An efficient hybrid method for solving fredholm integral equations using triangular functions

Year 2017, Volume: 5 Issue: 1, 213 - 224, 01.01.2017

Mohamed A. Ramadan , Mohamed R. Ali

Abstract

In this paper the orthogonal triangular function (TF)
based method is first applied to transform the Fredholm integral equations and
Fredholm system of integral equations to a coupled system of matrix algebraic
equations. The obtained system is a variant of coupled Sylvester matrix
equations. A finite iterative algorithm is then applied to solve this system to
obtain the coefficients used to get the form of approximate solution of the
unknown functions of the integral problems. Some numerical examples are solved
to illustrate the accuracy and the efficiency of the proposed hybrid method.
The obtained numerical results are compared with other numerical methods and
the exact solutions.

Keywords

Fredholm integral equation, triangular functions, generalized Sylvester matrix equation, generalized iterative algorithm

References

• A.M.Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications, Springer, New York, NY, USA (2011).
• Suayip Y “A collocation method based on Bernstein polynomials to solve nonlinear Fredholm– Volterra integro-differential equations,” Applied Mathematics and Computation, Vol.273, No. 2, pp. 142–154 (2016).
• Suayip Y “Numerical solutions of system of linear Fredholm–Volterra integro-differential equations by the Bessel collocation method and error estimation,” Applied Mathematics and Computation, Vol.250, No. 1, pp. 320–338 (2015).
• Suayip Y “Laguerre approach for solving pantograph-type Volterra integro-differential equations,” Applied Mathematics and Computation, vol.232, pp. 1183–1199 (2014).
• Suayip Y “A Numerical Approximation For Volterra’S Population Growth Model With Fractional Order,” Applied Mathematical Modelling, Vol.37, No. 2, pp. 3216–3227 (2013).
• Suayip Y “Numerical Solutions Of Integro-Differential Equations And Application Of A Population Model With An Improved Legendre Method,” Applied Mathematical Modelling, Vol.37, pp. 3216–3227 (2013).
• Suayip Y Niyazi Ş, Mehmet S “Numerical Solutions Of Systems Of Linear Fredholm Integro-Differential Equations With Bessel Polynomial Bases", Computers and Mathematics with Applications, Vol.61, pp.3079-3096 (2011).
• K. Maleknejad and M. N. Sahlan, “The method of moments for solution of second kind Fredholm integral equations based on B-spline wavelets,” International Journal of Computer Mathematics, Vol. 87, No. 7, pp. 1602–1616 (2010).
• J. H. He, “Variational iteration method-a kind of non-linear analytical technique” International Journal of Non-Linear Mechanics, Vol. 34, No. 4, pp. 699–708 (1999).
• J. H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, Vol. 20, No. 10, pp. 1141–1199 (2006).
• J. H. He, “Variational iteration method—some recent results and new interpretations,” Journal of Computational and Applied Mathematics, Vol. 207, No. 1, pp. 3–17 (2007).
• K. Maleknejad, M. Shahrezaee, and H. Khatami, “Numerical solution of integral equations system of the second kind by block-pulse functions,” Applied Mathematics and Computation, Vol. 166, No. 1, pp. 15–24 (2005).
• K. Maleknejad, N. Aghazadeh, and M. Rabbani, “Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method,” Applied Mathematics and Computation, Vol. 175, No. 2, pp. 1229–1234 (2006).
• K. Maleknejad and F. Mirzaee, “Numerical solution of linear Fredholm integral equations system by rationalized Haar functions method,” International Journal of Computer Mathematics, Vol. 80, No. 11, pp. 1397–1405 (2003).
• X.Y. Lin, J.-S. Leng, and Y.-J. Lu, “A Haar wavelet solution to Fredholm equations,” in Proceedings of the International Conference on Computational Intelligence and Software Engineering (CiSE ’09), pp. 1–4 (2009).
• M. J. Emamzadeh and M. T. Kajani, “Nonlinear Fredholm integral equation of the second kind with quadrature methods,” Journal of Mathematical Extension, Vol. 4, No. 2, pp. 51–58 (2010).
• M. Lakestani, M. Razzaghi, and M. Dehghan, “Solution of nonlinear Fredholm-Hammerstein integral equations by using semiorthogonal spline wavelets,” Mathematical Problems in Engineering, Vol. 2, No. 1, pp. 113–121 (2005).
• Y. Mahmoudi, “Wavelet Galerkin method for numerical solution of nonlinear integral equation,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1119–1129 (2005).
• J. Biazar and H. Ebrahimi, “Iterationmethod for Fredholm integral equations of second kind,” Iranian Journal of Optimization, Vol. 1, pp. 13–23 (2009).
• D. D. Ganji, G. A. Afrouzi, H. Hosseinzadeh, and R. A. Talarposhti, “Application of homotopy-perturbation method to the second kind of nonlinear integral equations,” Physics Letters A, Vol. 371, No. 1-2, pp. 20–25 (2007).
• M. Javidi and A. Golbabai, “Modified homotopy perturbation method for solving non-linear Fredholm integral equations,” Chaos, Solitons and Fractals, Vol. 40, No. 3, pp. 1408–1412 (2009).
• E. Babolian, J. Biazar, and A. R. Vahidi, “The decomposition method applied to systems of Fredholm integral equations of the second kind,” Applied Mathematics and Computation, Vol. 148, No. 2, pp. 443–452 (2004).
• A. Deb, A. Dasgupta and G. Sarkar, “A new set of orthogonal functions and its application to the analysis of dynamic systems,” Journal of The Franklin Institute, Vol. 343, No. 1, pp. 1–26 (2006).
• E. Babolian, R. Mokhtari, M. Salmani, “Using direct method for solving variational problems via triangular orthogonal functions,” Applied Mathematics and Computation, Vol. 191, No. 2, pp. 206–217 (2007).
• E. Babolian, Z. Masouri, S. Hatamzadeh-Varmazyar, “Numerical solution of nonlinear Volterra-Fredholm integro-di_erential equations via direct method using triangular functions, Computers and Mathematics with Applications,” Vol. 58, pp. 239-247, (2009).
• A. Deb, G. Sarkar, A. Sengupta, “Triangular orthogonal functions for the analysis of continuous time systems,” New Delhi: Elsevier, (2007).
• E. Babolian, HR. Marzban and M. Salmani, “Using triangular orthogonal functions for solving Fredholm integral equations of the second kind,”Applied Mathematics and Computation, Vol. 201, No. 2, pp. 452–464 (2008).
• M. A. Ramadan, T. S. El-Danaf and A. M. E. Bayoumi ,“A finite iterative algorithm for the solution of Sylvester-conjugate matrix equation AV+BW=E¯V F+C and AV+B¯W=E¯V F+C,”Journal of http://www.sciencedirect.com/science/journal/08957177Mathematical and Computer Modelling, http://www.sciencedirect.com/science/journal/08957177/58/11Vol. 58, No 11, pp 1738–1754, ( 2013).
• M. A. Ramadan and A. M. E. Bayoumi, “Explicit and iterative methods for solving the matrix equation AV + BW = EVF + C ,” Asian Journal of Control, Vol. 16, No. 2, pp. 965–974 (2014).
• K. Maleknejad , M. Shahrezaee , H. Khatami , “Numerical solution of integral equations system of the second kind by block-pulse functions,” Applied Mathematics and Computation, Vol. 166, No. 1, pp. 15–24 (2005).