Document Type : Original Article

Authors

1 Department of Applied Mathematics, Ayandegan Institute of Higher Education, Tonekabon, Iran.

2 Department of Applied Mathematics and Computer Science‎, ‎Faculty of Mathematical Sciences‎, ‎University of Guilan, Iran.

Abstract

In this work, we consider a collocation method for solving the pantograph-type Volterra Hammerstein integral equations based on the first kind Chebyshev polynomials. We use the Lagrange interpolating polynomial to approximate the solution. The convergence of the presented method has been analyzed by over estimating for error. Finally, some illustrative examples are given to test the accuracy of the method. The presented method is compared with the Legendre Tau method.

Keywords

  • Bellen, A., & Zennaro, M. (2013). Numerical methods for delay differential equations. Oxford University Press.‏
  • Iserles, A., & Liu, Y. (1994). On pantograph integro-differential equations. The journal of integral equations and applications, 6(2), 213-237.‏ https://www.jstor.org/stable/26163088
  • Iserles, A., & Liu, Y. (1997). Integro-differential equations and generalized hypergeometric functions. Journal of mathematical analysis and applications208(2), 404-424.‏ https://doi.org/10.1006/jmaa.1997.5322
  • Iserles, A., & Liu, Y. (1997). On neutral functional–differential equations with proportional delays. Journal of mathematical analysis and applications207(1), 73-95.‏ https://doi.org/10.1006/jmaa.1997.5262
  • Ishiwata, E., & Muroya, Y. (2009). On collocation methods for delay differential and Volterra integral equations with proportional delay. Frontiers of mathematics in China4(1), 89-111.‏ https://doi.org/10.1007/s11464-009-0004-x
  • Muroya, Y., Ishiwata, E., & Brunner, H. (2003). On the attainable order of collocation methods for pantograph integro-differential equations. Journal of computational and applied mathematics152(1-2), 347-366.‏ https://doi.org/10.1016/S0377-0427(02)00716-1
  • Tohidi, E., Bhrawy, A. H., & Erfani, K. (2013). A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation. Applied mathematical modelling37(6), 4283-4294.‏ https://doi.org/10.1016/j.apm.2012.09.032
  • Brunner, H. (2004). Collocation methods for Volterra integral and related functional differential equations(Vol. 15). Cambridge University Press.‏
  • Cai, H., & Qi, J. (2016). A legendre-galerkin method for solving general volterra functional integral equations. Numerical algorithms73(4), 1159-1180.‏ https://doi.org/10.1007/s11075-016-0134-7
  • Dönmez Demir, D., Kürkçü, Ö. K., & Sezer, M. (2021). Pell–lucas series approach for a class of fredholm-type delay integro-differential equations with variable delays. Mathematical sciences15(1), 55-64.‏
  • Ghomanjani, F., Farahi, M. H., & Pariz, N. (2017). A new approach for numerical solution of a linear system with distributed delays, volterra delay-integro-differential equations, and nonlinear Volterra-fredholm integral equation by bezier curves. Computational and applied mathematics36(3), 1349-1365.‏ https://doi.org/10.1007/s40314-015-0296-2
  • Ali, I., Brunner, H., & Tang, T. (2009). A spectral method for pantograph-type delay differential equations and its convergence analysis. Journal of computational mathematics, 27(2/3), 254-265.‏ https://www.jstor.org/stable/43693505
  • Ali, I., Brunner, H., & Tang, T. (2009). Spectral methods for pantograph-type differential and integral equations with multiple delays. Frontiers of mathematics in China4(1), 49-61.‏ https://doi.org/10.1007/s11464-009-0010-z
  • Laeli Dastjerdi, H., & Nili Ahmadabadi, M. (2017). Moving least squares collocation method for Volterra integral equations with proportional delay. International journal of computer mathematics94(12), 2335-2347.‏ https://doi.org/10.1080/00207160.2017.1283024
  • Mansouri, L., & Azimzadeh, Z. (2022). Numerical solution of fractional delay Volterra integro-differential equations by Bernstein polynomials. Mathematical sciences, 1-12.‏ https://doi.org/10.1007/s40096-022-00463-3
  • Mokhtary, P., Moghaddam, B. P., Lopes, A. M., & Machado, J. A. (2020). A computational approach for the non-smooth solution of non-linear weakly singular Volterra integral equation with proportional delay. Numerical algorithms83(3), 987-1006.‏ https://doi.org/10.1007/s11075-019-00712-y
  • Nili Ahmadabadi, M., & Laeli Dastjerdi, H. (2020). Numerical treatment of nonlinear Volterra integral equations of Urysohn type with proportional delay. International journal of computer mathematics97(3), 656-666.‏ https://doi.org/10.1080/00207160.2019.1585538
  • Okeke, G. A., & Efut Ofem, A. (2022). A novel iterative scheme for solving delay differential equations and nonlinear integral equations in banach spaces. Mathematical methods in the applied sciences45(9), 5111-5134.‏ https://doi.org/10.1002/mma.8095
  • Rahimkhani, P., Ordokhani, Y., & Babolian, E. (2017). A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations. Numerical algorithms74(1), 223-245.‏ https://doi.org/10.1007/s11075-016-0146-3
  • Du, J., Lu, C., & Jiang, Y. (2022). Rothe's method for solving multi‐term caputo–katugampola fractional delay integral diffusion equations. Mathematical methods in the applied sciences, 45(12), 7426-7442. https://doi.org/10.1002/mma.8250
  • Sajjadi, S. A., & Pishbin, S. (2021). Convergence analysis of the product integration method for solving the fourth kind integral equations with weakly singular kernels. Numerical algorithms86(1), 25-54.‏ https://doi.org/10.1007/s11075-020-00877-x
  • Sajjadi, S. A., Najafi, H. S., & Aminikhah, H. (2022). An error estimation of a nyström type method for integral-algebraic equations of index-1. Mathematical sciences, 1-13.‏ https://doi.org/10.1007/s40096-022-00467-z
  • Sajjadi, S. A., Najafi, H. S., & Aminikhah, H. (2022). A numerical algorithm for solving index-1 weakly singular integral-algebraic equations with non-smooth solutions. Applicable analysis, 1-18.‏ https://doi.org/10.1080/00036811.2022.2091551
  • Taghizadeh, E., & Matinfar, M. (2019). Modified numerical approaches for a class of Volterra integral equations with proportional delays. Computational and applied mathematics38(2), 1-19.‏ https://doi.org/10.1007/s40314-019-0819-3
  • Wei, Y., & Chen, Y. (2012). Legendre spectral collocation methods for pantograph Volterra delay-integro-differential equations. Journal of scientific computing53(3), 672-688.‏ https://doi.org/10.1007/s10915-012-9595-6
  • Yüzbaşı, Ş. (2014). Laguerre approach for solving pantograph-type Volterra integro-differential equations. Applied mathematics and computation232, 1183-1199.‏ https://doi.org/10.1016/j.amc.2014.01.075
  • Ansari, H., & Mokhtary, P. (2019). Computational Legendre tau method for Volterra Hammerstein pantograph integral equations. Bulletin of the Iranian mathematical society45(2), 475-493.‏ https://doi.org/10.1007/s41980-018-0144-4
  • Mason, J. C., & Handscomb, D. C. (2002). Chebyshev polynomials. Chapman and Hall/CRC.‏
  • Canuto, C., Hussaini, M. Y., Quarteroni, A., & Zang, T. A. (2007). Spectral methods: fundamentals in single domains. Springer science & business media.‏
  • Chen, Y., & Tang, T. (2010). Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Mathematics of computation79(269), 147-167.‏ https://www.ams.org/journals/mcom/2010-79-269/S0025-5718-09-02269-8/