Comp. Appl. Math. 365, 5 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sahihi, H., Allahviranloo, T., Abbasbandy, S.: Solving system of second-order BVPs using a new algorithm based on reproducing kernel Hilbert space. Appl. Numer. Math. 151, 27–39 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  36. Saw, V., Kumar, S.: The approximate solution for multi-term the fractional order initial value problem using Collocation method based on shifted Chebyshev polynomials of the first kind. Inf. Technol. Appl. Math. 699, 53–67 (2019)

    Article  MATH  Google Scholar 

  37. Secer, A., Bakir, Y.: Chebyshev Wavelet collocation Method for Ginzburg-Landau Equation. Therm. Sci. 23, 57–65 (2019)

    Article  Google Scholar 

  38. Sezer, M., Kaynak, M.: Chebyshev polynomial solutions of linear differential equations. Int. Math. Educ. Sci. Technol. 27, 607–618 (1996)

    Article  MATH  Google Scholar 

  39. Saw, V., Kumar, S.: Second kind Chebyshev polynomials for solving space fractional advection-dispersion equation using Collocation method. Iran. J. Sci. Technol. 43, 1027–1037 (2019)

    Article  MathSciNet  Google Scholar 

  40. Wang, L., Chen, Y., Liu, D.: Numerical algorithm to solve generalized fractional pantograph equations with variable coefficients based on shifted Chebyshev polynomials. Int. J. Comput. Math. 2019, 8 (2019)

    MathSciNet  MATH  Google Scholar 

  41. Xie, J., Yi, M.: Numerical research of nonlinear system of fractional Volterra-Fredholm integral-differential equations via Block-Pulse functions and error analysis. J. Comp. Appl. Math. 345, 159–167 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yiǧit, G., Bayram, M.: Chebyshev differential quadrature for numerical solutions of third and fourth-order singular perturbation problems, Proc. Natl. Acad. Sci. India, Sect. A Phys. Sci. (2019)

  43. Yousefi, A., Javadi, S., Babolian, E., Moradi, E.: Convergence analysis of the Chebyshev-Legendre spectral method for a class of Fredholm fractional integro-differential equations. J. Comput. Appl. Math. 358, 97–110 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zaky, M., Ameen, I.: A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and related Volterra-Fredholm integral equations with smooth solutions. Numer. Algor. 2019, 5 (2019)

    Google Scholar 

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Acknowledgements

The authors are extremely grateful to Dr. Amgad Abdrabou for some feedback about referees suggestions and to the referees for their helpful suggestions and comments.

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Correspondence to Mohamed El-Gamel.

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El-Gamel, M., Mohamed, O. Nonlinear second order systems of Fredholm integro-differential equations. SeMA 79, 383–396 (2022). https://doi.org/10.1007/s40324-021-00258-x

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