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Effective approximation method for solving linear Fredholm-Volterra integral equations

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  • 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:

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  • 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
     | Show Table
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    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
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    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
     | Show Table
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    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
  • [1] K. E. Atkinson, The Numerical Solution of Integral Equation of the Second Kind Cambridge University Press, United Kingdom, 1997. doi: 10.1017/CBO9780511626340.
    [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.
    [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.
    [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.
    [5] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equation Cambridge University Press, United Kingdom, 2004.
    [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.
    [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.
    [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.
    [9] H. Hochstadt, Integral Equations A Wiley-Interscience Publication, New York, 1973.
    [10] R. P. Kanwal, Linear Integral Equations Academic Press, New York, 1997. doi: 10.1007/978-1-4612-0765-8.
    [11] P. K. Kythe and M. R. Schaferkotter, Handbook of Computational Methods for Integration Chapman & Hall/CRC Press, New York, 2005.
    [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. 
    [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.
    [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.
    [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.
    [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.
    [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.
    [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. 
    [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. 
    [20] M. Rahman, Integral Equations and Their Applications WIT Press, Boston, 2007.
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