# American Institute of Mathematical Sciences

• Previous Article
The soft landing problem for an infinite system of second order differential equations
• NACO Home
• This Issue
• Next Article
Homogenization of optimal control problems on curvilinear networks with a periodic microstructure --Results on $\boldsymbol{S}$-homogenization and $\boldsymbol{Γ}$-convergence
March  2017, 7(1): 77-88. doi: 10.3934/naco.2017004

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

 1 Faculty of Science and Technology, University Sains Islam Malaysia (USIM), Negeri Sembilan, Malaysia 2 Institute for Mathematical Research (INSPEM), University Putra Malaysia (UPM), Selangor, Malaysia 3 Department of Mathematics, Faculty of Science, University Putra Malaysia (UPM) 4 Institute for Mathematical Research (INSPEM), UPM, Selangor, Malaysia

* Corresponding author: Z. K. Eshkuvatov, zainidin@usim.edu.my

Received  March 2016 Revised  February 2017 Published  February 2017

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.

Citation: Z. K. Eshkuvatov, M. Kammuji, Bachok M. Taib, N. M. A. Nik Long. Effective approximation method for solving linear Fredholm-Volterra integral equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 77-88. doi: 10.3934/naco.2017004
##### References:

show all references

##### References:
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
 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
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
 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
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)[6] 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
 s n=2 Qen(s)(11) Cen(s)[6] 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
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)[13] Qen(s)(11) Men(s)[13] Qen(s)(11) Men(s)[13] 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
 s n=5 n=7 n=9 Qen(s)(11) Men(s)[13] Qen(s)(11) Men(s)[13] Qen(s)(11) Men(s)[13] 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
The error comparisons between $Qe_{n} (s)$ and $Me_{n} (s)$ for lager ''n''
 s Men(s)[13] 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
 s Men(s)[13] 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
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)[18] Qen(s)(11) Muen(s)[18] 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
 s n=5 n=10 Qen(s)(11) Muen(s)[18] Qen(s)(11) Muen(s)[18] 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
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)[18] 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
 s n=9 Qen(s)(11) Muen(s)[18] 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
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)[18] 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
 s n=5 Qen(s)(11) Muen(s)[18] 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
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)[18] 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
 s n=9 Qen(s)(11) Muen(s)[18] 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
 [1] Zhong-Qing Wang, Li-Lian Wang. A Legendre-Gauss collocation method for nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 685-708. doi: 10.3934/dcdsb.2010.13.685 [2] Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299 [3] Zainidin Eshkuvatov. Homotopy perturbation method and Chebyshev polynomials for solving a class of singular and hypersingular integral equations. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 337-350. doi: 10.3934/naco.2018022 [4] Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667 [5] Yin Yang, Yunqing Huang. Spectral Jacobi-Galerkin methods and iterated methods for Fredholm integral equations of the second kind with weakly singular kernel. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 685-702. doi: 10.3934/dcdss.2019043 [6] Yinnian He, R. M.M. Mattheij. Reformed post-processing Galerkin method for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 369-387. doi: 10.3934/dcdsb.2007.8.369 [7] Kaitai Li, Yanren Hou. Fourier nonlinear Galerkin method for Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 497-524. doi: 10.3934/dcds.1996.2.497 [8] Zhongying Chen, Bin Wu, Yuesheng Xu. Fast numerical collocation solutions of integral equations. Communications on Pure & Applied Analysis, 2007, 6 (3) : 643-666. doi: 10.3934/cpaa.2007.6.643 [9] Jie Tang, Ziqing Xie, Zhimin Zhang. The long time behavior of a spectral collocation method for delay differential equations of pantograph type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 797-819. doi: 10.3934/dcdsb.2013.18.797 [10] Ben-Yu Guo, Zhong-Qing Wang. A spectral collocation method for solving initial value problems of first order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1029-1054. doi: 10.3934/dcdsb.2010.14.1029 [11] Stanisław Migórski, Shengda Zeng. The Rothe method for multi-term time fractional integral diffusion equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 719-735. doi: 10.3934/dcdsb.2018204 [12] Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817 [13] Netra Khanal, Ramjee Sharma, Jiahong Wu, Juan-Ming Yuan. A dual-Petrov-Galerkin method for extended fifth-order Korteweg-de Vries type equations. Conference Publications, 2009, 2009 (Special) : 442-450. doi: 10.3934/proc.2009.2009.442 [14] Juan-Ming Yuan, Jiahong Wu. A dual-Petrov-Galerkin method for two integrable fifth-order KdV type equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1525-1536. doi: 10.3934/dcds.2010.26.1525 [15] Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method. Evolution Equations & Control Theory, 2014, 3 (1) : 147-166. doi: 10.3934/eect.2014.3.147 [16] Yanzhao Cao, Li Yin. Spectral Galerkin method for stochastic wave equations driven by space-time white noise. Communications on Pure & Applied Analysis, 2007, 6 (3) : 607-617. doi: 10.3934/cpaa.2007.6.607 [17] Mickaël D. Chekroun, Michael Ghil, Honghu Liu, Shouhong Wang. Low-dimensional Galerkin approximations of nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4133-4177. doi: 10.3934/dcds.2016.36.4133 [18] Imtiaz Ahmad, Siraj-ul-Islam, Mehnaz, Sakhi Zaman. Local meshless differential quadrature collocation method for time-fractional PDEs. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020223 [19] Zhonghui Li, Xiangyong Chen, Jianlong Qiu, Tongshui Xia. A novel Chebyshev-collocation spectral method for solving the transport equation. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2020080 [20] Jie Shen, Li-Lian Wang. Laguerre and composite Legendre-Laguerre Dual-Petrov-Galerkin methods for third-order equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1381-1402. doi: 10.3934/dcdsb.2006.6.1381

Impact Factor: