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

Noui Djaidja; Mostefa Nadir

doi:10.3934/naco.2020039

Article Contents

2021, Volume 11, Issue 4: 487-493. Doi: 10.3934/naco.2020039

This issue Previous Article Next Article Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable

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

Noui Djaidja and
Mostefa Nadir

Department of Mathematics, University of Msila. Algeria

Received: May 2020

Revised: September 2020

Early access: September 2020

Published: December 2021

Abstract / Introduction Full Text(HTML) Figure(0) / Table(5) Related Papers Cited by

Abstract

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 }}. $

Keywords:

First-kind integral equations of Volterra,
Lavrentiev method,
numerical quadrature,
discrete appoximations.

Mathematics Subject Classification: Primary: ; Secondary: 65N20, 65M25, 65D30.

Citation:

FullText(HTML)

Table .

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

| Show Table

DownLoad: CSV

Table .

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

| Show Table

DownLoad: CSV

Table .

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

| Show Table

DownLoad: CSV

Table .

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

| Show Table

DownLoad: CSV

Table .

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

| Show Table

DownLoad: CSV

Related Papers

Cited by

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.