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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:
[1]

K. E. Atkinson, The Numerical Solution of Integral Equation of the Second Kind Cambridge University Press, United Kingdom, 1997. doi: 10.1017/CBO9780511626340.  Google Scholar

[2]

E. Babolian and F. Fattahzadeh, Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration, Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration, Applied Mathematics and Computation, 188 (2007), 1016-1022.  doi: 10.1016/j.amc.2006.10.073.  Google Scholar

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E. Babolian and A. Davari, Numerical implementation of Adomian decomposition method for linear Volterra integral equations of the second kind, Numerical implementation of Adomian decomposition method for linear Volterra integral equations of the second kind, Applied Mathematics and Computation, 165 (2005), 223-227.  doi: 10.1016/j.amc.2004.04.065.  Google Scholar

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E. Babolian and Z. Masouri, Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions, Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions, Journal of Computational and Applied Mathematics, 220 (2008), 51-57.  doi: 10.1016/j.cam.2007.07.029.  Google Scholar

[5]

H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equation Cambridge University Press, United Kingdom, 2004. Google Scholar

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A. Chakrabarti and S. C. Martha, Approximate solutions of Fredholm integral equations of the second kind, Approximate solutions of Fredholm integral equations of the second kind, Applied Mathematics and Computation, 211 (2009), 459-466.  doi: 10.1016/j.amc.2009.01.088.  Google Scholar

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D. Elliott, A Chebyshev series method for the numerical solution of Fredholm integral equation, A Chebyshev series method for the numerical solution of Fredholm integral equation, Comp. J, 6 (1963), 102-111.  doi: 10.1093/comjnl/6.1.102.  Google Scholar

[8]

Z. K. EshkuvatovA. AhmedovN. M. A. Nik Long and O. Shafiq, Approximate solution of the system of nonlinear integral equation by Newton--Kantorovich method, Approximate solution of the system of nonlinear integral equation by Newton--Kantorovich method, Applied Mathematics and Computation, 217 (2010), 3717-3725.  doi: 10.1016/j.amc.2010.09.068.  Google Scholar

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H. Hochstadt, Integral Equations A Wiley-Interscience Publication, New York, 1973.  Google Scholar

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R. P. Kanwal, Linear Integral Equations Academic Press, New York, 1997. doi: 10.1007/978-1-4612-0765-8.  Google Scholar

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P. K. Kythe and M. R. Schaferkotter, Handbook of Computational Methods for Integration Chapman & Hall/CRC Press, New York, 2005.  Google Scholar

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S. J. Majeed and H. H. Omran, Numerical methods for solving linear Fredholm-Volterra Integral Equations, Numerical methods for solving linear Fredholm-Volterra Integral Equations, Journal of Al-Nahrian University, 11 (2008), 131-134.   Google Scholar

[13]

K. MaleknejadK. Nouri and M. Yousefi, Discussion on convergence of Legendre polynomial for numerical solution of integral equations, Discussion on convergence of Legendre polynomial for numerical solution of integral equations, Applied Mathematics and Computation, 193 (2007), 335-339.  doi: 10.1016/j.amc.2007.03.062.  Google Scholar

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K. MaleknejadE. Hashemizadeh and R. Ezzati, A new approach to the numerical solution of Volterra integral equations by using Bernstein's approximation, A new approach to the numerical solution of Volterra integral equations by using Bernstein's approximation, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 647-655.  doi: 10.1016/j.cnsns.2010.05.006.  Google Scholar

[15]

K. Maleknejad and M. Yousefi, Numerical solution of the integral equation of the second kind by using wavelet bases of Hermite cubic splines, Numerical solution of the integral equation of the second kind by using wavelet bases of Hermite cubic splines, Applied Mathematics and Computation, 183 (2006), 134-141.  doi: 10.1016/j.amc.2006.05.104.  Google Scholar

[16]

K. Maleknejad and A. S. Shamloo, Numerical solution of singular Volterra integral equations system of convolution type by using operational matrices, Numerical solution of singular Volterra integral equations system of convolution type by using operational matrices, Applied Mathematics and Computation, 195 (2008), 500-505.  doi: 10.1016/j.amc.2007.05.001.  Google Scholar

[17]

B. N. Mandal and S. Bhattacharya, Numerical solution of some classes of integral equations using Bernstein polynomials, Numerical solution of some classes of integral equations using Bernstein polynomials, Appl. Math. Comput., 190 (2007), 1707-1716.  doi: 10.1016/j.amc.2007.02.058.  Google Scholar

[18]

M. M. Mustafa and I. N. Ghanim, Numerical Solution of Linear Volterra-Fredholm Integral Equations Using Lagrange Polynomials, Numerical Solution of Linear Volterra-Fredholm Integral Equations Using Lagrange Polynomials, Mathematical Theory and Modeling, 4 (2014), 137-146.   Google Scholar

[19]

N. M. A. Nik LongZ. K. EshkuvatovM. Yaghobifar and M. Hasan, Numerical Solution of Infinite Boundary Integral Equation by using Galerkin Method with Laguerre Polynomials, Numerical Solution of Infinite Boundary Integral Equation by using Galerkin Method with Laguerre Polynomials, International Journal of Computational and Mathematical Sciences, 3 (2009), 21-24.   Google Scholar

[20]

M. Rahman, Integral Equations and Their Applications WIT Press, Boston, 2007.  Google Scholar

show all references

References:
[1]

K. E. Atkinson, The Numerical Solution of Integral Equation of the Second Kind Cambridge University Press, United Kingdom, 1997. doi: 10.1017/CBO9780511626340.  Google Scholar

[2]

E. Babolian and F. Fattahzadeh, Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration, Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration, Applied Mathematics and Computation, 188 (2007), 1016-1022.  doi: 10.1016/j.amc.2006.10.073.  Google Scholar

[3]

E. Babolian and A. Davari, Numerical implementation of Adomian decomposition method for linear Volterra integral equations of the second kind, Numerical implementation of Adomian decomposition method for linear Volterra integral equations of the second kind, Applied Mathematics and Computation, 165 (2005), 223-227.  doi: 10.1016/j.amc.2004.04.065.  Google Scholar

[4]

E. Babolian and Z. Masouri, Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions, Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions, Journal of Computational and Applied Mathematics, 220 (2008), 51-57.  doi: 10.1016/j.cam.2007.07.029.  Google Scholar

[5]

H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equation Cambridge University Press, United Kingdom, 2004. Google Scholar

[6]

A. Chakrabarti and S. C. Martha, Approximate solutions of Fredholm integral equations of the second kind, Approximate solutions of Fredholm integral equations of the second kind, Applied Mathematics and Computation, 211 (2009), 459-466.  doi: 10.1016/j.amc.2009.01.088.  Google Scholar

[7]

D. Elliott, A Chebyshev series method for the numerical solution of Fredholm integral equation, A Chebyshev series method for the numerical solution of Fredholm integral equation, Comp. J, 6 (1963), 102-111.  doi: 10.1093/comjnl/6.1.102.  Google Scholar

[8]

Z. K. EshkuvatovA. AhmedovN. M. A. Nik Long and O. Shafiq, Approximate solution of the system of nonlinear integral equation by Newton--Kantorovich method, Approximate solution of the system of nonlinear integral equation by Newton--Kantorovich method, Applied Mathematics and Computation, 217 (2010), 3717-3725.  doi: 10.1016/j.amc.2010.09.068.  Google Scholar

[9]

H. Hochstadt, Integral Equations A Wiley-Interscience Publication, New York, 1973.  Google Scholar

[10]

R. P. Kanwal, Linear Integral Equations Academic Press, New York, 1997. doi: 10.1007/978-1-4612-0765-8.  Google Scholar

[11]

P. K. Kythe and M. R. Schaferkotter, Handbook of Computational Methods for Integration Chapman & Hall/CRC Press, New York, 2005.  Google Scholar

[12]

S. J. Majeed and H. H. Omran, Numerical methods for solving linear Fredholm-Volterra Integral Equations, Numerical methods for solving linear Fredholm-Volterra Integral Equations, Journal of Al-Nahrian University, 11 (2008), 131-134.   Google Scholar

[13]

K. MaleknejadK. Nouri and M. Yousefi, Discussion on convergence of Legendre polynomial for numerical solution of integral equations, Discussion on convergence of Legendre polynomial for numerical solution of integral equations, Applied Mathematics and Computation, 193 (2007), 335-339.  doi: 10.1016/j.amc.2007.03.062.  Google Scholar

[14]

K. MaleknejadE. Hashemizadeh and R. Ezzati, A new approach to the numerical solution of Volterra integral equations by using Bernstein's approximation, A new approach to the numerical solution of Volterra integral equations by using Bernstein's approximation, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 647-655.  doi: 10.1016/j.cnsns.2010.05.006.  Google Scholar

[15]

K. Maleknejad and M. Yousefi, Numerical solution of the integral equation of the second kind by using wavelet bases of Hermite cubic splines, Numerical solution of the integral equation of the second kind by using wavelet bases of Hermite cubic splines, Applied Mathematics and Computation, 183 (2006), 134-141.  doi: 10.1016/j.amc.2006.05.104.  Google Scholar

[16]

K. Maleknejad and A. S. Shamloo, Numerical solution of singular Volterra integral equations system of convolution type by using operational matrices, Numerical solution of singular Volterra integral equations system of convolution type by using operational matrices, Applied Mathematics and Computation, 195 (2008), 500-505.  doi: 10.1016/j.amc.2007.05.001.  Google Scholar

[17]

B. N. Mandal and S. Bhattacharya, Numerical solution of some classes of integral equations using Bernstein polynomials, Numerical solution of some classes of integral equations using Bernstein polynomials, Appl. Math. Comput., 190 (2007), 1707-1716.  doi: 10.1016/j.amc.2007.02.058.  Google Scholar

[18]

M. M. Mustafa and I. N. Ghanim, Numerical Solution of Linear Volterra-Fredholm Integral Equations Using Lagrange Polynomials, Numerical Solution of Linear Volterra-Fredholm Integral Equations Using Lagrange Polynomials, Mathematical Theory and Modeling, 4 (2014), 137-146.   Google Scholar

[19]

N. M. A. Nik LongZ. K. EshkuvatovM. Yaghobifar and M. Hasan, Numerical Solution of Infinite Boundary Integral Equation by using Galerkin Method with Laguerre Polynomials, Numerical Solution of Infinite Boundary Integral Equation by using Galerkin Method with Laguerre Polynomials, International Journal of Computational and Mathematical Sciences, 3 (2009), 21-24.   Google Scholar

[20]

M. Rahman, Integral Equations and Their Applications WIT Press, Boston, 2007.  Google Scholar

Table 1.  The error term $Qe_{n} \left(s\right)=\left|x\left(s\right)-Qx_{n} \left(s\right)\right|$ for Example 1
sQen(s)(11)
n=5 n=11 n=20
13.000e-191.000e-194.000e-19
0.82.000e-191.000e-194.000e-19
0.61.000e-190.000e-03.000e-19
0.40.000e-00.000e-01.000e-19
0.25.000e-207.000e-207.000e-20
0.13.000e-201.000e-197.000e-20
06.000e-201.100e-196.000e-20
sQen(s)(11)
n=5 n=11 n=20
13.000e-191.000e-194.000e-19
0.82.000e-191.000e-194.000e-19
0.61.000e-190.000e-03.000e-19
0.40.000e-00.000e-01.000e-19
0.25.000e-207.000e-207.000e-20
0.13.000e-201.000e-197.000e-20
06.000e-201.100e-196.000e-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
sQen(s)(11)
n=5 n=11 n=20
1.03.571e-32.676e-33.999e-4
0.99.534e-31.518e-32.031e-4
0.71.445e-31.501e-31.800e-4
0.51.014e-28.541e-41.368e-4
0.33.406e-35.103e-45.103e-4
0.12.674e-23.278e-33.673e-4
0.04.155e-12.331e-11.244e-1
sQen(s)(11)
n=5 n=11 n=20
1.03.571e-32.676e-33.999e-4
0.99.534e-31.518e-32.031e-4
0.71.445e-31.501e-31.800e-4
0.51.014e-28.541e-41.368e-4
0.33.406e-35.103e-45.103e-4
0.12.674e-23.278e-33.673e-4
0.04.155e-12.331e-11.244e-1
Table 3.  Error comparison between $ Qe_{n} \left(s\right)$ and $ Ce_{n} \left(s\right)$ for Example 2
sn=2
Qen(s)(11)Cen(s)[6]
1.009.943e-29.500e-2
0.751.874e-36.380e-2
0.503.183e-23.690e-2
0.259.565e-28.200e-2
0.007.925e-16.927e-1
sn=2
Qen(s)(11)Cen(s)[6]
1.009.943e-29.500e-2
0.751.874e-36.380e-2
0.503.183e-23.690e-2
0.259.565e-28.200e-2
0.007.925e-16.927e-1
Table 4.  The comparison of error terms $Qe_{n} \left(s\right)$ and $Me_{n} \left(s\right)$ for Example 3
sn=5n=7n=9
Qen(s)(11)Men(s)[13]Qen(s)(11)Men(s)[13]Qen(s)(11)Men(s)[13]
0.9996.935e-11.829e-12.908e-11.109e-14.768e-21.562e-2
0.7532.829e-11.026e06.962e-22.700e-11.581e-21.445e-2
0.3524.374e-11.009e01.355e-11.502e-11.387e-23.929e-2
0.0016.416e-13.025e-11.412e-11.401e-11.750e-21.152e-2
-0.0016.411e-13.058e-11.412e-11.386e-11.750e-21.123e-2
-0.3525.173e-14.409e-11.370e-14.024e-21.388e-22.596e-2
-0.7534.027e-11.165e-17.146e-21.532e-11.580e-26.695e-2
-0.9997.916e-12.605e-12.913e-11.226e-14.768e-22.196e-2
sn=5n=7n=9
Qen(s)(11)Men(s)[13]Qen(s)(11)Men(s)[13]Qen(s)(11)Men(s)[13]
0.9996.935e-11.829e-12.908e-11.109e-14.768e-21.562e-2
0.7532.829e-11.026e06.962e-22.700e-11.581e-21.445e-2
0.3524.374e-11.009e01.355e-11.502e-11.387e-23.929e-2
0.0016.416e-13.025e-11.412e-11.401e-11.750e-21.152e-2
-0.0016.411e-13.058e-11.412e-11.386e-11.750e-21.123e-2
-0.3525.173e-14.409e-11.370e-14.024e-21.388e-22.596e-2
-0.7534.027e-11.165e-17.146e-21.532e-11.580e-26.695e-2
-0.9997.916e-12.605e-12.913e-11.226e-14.768e-22.196e-2
Table 5.  The error comparisons between $Qe_{n} (s)$ and $ Me_{n} (s)$ for lager ''n''
sMen(s)[13]Qen(s)(11)
n=13n=15n=19n=13n=15n=19n=20
0.9996.092e-31.863e-29.825e03.002e-41.406e-51.385e-86.319e-10
0.7536.688e-34.460e-21.083e09.126e-52.509e-78.004e-106.528e-11
0.3526.233e-41.067e-23.697e-14.099e-53.483e-61.565e-95.362e-11
0.0013.081e-34.409e-21.212e08.363e-53.644e-63.330e-94.196e-15
-0.0013.111e-34.361e-21.208e08.363e-53.644e-63.330e-94.212e-15
-0.3523.111e-32.288e-25.682e-14.099e-53.483e-61.565e-95.362e-11
-0.7539.683e-35.025e-29.905e-19.126e-52.509e-78.004e-106.528e-11
-0.9992.291e-31.083e-17.677e03.002e-41.406e-51.385e-86.319e-10
sMen(s)[13]Qen(s)(11)
n=13n=15n=19n=13n=15n=19n=20
0.9996.092e-31.863e-29.825e03.002e-41.406e-51.385e-86.319e-10
0.7536.688e-34.460e-21.083e09.126e-52.509e-78.004e-106.528e-11
0.3526.233e-41.067e-23.697e-14.099e-53.483e-61.565e-95.362e-11
0.0013.081e-34.409e-21.212e08.363e-53.644e-63.330e-94.196e-15
-0.0013.111e-34.361e-21.208e08.363e-53.644e-63.330e-94.212e-15
-0.3523.111e-32.288e-25.682e-14.099e-53.483e-61.565e-95.362e-11
-0.7539.683e-35.025e-29.905e-19.126e-52.509e-78.004e-106.528e-11
-0.9992.291e-31.083e-17.677e03.002e-41.406e-51.385e-86.319e-10
Table 6.  Comparison of error terms $Qe_{n} \left(s\right)$ and $Mue_{n} \left(s\right)$ for Example 4
sn=5n=10
Qen(s)(11)Muen(s)[18]Qen(s)(11)Muen(s)[18]
0.06.117e-505.662e-110
0.49.766e-64.593e-61.379e-112.213e-12
0.84.252e-68.389e-61.037e-115.683e-12
1.23.841e-61.378e-51.059e-111.009e-11
1.67.448e-62.153e-51.470e-111.612e-11
2.03.880e-53.179e-56.304e-112.376e-11
sn=5n=10
Qen(s)(11)Muen(s)[18]Qen(s)(11)Muen(s)[18]
0.06.117e-505.662e-110
0.49.766e-64.593e-61.379e-112.213e-12
0.84.252e-68.389e-61.037e-115.683e-12
1.23.841e-61.378e-51.059e-111.009e-11
1.67.448e-62.153e-51.470e-111.612e-11
2.03.880e-53.179e-56.304e-112.376e-11
Table 7.  Comparison of error terms $Qe_{n} \left(s\right)$ and $Mue_{n} \left(s\right)$ for Example 4
sn=9
Qen(s)(11)Muen(s)[18]
0.09.194e-100
0.22222.775e-105.694e-11
0.44442.343e-109.869e-11
0.66671.951e-101.478e-10
0.88897.917e-102.037e-10
1.11117.663e-102.698e-10
1.33331.775e-103.493e-10
1.55562.001e-104.458e-10
1.77782.224e-105.663e-10
2.00006.907e-107.011e-10
sn=9
Qen(s)(11)Muen(s)[18]
0.09.194e-100
0.22222.775e-105.694e-11
0.44442.343e-109.869e-11
0.66671.951e-101.478e-10
0.88897.917e-102.037e-10
1.11117.663e-102.698e-10
1.33331.775e-103.493e-10
1.55562.001e-104.458e-10
1.77782.224e-105.663e-10
2.00006.907e-107.011e-10
Table 8.  Comparison of error terms $Qe_{n} \left(s\right)$ and $Mue_{n} \left(s\right)$ for Example 5
sn=5
Qen(s)(11)Muen(s)[18]
0.02.304e-60
0.24.121e-71.263e-6
0.42.096e-72.555e-6
0.61.903e-73.879e-5
0.84.451e-75.506e-5
1.02.710e-67.751e-5
sn=5
Qen(s)(11)Muen(s)[18]
0.02.304e-60
0.24.121e-71.263e-6
0.42.096e-72.555e-6
0.61.903e-73.879e-5
0.84.451e-75.506e-5
1.02.710e-67.751e-5
Table 9.  Comparison of error terms $Qe_{n} \left(s\right)$ and $Mue_{n} \left(s\right)$ for Example 5
sn=9
Qen(s)(11)Muen(s)[18]
0.09.194e-100
0.11112.775e-109.133e-13
0.22222.343e-101.842e-12
0.33331.951e-102.753e-12
0.44447.917e-103.678e-12
0.55567.663e-104.638e-12
0.66671.775e-105.685e-12
0.77782.001e-106.871e-12
0.88892.224e-108.292e-12
1.06.907e-101.005e-11
sn=9
Qen(s)(11)Muen(s)[18]
0.09.194e-100
0.11112.775e-109.133e-13
0.22222.343e-101.842e-12
0.33331.951e-102.753e-12
0.44447.917e-103.678e-12
0.55567.663e-104.638e-12
0.66671.775e-105.685e-12
0.77782.001e-106.871e-12
0.88892.224e-108.292e-12
1.06.907e-101.005e-11
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