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# Effective approximation method for solving linear Fredholm-Volterra integral equations

• An efficient approximate method for solving Fredholm-Volterra integral equations of the third kind is presented. As a basis functions truncated Legendre series is used for unknown function and Gauss-Legendre quadrature formula with collocation method are applied to reduce problem into linear algebraic equations. The existence and uniqueness solution of the integral equation of the 3rd kind are shown as well as rate of convergence is obtained. Illustrative examples revels that the proposed method is very efficient and accurate. Finally, comparison results with the previous work are also given.

Mathematics Subject Classification: Primary: 45B05; Secondary: 45A05, 45L05.

 Citation: • • Table 1.  The error term $Qe_{n} \left(s\right)=\left|x\left(s\right)-Qx_{n} \left(s\right)\right|$ for Example 1

 s Qen(s)(11) n=5 n=11 n=20 1 3.000e-19 1.000e-19 4.000e-19 0.8 2.000e-19 1.000e-19 4.000e-19 0.6 1.000e-19 0.000e-0 3.000e-19 0.4 0.000e-0 0.000e-0 1.000e-19 0.2 5.000e-20 7.000e-20 7.000e-20 0.1 3.000e-20 1.000e-19 7.000e-20 0 6.000e-20 1.100e-19 6.000e-20

Table 2.  The error term $Qe_{n} \left(s\right)=\left|x\left(s\right)-Qx_{n} \left(s\right)\right|$ for Example 2

 s Qen(s)(11) n=5 n=11 n=20 1.0 3.571e-3 2.676e-3 3.999e-4 0.9 9.534e-3 1.518e-3 2.031e-4 0.7 1.445e-3 1.501e-3 1.800e-4 0.5 1.014e-2 8.541e-4 1.368e-4 0.3 3.406e-3 5.103e-4 5.103e-4 0.1 2.674e-2 3.278e-3 3.673e-4 0.0 4.155e-1 2.331e-1 1.244e-1

Table 3.  Error comparison between $Qe_{n} \left(s\right)$ and $Ce_{n} \left(s\right)$ for Example 2

 s n=2 Qen(s)(11) Cen(s) 1.00 9.943e-2 9.500e-2 0.75 1.874e-3 6.380e-2 0.50 3.183e-2 3.690e-2 0.25 9.565e-2 8.200e-2 0.00 7.925e-1 6.927e-1

Table 4.  The comparison of error terms $Qe_{n} \left(s\right)$ and $Me_{n} \left(s\right)$ for Example 3

 s n=5 n=7 n=9 Qen(s)(11) Men(s) Qen(s)(11) Men(s) Qen(s)(11) Men(s) 0.999 6.935e-1 1.829e-1 2.908e-1 1.109e-1 4.768e-2 1.562e-2 0.753 2.829e-1 1.026e0 6.962e-2 2.700e-1 1.581e-2 1.445e-2 0.352 4.374e-1 1.009e0 1.355e-1 1.502e-1 1.387e-2 3.929e-2 0.001 6.416e-1 3.025e-1 1.412e-1 1.401e-1 1.750e-2 1.152e-2 -0.001 6.411e-1 3.058e-1 1.412e-1 1.386e-1 1.750e-2 1.123e-2 -0.352 5.173e-1 4.409e-1 1.370e-1 4.024e-2 1.388e-2 2.596e-2 -0.753 4.027e-1 1.165e-1 7.146e-2 1.532e-1 1.580e-2 6.695e-2 -0.999 7.916e-1 2.605e-1 2.913e-1 1.226e-1 4.768e-2 2.196e-2

Table 5.  The error comparisons between $Qe_{n} (s)$ and $Me_{n} (s)$ for lager ''n''

 s Men(s) Qen(s)(11) n=13 n=15 n=19 n=13 n=15 n=19 n=20 0.999 6.092e-3 1.863e-2 9.825e0 3.002e-4 1.406e-5 1.385e-8 6.319e-10 0.753 6.688e-3 4.460e-2 1.083e0 9.126e-5 2.509e-7 8.004e-10 6.528e-11 0.352 6.233e-4 1.067e-2 3.697e-1 4.099e-5 3.483e-6 1.565e-9 5.362e-11 0.001 3.081e-3 4.409e-2 1.212e0 8.363e-5 3.644e-6 3.330e-9 4.196e-15 -0.001 3.111e-3 4.361e-2 1.208e0 8.363e-5 3.644e-6 3.330e-9 4.212e-15 -0.352 3.111e-3 2.288e-2 5.682e-1 4.099e-5 3.483e-6 1.565e-9 5.362e-11 -0.753 9.683e-3 5.025e-2 9.905e-1 9.126e-5 2.509e-7 8.004e-10 6.528e-11 -0.999 2.291e-3 1.083e-1 7.677e0 3.002e-4 1.406e-5 1.385e-8 6.319e-10

Table 6.  Comparison of error terms $Qe_{n} \left(s\right)$ and $Mue_{n} \left(s\right)$ for Example 4

 s n=5 n=10 Qen(s)(11) Muen(s) Qen(s)(11) Muen(s) 0.0 6.117e-5 0 5.662e-11 0 0.4 9.766e-6 4.593e-6 1.379e-11 2.213e-12 0.8 4.252e-6 8.389e-6 1.037e-11 5.683e-12 1.2 3.841e-6 1.378e-5 1.059e-11 1.009e-11 1.6 7.448e-6 2.153e-5 1.470e-11 1.612e-11 2.0 3.880e-5 3.179e-5 6.304e-11 2.376e-11

Table 7.  Comparison of error terms $Qe_{n} \left(s\right)$ and $Mue_{n} \left(s\right)$ for Example 4

 s n=9 Qen(s)(11) Muen(s) 0.0 9.194e-10 0 0.2222 2.775e-10 5.694e-11 0.4444 2.343e-10 9.869e-11 0.6667 1.951e-10 1.478e-10 0.8889 7.917e-10 2.037e-10 1.1111 7.663e-10 2.698e-10 1.3333 1.775e-10 3.493e-10 1.5556 2.001e-10 4.458e-10 1.7778 2.224e-10 5.663e-10 2.0000 6.907e-10 7.011e-10

Table 8.  Comparison of error terms $Qe_{n} \left(s\right)$ and $Mue_{n} \left(s\right)$ for Example 5

 s n=5 Qen(s)(11) Muen(s) 0.0 2.304e-6 0 0.2 4.121e-7 1.263e-6 0.4 2.096e-7 2.555e-6 0.6 1.903e-7 3.879e-5 0.8 4.451e-7 5.506e-5 1.0 2.710e-6 7.751e-5

Table 9.  Comparison of error terms $Qe_{n} \left(s\right)$ and $Mue_{n} \left(s\right)$ for Example 5

 s n=9 Qen(s)(11) Muen(s) 0.0 9.194e-10 0 0.1111 2.775e-10 9.133e-13 0.2222 2.343e-10 1.842e-12 0.3333 1.951e-10 2.753e-12 0.4444 7.917e-10 3.678e-12 0.5556 7.663e-10 4.638e-12 0.6667 1.775e-10 5.685e-12 0.7778 2.001e-10 6.871e-12 0.8889 2.224e-10 8.292e-12 1.0 6.907e-10 1.005e-11
• Tables(9)

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