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Homotopy approach for solving two-dimensional integral equations of the second kind

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Języki publikacji
EN
Abstrakty
EN
In this paper, the two-dimensional linear and nonlinear integral equations of the second kind is analyzed. The homotopy analysis method (HAM) is used for determining the solution of the investigated equation. In this method, a solution is sought in the series form. It is shown that if this series is convergent, its sum gives the solution of the considered equation. The sufficient condition for the convergence of the series is also presented. Additionally, the error of approximate solution, obtained as partial sum of the series, is estimated. Application of the HAM is illustrated by examples.
Rocznik
Strony
19--41
Opis fizyczny
Bibliogr. 51 poz., tab., wykr.
Twórcy
autor
  • Institute of Mathematics Silesian University of Technology Kaszubska 23, 44-100 Gliwice, Poland
autor
  • Institute of Mathematics Silesian University of Technology Kaszubska 23, 44-100 Gliwice, Poland
autor
  • Institute of Mathematics Silesian University of Technology Kaszubska 23, 44-100 Gliwice, Poland
autor
  • Institute of Mathematics Silesian University of Technology Kaszubska 23, 44-100 Gliwice, Poland
Bibliografia
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  • [12] R. Brociek, E. Hetmaniok, J. Matlak, D. Słota. Application of the homotopy analysis method in solving the systems of linear and nonlinear integral equations. Math. Model. Anal., 21(3): 350–370, 2016.
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  • [17] M. El-Borai, M. Abdou, M. Basseem. An analysis of two dimensional integral equations of the second kind. Le Matematiche, 62: 15–39, 2007.
  • [18] B. Ghanbari. On the convergence of the homotopy analysis method for solving Fredholm integral equations. Walailak J. Sci. & Tsch., 10(4):395–403, 2013.
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  • [24] Y. Khan, K. Sayevand, M. Fardi, M. Ghasemi. A novel computing multi-parametric homotopy approach for system of linear and nonlinear Fredholm integral equations. Appl. Math. Comput., 249: 229–236, 2014.
  • [25] S. Liao. Homotopy analysis method: a new analytic method for nonlinear problems. Appl. Math. Mech. – Engl. Edition, 19: 957–962, 1998.
  • [26] S. Liao. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and Hall – CRC Press, Boca Raton, 2003.
  • [27] S. Liao. On the homotopy analysis method for nonlinear problems. Appl. Math. Comput., 147: 499–513, 2004.
  • [28] S. Liao. Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. Numer. Simulat., 14: 983–997, 2009.
  • [29] S. Liao. Homotopy Analysis Method in Nonlinear Differential Equations. Springer/Higher Education Press, Berlin/Beijing, 2012.
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  • [31] S. Mesloub, S. Obaida. On the application of the homotopy analysis method for a nonlocal mixed problem with Bessel operator. Appl. Math. Comput., 219: 3477–3485, 2012.
  • [32] Z. Odibat. A study on the convergence of homotopy analysis method. Appl. Math. Comput., 217: 782–789, 2010.
  • [33] A. Rezania, A. Ghorbali, G. Domairry, H. Bararnia. Consideration of transient heat conduction in a semi-infinite medium using homotopy analysis method. Appl. Math. Mech. – Engl. Edition, 29: 1625–1632, 2008.
  • [34] M. Russo, R. Van Gorder. Control of error in the homotopy analysis of nonlinear Klein-Gordon initial value problems. Appl. Math. Comput., 219: 6494–6509, 2013.
  • [35] A. Shayganmanesh. Generalizing homotopy analysis method to solve system of integral equations. J. Math. Extension, 5: 21–30, 2010.
  • [36] A. Shidfar, A. Babaei, A. Molabahrami. Solving the inverse problem of identifying an unknown source term in a parabolic equation. Comput. Math. Appl., 60: 1209–1213, 2010.
  • [37] A. Shidfar, A. Babaei, A. Molabahrami, M. Alinejadmofrad. Approximate analytical solutions of the nonlinear reaction-diffusion-convection problems. Math. Comput. Modelling, 53: 261–268, 2011.
  • [38] A. Shidfar, M. Garshasbi. A weighted algorithm based on the homotopy analysis method: Application to inverse heat conduction problems. Commun. Nonlinear Sci. Numer. Simulat., 14: 2908–2915, 2009.
  • [39] A. Shidfar, A. Molabahrami. Solving a system of integral equations by an analytic method. Math. Comput. Modelling, 54: 828–835, 2011.
  • [40] A. Shidfar, A. Molabahrami, A. Babaei, A. Yazdanian. A series solution of the nonlinear Volterra and Fredholm integro-differential equations. Commun. Nonlinear Sci. Numer. Simulat., 15: 205–215, 2010.
  • [41] A. Tari, M. Rahimi, S. Shahmorad, F. Talati. Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method. J. Comput. Appl. Math., 228: 70–76, 2009.
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  • [43] M. Turkyilmazoglu. Convergence of the homotopy analysis method. arXiv p. 1006.4460v1, 2010.
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  • [45] M. Turkyilmazoglu. Some issues on HPM and HAM: a convergence scheme. Math. Comput. Modelling, 53: 1929–1936, 2011.
  • [46] R. Van Gorder, K. Vajravelu. On the selection of auxiliary functions, operators, and convergence control parameters in the application of the homotopy analysis method to nonlinear differential equations: a general approach. Commun. Nonlinear Sci. Numer. Simulat., 14: 4078–4089, 2009.
  • [47] H. Vosughi, E. Shivanian, S. Abbasbandy. A new analytical technique to solve Volterra’s integral equations. Math. Methods Appl. Sci., 34: 1243–1253, 2011.
  • [48] Q. Wang. Approximate solution for system of differential-difference equations by means of the homotopy analysis method. Appl. Math. Comput., 217: 4122–4128, 2010.
  • [49] H. Xu. An effective treatment of nonlinear differential equations with linear boundary conditions using the homotopy analysis method. Math. Comput. Modelling, 49: 770–779, 2009.
  • [50] K. Yabushita, M. Yamashita, K. Tsuboi. An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method. J. Phys. A: Math. Theor., 40: 8403–8416, 2007.
  • [51] M. Zurigat, S. Momani, Z. Odibat, A. Alawneh. The homotopy analysis method for handling systems of fractional differential equations. Appl. Math. Modelling, 34: 24–35, 2010.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-c5ec1da2-a075-4e04-86f1-170fde64d16c
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