Abstract
Second-order Volterra integro-differential equation is solved by the linear barycentric rational collocation method. Following the barycentric interpolation method of Lagrange polynomial and Chebyshev polynomial, the matrix form of the collocation method is obtained from the discrete Volterra integro-differential equation. With the help of the convergence rate of the linear barycentric rational interpolation, the convergence rate of linear barycentric rational collocation method for solving Volterra integro-differential equation is proved. At last, several numerical examples are provided to validate the theoretical analysis.
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References
Abdi A, Hossseint SA (2019) The barycentric rational difference-quadrature scheme for systems of volterra integro-differential equations. SIAM J Sci Comput 40(3):A1936–A1960
Abdi A, Berrut J-P, Hosseini SA (2001) The linear barycentric rational method for a class of delay Volterra integro-differential equations. pp 1195–1210
Bayramov NR, Kraus JK (2015) On the stable solution of transient convection–diffusion equations. J Comput Appl Math 280:275–293
Berrut P, Klein G (2014) Recent advances in linear barycentric rational interpolation. J Comput Appl Math 259(Part A):95–107
Berrut P, Floater MS, Klein G (2011) Convergence rates of derivatives of a family of barycentric rational interpolants. Appl Numer Math 61(9):989–1000
Berrut JP, Hosseini SA, Klein G (2014) The linear barycentric rational quadrature method for Volterra integral equations. SIAM J Sci Comput 36(1):105–123
Cirillo E, Hormann K (2019) On the Lebesgue constant of barycentric rational Hermite interpolants at equidistant nodes. J Comput Appl Math 349:292–301
Delves LM, Mohamed JL (1985) Computational methods for integral equations. Cambridge University Press, Cambridge
Floater MS, Kai H (2007) Barycentric rational interpolation with no poles and high rates of approximation. Numer Math 107(2):315–331
Garey LE, Shaw RE (1991) Algorithms for the solution of second order Volterra integro-differential equations. Comput Math Appl 22(3):27–34
Hosseini SM, Shahmorad S (2003) Numerical solution of a class of integro-differential equations by the tau method with an error estimation. Appl Math Comput 136:559–570
Klein G, Berrut JP (2012a) Linear rational finite differences from derivatives of barycentric rational interpolants. SIAM J Numer Anal 50(2):643–656
Klein G, Berrut J-P (2012b) Linear barycentric rational quadrature. BIT Numer Math 52:407–424
Li S, Wang Z (2012) High precision meshless barycentric interpolation collocation method-algorithmic program and engineering application. Science Publishing, New York
Maleknejad K, Aghazadeh N (2005) Numerical solutions of Volterra integral equations of the second kind with convolution kernel by using Taylor-series expansion method. Appl Math Comput 161(3):915–922
Ortiz EL, Samara L (1981) An operational approach to the tau method for the numerical solution of nonlinear differential equations. Computing 27:15–25
Pour-Mahmoud J, Rahimi-Ardabili MY, Shahmorad S (2005) Numerical solution of the system of Fredholm integro-differential equations by the tau method. Appl Math Comput 168:465–478
Razzaghi M, Yousefi S (2005) Legendre wavelets method for the nonlinear Volterra Fredholm integral equations. Math Comput Simul 70:1–8
Shen J, Tang T, Wang L (2011) Spectral methods algorithms, analysis and applications. Springer, Berlin
Wang Z, Li S (2015) Barycentric interpolation collocation method for nonlinear problems. National Defense Industry Press, Beijing
Wang Z, Xu Z, Li J (2018) Mixed barycentric interpolation collocation method of displacement-pressure for incompressible plane elastic problems. Chin J Appl Mech 35(3):195–201
Wang Z, Zhang L, Xu Z, Li J (2018) Barycentric interpolation collocation method based on mixed displacement-stress formulation for solving plane elastic problems. Chin J Appl Mech 35(2):304–309
Yalcinbas S, Sezer M, Sorkun H (2009) Legendre polynomial solutions of high-order linear Fredholm integro-differential equations. Appl Numer Math 210:334–349
Acknowledgements
The work of Jin Li was supported by Natural Science Foundation of Shandong Province (Grant No. ZR2016JL006) Natural Science Foundation of Hebei Province (Grant No. A2019209533), National Natural Science Foundation of China (Grant Nos. 11471195, 11771398) and China Postdoctoral Science Foundation (Grant No. 2015T80703).
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Communicated by Hui Liang.
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Li, J., Cheng, Y. Linear barycentric rational collocation method for solving second-order Volterra integro-differential equation. Comp. Appl. Math. 39, 92 (2020). https://doi.org/10.1007/s40314-020-1114-z
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DOI: https://doi.org/10.1007/s40314-020-1114-z
Keywords
- Linear barycentric rational interpolation
- Collocation method
- Volterra integro-differential equation
- Convergence rate
- Barycentric interpolation method