An operational matrix method for solving a class of nonlinear Volterra integral equations arising in steady activation of a skeletal muscle

Iranian Journal of Numerical Analysis and Optimization

دوره 13، شماره 2 - شماره پیاپی 25، شهریور 2023، صفحه 336-353   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2022.73126.1069

نویسندگان

S. Torkaman^* ؛ M. Heydari؛ G.B. Loghmani

Department of Mathematical Sciences, Yazd University, Yazd, Iran.

چکیده

The problem of the steady activation of a skeletal muscle is one of the ap-plicable phenomena in real life that can be modeled by a Volterra integral equation. The current research aims to investigate this problem by using an effective operational matrix-based method. For this purpose, the opera-tional matrix of integration is derived for the barycentric rational cardinal basis functions. Then, by utilizing the obtained operational matrix and without using any collocation points, the governing integral equation is re-duced to a system of nonlinear algebraic equations. Convergence analysis of the proposed numerical method is studied thoroughly. Moreover, the obtained numerical results based on the proposed method with acceptable computational times support the theoretical results and reveal the accuracy and efficiency of the method.

کلیدواژه‌ها

Volterra integral equations؛ Linear barycentric rational inter-polants؛ Operational matrix

مراجع

[1] Abdi, A. and Hosseini, A. The barycentric rational difference-quadrature scheme for systems of Volterra integro-differential equations, SIAM J. Sci. Comput. 40 (2018), A1936–A1960.
[2] Berrut, J.P. Rational function for guaranteed and experimentally well-conditioned global interpolation, Comput. Math. Appl. 15 (1988), 1–16.
[3] Berrut, J. P., Hosseini, S. A. and Klein, G. The linear barycentric rational quadrature method for volterra integral equations, SIAM J. Sci. Comput. 36 (2014), A105–A123.
[4] Floater, M.S. and Hormann, K. Barycentric rational interpolation with no poles and high rates of approximation, Numer. Math. 107(2) (2007), 315–331.
[5] Gabriel, J.P., Studer, L.M., Rüegg, D.G. and Schnetzer, M.A. A math-ematical model for the steady activation of a skeletal muscle, SIAM J. Appl. Math. 68 (2008), 869–889.
[6] Klein, G. and Berrut, J.P. Linear barycentric rational quadrature, BIT, 52 (2012), 407–424.
[7] Liu, H., Huang, J., Pan, Y. and Zhang, J. Barycentric interpolation collocation methods for solving linear and nonlinear high-dimensional Fredholm integral equations, J. Comput. Appl. Math. 327 (2018), 141–154.
[8] Maleknejad, K., Torabi, P. and Mollapourasl, R. Fixed point method for solving nonlinear quadratic Volterra integral equations, Comput. Math. Appl. 62 (2011), 2555–2566.
[9] Phillips, G.M. Interpolation and approximation by polynomials, CMS Books in Mathematics, Springer, New York, NY, 2003.
[10] Studer, L.M., Ruegg, D.G. and Gabrie, J.P. A model for steady isometric muscle activation, Biol. Cybern. 80 (1999), 339–355.
[11] Torkaman, S., Heydari, M., Loghmani, G. B. and Ganji, D.D. Barycen-tric rational interpolation method for numerical investigation of magne-tohydrodynamics nanofluid flow and heat transfer in nonparallel plates with thermal radiation, Heat Transfer-Asian Research, 49 (2020), 565–590.
[12] Torkaman, S., Heydari, M., Loghmani, G.B. and Wazwaz, A.M. Numer-ical investigation of three-dimensional nanofluid flow with heat and mass transfer on a nonlinearly stretching sheet using the barycentric functions, Int. J. Numer. Methods Heat Fluid Flow. 31 (2020), 783–808.

[13] Torkaman, S., Loghmani, G.B., Heydari, M. and Rashidi, M.M. Novel numerical solutions of nonlinear heat transfer problems using the lin-ear barycentric rational interpolation, Heat Transfer-Asian Research, 48(2019), 1318–1344.
[14] Torkaman, S., Loghmani, G.B., Heydari, M. and Rashidi, M.M. An ef-fective operational matrix method based on barycentric cardinal functions to study nonlinear MHD nanofluid flow and heat transfer, Int. J. Mech. Eng. 5 (2020,) 51–63.