The Regularization q-Homotopy Analysis Method for (1 and 2) - Dimensional Non-linear First Kind Fredholm Integral Equations

  • Shaheed N. Huseen Thi-Qar University
  • Rawan A. Shlaka University of Thi-Qar, Faculty of Computer Science and Mathematics, Mathematics Department, Thi-Qar
Keywords: Regularization method ; q-homotopy analysis method; Fredholm integral equations.

Abstract

In this paper, the solutions of (1 and 2)- dimensional non-linear first kind Fredholm integral equations are studied by combine the q-homotopy analysis method (q-HAM) [2-11] and the regularization method [16,17]. The utilization of this technique depends on converting the first kind Fredholm Integral Equations to the second kind of equations by applying the regularization method. Then q-HAM is employed to the resulting second kind of equations to obtain a solution. Some illustrative examples are given to demonstrate the validity and applicability of this technique.

Author Biography

Shaheed N. Huseen, Thi-Qar University

Faculty of Computer Science and Mathematics,
Mathematics Department, Thi-Qar, Iraq

References

[1] Alturk A `` The Regularization –Homotopy Method for the Two-Dimensional Fredholm Integral Equation of the first kind ``. Mathematical and Computational Applications. 2016.
[2] El-Tawil M.A , Huseen. S.N ``The q-Homotopy Analysis Method (q-HAM)`` .International Journal of Applied mathematics and mechanics. 15, 2012, 51-75.
[3] El-Tawil M.A , Huseen S.N ``On Convergence of The q-Homotopy Analysis Method``. Int. J. Contemp. Math. Sciences. 8, 2013, 481 – 497.
[4] Huseen S.N , Grace S.R ``Approximate Solutions of Nonlinear Partial Differential Equations by Modified q-Homotopy Analysis Method (mq-HAM) `` .HindawiPublishing Corporation, Journal of Applied Mathematics. 2013 Article ID 569674, 9 pages. http:// dx.doi.org/10.1155/ 2013/ 569674.
[5] Huseen S.N , Grace S.R and El-Tawil M.A ``The Optimal q-Homotopy Analysis Method (Oq-HAM)``. International Journal of Computers & Technology. 11,2013.
[6] Huseen S.N ``Solving the K(2,2) Equation by Means of the q-Homotopy Analysis Method (q-HAM)``. International Journal of Innovative Science, Engineering & Technology. 2,August 2015.
[7] Huseen S.N. ``Series Solutions of Fractional Initial-Value Problems by q-Homotopy Analysis Method``. International Journal of Innovative Science, Engineering & Technology. 3,January 2016.
[8] Huseen S.N ``Application of optimal q-homotopy analysis method to second order initial and boundary value problems``. Int J Sci Innovative Math Res (IJSIMR). 3(1),2015,18–24.
[9] Huseen S.N ``A Numerical Study of One-Dimensional Hyperbolic Telegraph Equation``. Journal of Mathematics and System Science. 7 ,2017, 62-72.
[10] Huseen S.N , Nada M. Ayay ``A New Technique of The q-Homotopy Analysis Method for Solving Non-Linear Initial Value Problems``. Journal of Progressive Research in Mathematics. 14,2018,pp 2292-2307.
[11] Iyiola O.S ; Soh M.E and Enyi C.D. ``Generalised homotopy analysis method (q- Ham) for solving foam drainage equation of time fractional type``. Mathematics in Engineering , Science and ARE. space. 4,2013,429-440.
[12] Phillips D.L. `` A Technique for the Numerical solution of Certain integral equation of the first kind ``. J.ACM. 9.1, 1962, 84-96.
[13] Tikhonov A.N. ``On the solution of in correctly posed problem and the method of the regularization ``. Soviet math. 4,1963,1305-1308.
[14] Tikhonov A.N , Goncharsky. A.V , Stepanov. V.V , Yagola. A.G ``Numerical Method for the solution of ill – posed problems``. Springer. 1995 .
[15] Wazwaz A.M ``Linear and Non liner integral equation Method and Application``. Springer-Verlag , Berlin .2011.
[16] Wazwaz A.M. ``The Regularization method for Fredholm integral equation of the first kind ``. Computers and Mathematics with Applications. 61,2011,2981-2986.
[17] Ziyaee F. and Tari A. `` Regularization method for the Two-dimensional Fredholm integral equation of the first kind ``. International journal of Nonlinear Science. 18,2014,189-194.
Published
2019-11-21
How to Cite
Huseen, S., & Shlaka, R. (2019). The Regularization q-Homotopy Analysis Method for (1 and 2) - Dimensional Non-linear First Kind Fredholm Integral Equations. Journal of Progressive Research in Mathematics, 15(3), 2721-2743. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/1789
Section
Articles