A new approach for Volterra functional integral equations with non-vanishing delays and fractional Bagley-Torvik equation
Keywords:Integral equations, Numerical solution, Fractional Bagley-Torvik equation
A numerical technique for Volterra functional integral equations (VFIEs) with non-vanishing delays and fractional Bagley-Torvik equation is displayed in this work. The technique depends on Bernstein polynomial approximation. Numerical examples are utilized to evaluate the accurate results. The findings for examples figs and tables show that the technique is accurate and simple to use.
W. Ming and C. Huang, “Collocation methods for Volterra functional integral equations with non-vanishing delays”, Applied mathematics and computation, vol. 296, pp. 198–214, 2017. https://doi.org/10.1016/j.amc.2016.10.021
Q. Huang, H. Xie, and H. Brunner, “The $hp$ discontinuous Galerkin method for delay differential equations with nonlinear Vanishing delay”, SIAM journal on scientific computing, vol. 35, no. 3, pp. A1604–A1620, 2013. https://doi.org/10.1137/120901416
H. Xie, R. Zhang, and H. Brunner, “Collocation methods for general Volterra functional integral equations with vanishing delays”, SIAM journal on scientific computing, vol. 33, no. 6, pp. 3303-3332, 2011. https://doi.org/10.1137/100818595
Q. Huang, H. Xie, and H. Brunner, “Super convergence of a discontinuous Galerkin solutions for delay differential equations of pantograph type”, SIAM journal on scientific computing, vol. 33, no. 5, pp. 2664-2684, 2011. https://doi.org/10.1137/110824632
K. Harada and E. Nakamae, “Application of the Bézier curve to data interpolation”, Computer-aided design, vol. 14, no. 1, pp. 55-59, 1982. https://doi.org/10.1016/0010-4485(82)90013-6
G. Nürnberger and F. Zeilfelder, “Developments in bivariate spline interpolation”, Journal of Computational and Applied Mathematics, vol. 121, no. 1-2, pp. 125-152, 2000. https://doi.org/10.1016/S0377-0427(00)00346-0
J. Zheng, T. W. Sederberg, and R. W. Johnson, “Least squares methods for solving differential equations using Bézier control points”, Applied numerical mathematics, vol. 48, no. 2, pp. 237-252, 2004. https://doi.org/10.1016/j.apnum.2002.01.001
F. Ghomanjani and M. H. Farahi, “The Bézier control points method for solving delay differential equation”, Intelligent control and automation, vol. 3, no. 2, pp. 188-196, 2012. http://dx.doi.org/10.4236/ica.2012.32021
F. Ghomanjani, M. H. Farahi, and M. Gachpazan, “Bézier control points method to solve constrained quadratic optimal control of time varying linear systems”, Computational and applied mathematics, vol. 31, no. 3, pp. 433-456, 2012. https://doi.org/10.1590/S1807-03022012000300001
F. Ghomanjani and M. H. Farahi, “Optimal control of switched systems based on Bézier control points”, International journal of intelligent systems and applications, vol. 4, no. 7, pp. 16-22, 2012. https://doi.org/10.5815/ijisa.2012.07.02
F. Ghomanjani, M. H. Farahi, and A. V. Kamyad, “Numerical solution of some linear optimal control systems with pantograph delays”, IMA journal of mathematical control and information, vol. 32, no. 2, pp. 225-243. 2015. https://doi.org/10.1093/imamci/dnt037
L. J. Xie, C. L. Zhou, and S. Xu, “An effective numerical method to solve a class of nonlinear singular boundary value problems using improved differential transform method”, SpringerPlus, vol. 5, Art. ID. 1066, 2016. https://doi.org/10.1186/s40064-016-2753-9
H. L. Dastjerdi and M. N. Ahmadabadi, “Moving least squares collocation method for Volterra integral equations with proportional delay”, International journal of computer mathematics, vol. 94, no. 12, pp. 2335–2347, 2017. https://doi.org/10.1080/00207160.2017.1283024
E. Barkai, R. Metzler, and J. Klafter, “From continuous time random walks to the fractional Fokker-Planck equation”, Physical review E, vol. 61, no. 1, pp. 132–138, 2000. https://doi.org/10.1103/physreve.61.132
D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “Application of a fractional advection-dispersion equation”. Water resources research, vol. 36, no. 6, pp. 1403-1412, 2000. [On line]. Available: https://bit.ly/3xRoJve
R. Hilfer, Ed., Applications of fractional calculus in physics. Singapore: World Scientific, 2000. https://doi.org/10.1142/3779
F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and fractional calculus in continuum mechanics, A. Carpinteri and F. Mainardi, Eds. Vienna: Springer, 1997, pp. 291–348. https://doi.org/10.1007/978-3-7091-2664-6_7
K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations. New York, NY: John Wiley & Sons, 1993.
A. I. Saichev and G. M. Zaslavsky, “Fractional kinetic equations: solutions and applications”, Chaos: an interdisciplinary journal of nonlinear science, vol. 7, no. 4, pp. 753–764, 1997, https://doi.org/10.1063/1.166272
H. Ding and C. Li, “Numerical algorithms for the fractional diffusion-wave equation with reaction term”, Abstract and applied analysis, vol. 2013, pp. Art ID. 493406, 2013. https://doi.org/10.1155/2013/493406
P. J. Torvik and R. L. Bagley, “On the appearance of the fractional derivative in the behavior of real materials”, Journal of applied mechanics, vol. 51, no. 2, pp. 294–298, 1984. https://doi.org/10.1115/1.3167615
I. Podlubny, Fractional differential equations. San Diego, CA: Elsevier, 1999.
T. M. Atanackovic and D. Zorica, “On the Bagley–Torvik equation”, Journal of applied mechanics, vol. 80, no. 4, 2013. https://doi.org/10.1115/1.4007850
S. Esmaeili and M. Shamsi, “A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations”, Communications in nonlinear science and numerical simulation, vol. 16, no. 9, pp. 3646–3654, 2011. https://doi.org/10.1016/j.cnsns.2010.12.008
Ş. Yüzbaşı, “Numerical solution of the Bagley-Torvik equation by the Bessel collocation method”, Mathematical methods in the applied sciences, vol. 36, no. 3, pp. 300–312, 2012. https://doi.org/10.1002/mma.2588
Y. Çenesiz, Y. Keskin, and A. Kurnaz, “The solution of the Bagley–Torvik equation with the generalized Taylor collocation method”, Journal of the Franklin Institute, vol. 347, no. 2, pp. 452–466, 2010. https://doi.org/10.1016/j.jfranklin.2009.10.007
T. Mekkaoui and Z. Hammouch, “Approximate analytical solutions to the Bagley-Torvik equation by the fractional iteration method”, Analele Universităţii din Craiova. Seria matematică, informatică (Online), vol. 39, no. 2, pp. 251-256, 2012. [On line]. Available: https://bit.ly/3roy5fE
K. Diethelm, J. Ford, “Numerical solution of the Bagley-Torvik equation”, BIT numerical mathematics, vol. 42, pp. 490-507, 2002. https://doi.org/10.1023/A:1021973025166
S. Mashayekhi and M. Razzaghi, “Numerical solution of the fractional Bagley-Torvik equation by using hybrid functions approximation,” Mathematical methods in the applied sciences, vol. 39, no. 3, pp. 353–365, 2015. https://doi.org/10.1002/mma.3486
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