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Article Contents

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

• 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 $\dfrac{\delta }{\sqrt{\alpha }}.$

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

 Citation:

• 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

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

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

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

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
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