Abstract
In this work we study the numerical solution to the Volterra integro-differential algebraic equation. Two numerical examples based on the Legendre collocation scheme are designed. It follows from the convergence proof and numerical experiments that the errors of the approximate solution and the errors of the approximate derivative of the solution decay exponentially.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11671157
Award Identifier / Grant number: 91430213
Funding source: Shandong Province
Award Identifier / Grant number: ZR2017MA005
Funding statement: This work is supported by National Natural Science Foundation of China (11671157, 91430213) and Shandong Province Natural Science Foundation of China (ZR2017MA005).
References
[1] I. Ali, H. Brunner and T. Tang, A spectral method for pantograph-type delay differential equations and its convergence analysis, J. Comput. Math. 27 (2009), no. 2–3, 254–265. Search in Google Scholar
[2] H. Brunner, Volterra Integral Equations. An Introduction to Theory and Applications, Cambridge Monogr. Appl. Comput. Math. 30, Cambridge University Press, Cambridge, 2017. 10.1017/9781316162491Search in Google Scholar
[3] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods. Fundamentals in Single Domains, Sci. Comput., Springer, Berlin, 2006. 10.1007/978-3-540-30726-6Search in Google Scholar
[4] Y. Chen and T. Tang, Spectral methods for weakly singular Volterra integral equations with smooth solutions, J. Comput. Appl. Math. 233 (2009), no. 4, 938–950. 10.1016/j.cam.2009.08.057Search in Google Scholar
[5] Y. Chen and T. Tang, Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel, Math. Comp. 79 (2010), no. 269, 147–167. 10.1090/S0025-5718-09-02269-8Search in Google Scholar
[6] A. I. Fedotov, Lebesgue constant estimation in multidimensional Sobolev space, Lobachevskii J. Math. 14 (2004), 25–32. Search in Google Scholar
[7] M. Hadizadeh, F. Ghoreishi and S. Pishbin, Jacobi spectral solution for integral algebraic equations of index-2, Appl. Numer. Math. 61 (2011), no. 1, 131–148. 10.1016/j.apnum.2010.08.009Search in Google Scholar
[8] C. Huang and Z. Zhang, Spectral collocation methods for differential-algebraic equations with arbitrary index, J. Sci. Comput. 58 (2014), no. 3, 499–516. 10.1007/s10915-013-9755-3Search in Google Scholar
[9] Y.-L. Jiang, Waveform relaxation methods of nonlinear integral-differential-algebraic equations, J. Comput. Math. 23 (2005), no. 1, 49–66. Search in Google Scholar
[10] J.-P. Kauthen, Implicit Runge–Kutta methods for some integrodifferential-algebraic equations, Appl. Numer. Math. 13 (1993), no. 1–3, 125–134. 10.1016/0168-9274(93)90136-FSearch in Google Scholar
[11] J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Math. Monogr. Ser. 3, Science Press Beijing, Beijing, 2006. Search in Google Scholar
[12] Y. Wei and Y. Chen, Legendre spectral collocation methods for pantograph Volterra delay-integro-differential equations, J. Sci. Comput. 53 (2012), no. 3, 672–688. 10.1007/s10915-012-9595-6Search in Google Scholar
[13] Y. Wei and Y. Chen, Legendre spectral collocation method for neutral and high-order Volterra integro-differential equation, Appl. Numer. Math. 81 (2014), 15–29. 10.1016/j.apnum.2014.02.012Search in Google Scholar
[14] J. Zhao and S. Wang, Jacobi spectral solution for weakly singular integral algebraic equations of index-1, Adv. Difference Equ. 2014 (2014), Paper No. 165. 10.1186/1687-1847-2014-165Search in Google Scholar
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