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A series solution of the nonlinear Volterra and Fredholm integro-differential equations. (English) Zbl 1221.65343
Summary: The homotopy analysis method is applied to obtained the series solution of the high-order nonlinear Volterra and Fredholm integro-differential problems with power-law nonlinearity. Two cases are considered, in the first case the set of base functions is introduced to represent solution of given nonlinear problem and in the other case, the set of base functions is not introduced. However, in both cases, the convergence-parameter provides us with a simple way to adjust and control the convergence region of solution series.
##### MSC:
 65R20 Integral equations (numerical methods) 45J05 Integro-ordinary differential equations
##### References:
 [1] Delves, L. M.; Mohamed, J. L.: Computational methods for integral equations, (1985) [2] Razzaghi, M.; Yousefi, S.: Legendre wavelets method for the nonlinear Volterra – Fredholm integral equations, Math comput simul 70, 1-8 (2005) · Zbl 1205.65342 · doi:10.1016/j.matcom.2005.02.035 [3] Ebadi, G.; Rahimi-Ardabili, M. Y.; Shahmorad, S.: Numerical solution of the nonlinear Volterra integro-differential equations by the tau method, Appl math comput 188, 1580-1586 (2007) · Zbl 1119.65123 · doi:10.1016/j.amc.2006.11.024 [4] Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University; 1992. [5] Liao, S. J.: An explicit, totally analytic approximate solution for Blasius viscous flow problems, Int J non-linear mech 34, 759-778 (1999) [6] Liao, S. J.: Beyond perturbation: introduction to the homotopy analysis method, (2003) [7] Liao, S. J.: On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J fluid mech 488, 189 (2003) · Zbl 1063.76671 · doi:10.1017/S0022112003004865 [8] Liao, S. J.: On the homotopy analysis method for nonlinear problems, Appl math comput 147, 499-513 (2004) · Zbl 1086.35005 · doi:10.1016/S0096-3003(02)00790-7 [9] Liao, S. J.: A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int J heat mass transfer 48, 2529-3259 (2005) · Zbl 1189.76142 · doi:10.1016/j.ijheatmasstransfer.2005.01.005 [10] Liao, S. J.: Notes on the homotopy analysis method: some definitions and theorems, Commun nonlinear sci numer simul 14, 983-997 (2009) · Zbl 1221.65126 · doi:10.1016/j.cnsns.2008.04.013 [11] Watson, L. T.: Globally convergent homotopy methods: a tutorial, Appl math comput 13BK, 369-396 (1989) · Zbl 0689.65033 [12] Watson, Layne T.; Scott, Melvin R.: Solving spline-collocation approximations to nonlinear two-point boundary-value problems by a homotopy method, Appl math comput 24, 3X-357 (1987) · Zbl 0635.65099 · doi:10.1016/0096-3003(87)90015-4 [13] Watson, Layne T.: Engineering applications of the chowyorke algorithm, Appl math comput 9, 111-133 (1981) · Zbl 0481.65029 · doi:10.1016/0096-3003(81)90010-2 [14] Watson, Layne T.; Haftka, Raphael T.: Modern homotopy methods in optimization, Comput meth appl mech eng 74, 289-305 (1989) · Zbl 0693.65046 · doi:10.1016/0045-7825(89)90053-4 [15] Wang, Y.; Bernstein, D. S.; Watson, L. T.: Probability-one homotopy algorithms for solving the coupled Lyapunov equations arising in reduced-order H2=H1 modeling estimation and control, Appl math comput 123, 155-185 (2001) · Zbl 1028.93011 · doi:10.1016/S0096-3003(00)00059-X [16] Molabahrami, A.; Khani, F.: The homotopy analysis method to solve the Burgers – Huxley equation, Nonlinear anal real 10, 589-600 (2009) · Zbl 1167.35483 · doi:10.1016/j.nonrwa.2007.10.014