# American Institute of Mathematical Sciences

December  2021, 11(4): 487-493. doi: 10.3934/naco.2020039

## Comparison between Taylor and perturbed method for Volterra integral equation of the first kind

 Department of Mathematics, University of Msila. Algeria

Received  May 2020 Revised  September 2020 Published  December 2021 Early access  September 2020

As it is known the equation $A\varphi = f$ with injective compact operator has a unique solution for all $f$ in the range $R(A).$Unfortunately, the right-hand side $f$ is never known exactly, so we can take an approximate data $f_{\delta }$ and used the perturbed problem $\alpha \varphi +A\varphi = f_{\delta }$ where the solution $\varphi _{\alpha \delta }$ depends continuously on the data $f_{\delta },$ and the bounded inverse operator $\left( \alpha I+A \right) ^{-1}$ approximates the unbounded operator $A^{-1}$ but not stable. In this work we obtain the convergence of the approximate solution of $\varphi _{\alpha \delta }$ of the perturbed equation to the exact solution $\varphi$ of initial equation provided $\alpha$ tends to zero with $\dfrac{\delta }{\sqrt{\alpha }}.$

Citation: Noui Djaidja, Mostefa Nadir. Comparison between Taylor and perturbed method for Volterra integral equation of the first kind. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 487-493. doi: 10.3934/naco.2020039
##### References:
 [1] H. Brunner, Discretization of Volterra integral equations of the first kind, Mathematics of Computation, 31 (1977), 708-716.  doi: 10.2307/2006002. [2] J. Kumar, P. Manchanda and Pooja, Numerical solution of Fredholm integral equations of the first kind using Legendre wavelets collocation method, International Journal of Pure and Applied Mathematics, 117 (2017), 33-43. [3] Hui Liang, The fine error estimation of collocation methods on uniform meshes for weakly singular volterra integral equations, Scientific Computing, 84 (2020), Article number: 12. doi: 10.1007/s10915-020-01266-1. [4] P. K. Lamm, A survey of regularization methods for first-kind volterra equations, Editors Springer (Vienna, New York), (2000), 53-82. [5] Pin Lyu and S. Vang, A high-order method with a temporal nonuniform mesh for a timefractional Benjamin-Bona-Mahony equation, J. Sci. Comput., 80 (2019), 1607-1628.  doi: 10.1007/s10915-019-00991-6. [6] K. Maleknejad, M. T. Kajani and Y. Mahmoudi, Numerical solution of linear Fredholm and Volterra integral equations of the second kind using Legendre wavelets, Journal of Sciences, Islamic Republic of Iran, 13 (2002), 161-166. [7] K. Maleknejad, M. Roodaki and H. Almasieh, Numerical solution of Volterra integral equations of first kind by using a recursive scheme, Journal of Mathematical Extension, 3 (2009), 113-121. [8] M. Nadir and A. Rahmoune, Modifed method for solving linear Volterra integral equations of the second kind using Simpson's rule, International Journal Mathematical Manuscripts, 1 (2007), 133-140. [9] M. Nadir and N. Djaidja, Approximation method for Volterra integral equation of the first kind, International Journal of Mathematics and Computation, 29 (2018), 67-72.  doi: 10.1093/comjnl/12.4.393. [10] N. A. Sidorov, M. V. Falaleev and D. N. Sidorov, Generalized solutions of Volterra integral equations of the first kind, Bull. Malays. Math. Sci. Soc., 29 (2006), 101-109. [11] Tao Tang, Superconvergence of numerical solutions to weakly singular Volterra integro-di erential equations, Numer. Math., 61 (1992), 373-382.  doi: 10.1007/BF01385515.

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##### References:
 [1] H. Brunner, Discretization of Volterra integral equations of the first kind, Mathematics of Computation, 31 (1977), 708-716.  doi: 10.2307/2006002. [2] J. Kumar, P. Manchanda and Pooja, Numerical solution of Fredholm integral equations of the first kind using Legendre wavelets collocation method, International Journal of Pure and Applied Mathematics, 117 (2017), 33-43. [3] Hui Liang, The fine error estimation of collocation methods on uniform meshes for weakly singular volterra integral equations, Scientific Computing, 84 (2020), Article number: 12. doi: 10.1007/s10915-020-01266-1. [4] P. K. Lamm, A survey of regularization methods for first-kind volterra equations, Editors Springer (Vienna, New York), (2000), 53-82. [5] Pin Lyu and S. Vang, A high-order method with a temporal nonuniform mesh for a timefractional Benjamin-Bona-Mahony equation, J. Sci. Comput., 80 (2019), 1607-1628.  doi: 10.1007/s10915-019-00991-6. [6] K. Maleknejad, M. T. Kajani and Y. Mahmoudi, Numerical solution of linear Fredholm and Volterra integral equations of the second kind using Legendre wavelets, Journal of Sciences, Islamic Republic of Iran, 13 (2002), 161-166. [7] K. Maleknejad, M. Roodaki and H. Almasieh, Numerical solution of Volterra integral equations of first kind by using a recursive scheme, Journal of Mathematical Extension, 3 (2009), 113-121. [8] M. Nadir and A. Rahmoune, Modifed method for solving linear Volterra integral equations of the second kind using Simpson's rule, International Journal Mathematical Manuscripts, 1 (2007), 133-140. [9] M. Nadir and N. Djaidja, Approximation method for Volterra integral equation of the first kind, International Journal of Mathematics and Computation, 29 (2018), 67-72.  doi: 10.1093/comjnl/12.4.393. [10] N. A. Sidorov, M. V. Falaleev and D. N. Sidorov, Generalized solutions of Volterra integral equations of the first kind, Bull. Malays. Math. Sci. Soc., 29 (2006), 101-109. [11] Tao Tang, Superconvergence of numerical solutions to weakly singular Volterra integro-di erential equations, Numer. Math., 61 (1992), 373-382.  doi: 10.1007/BF01385515.
 Val of $x$ Ex sol $\varphi$ Ap sol $\varphi _{T}$ Error$_{T}$ Ap sol $\varphi _{\alpha \delta }$ Error$_{\delta }$ 0.000 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.200 3.97e-01 3.97e-01 3.31e-04 3.97e-01 5.55e-17 0.400 7.78e-01 7.79e-01 3.17e-04 7.78e-01 5.55e-16 0.600 1.12e+00 1.13e+00 9.36e-04 1.12e+00 2.44e-15 0.800 1.43e+00 1.43e+00 1.52e-03 1.43e+00 3.99e-15 1.000 1.68e+00 1.68e+00 2.08e-03 1.68e+00 4.21e-15
 Val of $x$ Ex sol $\varphi$ Ap sol $\varphi _{T}$ Error$_{T}$ Ap sol $\varphi _{\alpha \delta }$ Error$_{\delta }$ 0.000 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.200 3.97e-01 3.97e-01 3.31e-04 3.97e-01 5.55e-17 0.400 7.78e-01 7.79e-01 3.17e-04 7.78e-01 5.55e-16 0.600 1.12e+00 1.13e+00 9.36e-04 1.12e+00 2.44e-15 0.800 1.43e+00 1.43e+00 1.52e-03 1.43e+00 3.99e-15 1.000 1.68e+00 1.68e+00 2.08e-03 1.68e+00 4.21e-15
 Val of $x$ Ex sol $\varphi$ Ap sol $\varphi _{T}$ Error$_{T}$ Ap sol $\varphi _{\alpha \delta }$ Error$_{\delta }$ 0.000 1.00e+00 1.00e+00 0.00e+00 1.00e+00 0.00e+00 0.200 8.18e-01 8.10e-01 7.84e-03 8.18e-01 3.89e-011 0.400 6.70e-01 6.63e-01 6.38e-03 6.70e-01 5.14e-011 0.600 5.48e-01 5.43e-01 5.39e-03 5.48e-01 5.17e-011 0.800 4.49e-01 4.44e-01 4.73e-03 4.49e-01 4.67e-011 1.000 3.67e-01 3.63e-01 4.29e-03 3.67e-01 4.01e-011
 Val of $x$ Ex sol $\varphi$ Ap sol $\varphi _{T}$ Error$_{T}$ Ap sol $\varphi _{\alpha \delta }$ Error$_{\delta }$ 0.000 1.00e+00 1.00e+00 0.00e+00 1.00e+00 0.00e+00 0.200 8.18e-01 8.10e-01 7.84e-03 8.18e-01 3.89e-011 0.400 6.70e-01 6.63e-01 6.38e-03 6.70e-01 5.14e-011 0.600 5.48e-01 5.43e-01 5.39e-03 5.48e-01 5.17e-011 0.800 4.49e-01 4.44e-01 4.73e-03 4.49e-01 4.67e-011 1.000 3.67e-01 3.63e-01 4.29e-03 3.67e-01 4.01e-011
 Val of $x$ Ex sol $\varphi$ Ap sol $\varphi _{T}$ Error$_{T}$ Ap sol $\varphi _{\alpha \delta }$ Error$_{\delta }$ 0.000 1.00e+00 1.00e+00 0.00e+00 1.00e+00 0.00e+00 0.200 9.80e-01 1.00e+01 2.15e-02 9.79e-01 7.12e-05 0.400 9.21e-01 9.51e-01 3.06e-02 9.20e-01 1.51e-04 0.600 8.25e-01 8.60e-01 3.55e-02 8.25e-01 2.26e-04 0.800 6.96e-01 7.31e-01 3.52e-02 6.96e-01 2.89e-04 1.000 5.40e-01 5.71e-01 3.09e-02 5.39e-01 3.42e-04
 Val of $x$ Ex sol $\varphi$ Ap sol $\varphi _{T}$ Error$_{T}$ Ap sol $\varphi _{\alpha \delta }$ Error$_{\delta }$ 0.000 1.00e+00 1.00e+00 0.00e+00 1.00e+00 0.00e+00 0.200 9.80e-01 1.00e+01 2.15e-02 9.79e-01 7.12e-05 0.400 9.21e-01 9.51e-01 3.06e-02 9.20e-01 1.51e-04 0.600 8.25e-01 8.60e-01 3.55e-02 8.25e-01 2.26e-04 0.800 6.96e-01 7.31e-01 3.52e-02 6.96e-01 2.89e-04 1.000 5.40e-01 5.71e-01 3.09e-02 5.39e-01 3.42e-04
 Val of $x$ Ex sol $\varphi$ Ap sol $\varphi _{T}$ Error$_{T}$ Ap sol $\varphi _{\alpha \delta }$ Error$_{\delta }$ 0.000 1.00e+00 1.00e+00 0.00e+00 1.00e+00 0.00e+00 0.200 8.35e-01 8.28e-01 7.06e-03 8.35e-01 1.36e-05 0.400 7.24e-01 7.19e-01 5.18e-03 7.24e-01 4.52e-05 0.600 6.50e-01 6.46e-01 4.01e-03 6.50e-01 8.47e-05 0.800 6.00e-01 5.97e-01 3.29e-03 6.00e-01 1.26e-04 1.000 5.67e-01 5.64e-01 2.83e-03 5.67e-01 1.66e-04
 Val of $x$ Ex sol $\varphi$ Ap sol $\varphi _{T}$ Error$_{T}$ Ap sol $\varphi _{\alpha \delta }$ Error$_{\delta }$ 0.000 1.00e+00 1.00e+00 0.00e+00 1.00e+00 0.00e+00 0.200 8.35e-01 8.28e-01 7.06e-03 8.35e-01 1.36e-05 0.400 7.24e-01 7.19e-01 5.18e-03 7.24e-01 4.52e-05 0.600 6.50e-01 6.46e-01 4.01e-03 6.50e-01 8.47e-05 0.800 6.00e-01 5.97e-01 3.29e-03 6.00e-01 1.26e-04 1.000 5.67e-01 5.64e-01 2.83e-03 5.67e-01 1.66e-04
 Val of $x$ Ex sol $\varphi$ Ap sol $\varphi _{T}$ Error$_{T}$ Ap sol $\varphi _{\alpha \delta }$ Error$_{\delta }$ 0.000 0.00e+00 NaN NaN 0.00e+00 0.00e+00 0.200 -1.81e-01 NaN NaN -1.84e-01 3.51e-03 0.400 -3.29e-01 NaN NaN -3.32e-01 2.83e-03 0.600 -4.51e-01 NaN NaN -4.53e-01 2.62e-03 0.800 -5.50e-01 NaN NaN -5.53e-01 2.65e-03 1.000 -6.32e-01 NaN NaN -6.35e-01 2.91e-03
 Val of $x$ Ex sol $\varphi$ Ap sol $\varphi _{T}$ Error$_{T}$ Ap sol $\varphi _{\alpha \delta }$ Error$_{\delta }$ 0.000 0.00e+00 NaN NaN 0.00e+00 0.00e+00 0.200 -1.81e-01 NaN NaN -1.84e-01 3.51e-03 0.400 -3.29e-01 NaN NaN -3.32e-01 2.83e-03 0.600 -4.51e-01 NaN NaN -4.53e-01 2.62e-03 0.800 -5.50e-01 NaN NaN -5.53e-01 2.65e-03 1.000 -6.32e-01 NaN NaN -6.35e-01 2.91e-03
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