Research Article

Year 2015,
Volume: 12 Issue: 2, - , 01.11.2015
### Abstract

### Keywords

### References

In this paper, we present a new approach to obtain the numerical solution of the linear twodimensional

Fredholm and Volterra integro-differential equations (2D-FIDE and 2D-VIDE). First, we introduce

the two-dimensional Chebyshev polynomials and construct their operational matrices of integration.

Then, both of them, two-dimensional Chebyshev polynomials and their operational matrix of integration, are

used to represent the matrix form of 2D-FIDE and 2D-VIDE. The main characteristic of this approach is

that it reduces 2D-FIDE and 2D-VIDE to a system of linear algebraic equations. Illustrative examples are

included to demonstrate the validity and applicability of the presented technique

Two-dimensional Fredholm and Volterra integro-differential equations Two-dimensional Chebyshev polynomials Operational matrix of integration

- [1] A. Akyuz Daciolu, Chebyshev polynomials solution of system of linear integral equations, Appl. Math. Comput. 151, (2004), 221–232.
- [2] A. Avudainayagam, C. Vani, Wavelet-Galerkin method for integro-differential equations, Appl. Numer. Math. 32, (2000), 247–254.
- [3] T. A. Burton, Volterra Integral and Differential Equations, Elsevier, Netherlands, (2005).
- [4] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, Science Publisher Inc., New York, (1978).
- [5] P. Darania, A. Ebadian, A method for the numerical solution of the integro-differential equations, Appl. Math. Comput. 188, (2007), 657–668.
- [6] L. M. Delves, J. L. Mohamed, Computational methods for integral equations, Cambridge University Press, Cambridge, (1985).
- [7] S. M. El-Sayed, M. R. Abdel-Aziz, A comparison of Adomian’s decomposition method and Wavelet-Galerkin method for solving integro-differential equations, Appl. Math. Comput. 136, (2003), 151–159.
- [8] R. Ezzati, S. Najafalizadeh, Application of Chebyshev polynomials for solving nonlinear Volterra-Fredholm integral equations system and convergence analysis, Indian Journal of Science and Technology, (2012), 2060–2064.
- [9] L. Fax, I. B. Parker, Chebyshev polynomials in Numerical Analysis, Oxford University Press, London, (1998).
- [10] K. Maleknejad, S. Sohrabi, and Y. Rostami, Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials, Appl. Math. Comput. 188, (2007), 123–128.
- [11] M. T. Rashed, Lagrange interpolation to compute the numerical solutions of differential and integro-differential equations, Appl. Math. Comput. 51, (2004), 869–878.
- [12] W. J. Xie, F. R. Lin, A fast numerical solution method for two-dimensional Fredholm integral equations of the second kind, Applied Numerical Mathematics, 7, (2009), 1709–1719.
- [13] S. Yalcinbas, Taylor polynomial solution of nonlinear Volterra-Fredholm integral equations, Appl. Math. Comput. 127, (2002), 195–206.

Year 2015,
Volume: 12 Issue: 2, - , 01.11.2015
### Abstract

### References

- [1] A. Akyuz Daciolu, Chebyshev polynomials solution of system of linear integral equations, Appl. Math. Comput. 151, (2004), 221–232.
- [2] A. Avudainayagam, C. Vani, Wavelet-Galerkin method for integro-differential equations, Appl. Numer. Math. 32, (2000), 247–254.
- [3] T. A. Burton, Volterra Integral and Differential Equations, Elsevier, Netherlands, (2005).
- [4] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, Science Publisher Inc., New York, (1978).
- [5] P. Darania, A. Ebadian, A method for the numerical solution of the integro-differential equations, Appl. Math. Comput. 188, (2007), 657–668.
- [6] L. M. Delves, J. L. Mohamed, Computational methods for integral equations, Cambridge University Press, Cambridge, (1985).
- [7] S. M. El-Sayed, M. R. Abdel-Aziz, A comparison of Adomian’s decomposition method and Wavelet-Galerkin method for solving integro-differential equations, Appl. Math. Comput. 136, (2003), 151–159.
- [8] R. Ezzati, S. Najafalizadeh, Application of Chebyshev polynomials for solving nonlinear Volterra-Fredholm integral equations system and convergence analysis, Indian Journal of Science and Technology, (2012), 2060–2064.
- [9] L. Fax, I. B. Parker, Chebyshev polynomials in Numerical Analysis, Oxford University Press, London, (1998).
- [10] K. Maleknejad, S. Sohrabi, and Y. Rostami, Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials, Appl. Math. Comput. 188, (2007), 123–128.
- [11] M. T. Rashed, Lagrange interpolation to compute the numerical solutions of differential and integro-differential equations, Appl. Math. Comput. 51, (2004), 869–878.
- [12] W. J. Xie, F. R. Lin, A fast numerical solution method for two-dimensional Fredholm integral equations of the second kind, Applied Numerical Mathematics, 7, (2009), 1709–1719.
- [13] S. Yalcinbas, Taylor polynomial solution of nonlinear Volterra-Fredholm integral equations, Appl. Math. Comput. 127, (2002), 195–206.

There are 13 citations in total.

Subjects | Engineering |
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Journal Section | Articles |

Authors | |

Publication Date | November 1, 2015 |

Published in Issue | Year 2015 Volume: 12 Issue: 2 |