A modified parametric iteration method for solving nonlinear second order BVPs

Abstract

Abstract. The original parametric iteration method (PIM) provides the solution of a nonlinearsecond order boundary value problem (BVP) as a sequence of iterations. Since the successive iterations of the PIM may be very complex so that the resulting integrals in its iterative relation may not be performed analytically. Also, the implementation of the PIM generally leads to calculation of unneeded terms, which more time is consumed in repeated calculations for series solutions. In order to overcome these difficulties, in this paper, a useful improvement of the PIM is proposed.The implementation of the modified method is demonstrated by solving several nonlinear second order BVPs. The results reveal that the new developed method is a promising analytical tool to solve the nonlinear second order BVPs and more promising because it can further be applied easily to solve nonlinear higher order BVPs with highly accurate.Mathematical subject classification: Primary: 34B15; Secondary: 41A10.

On convergence and error analysis of the parametric iteration method

“…In fact, he concluded the convergence of the VIM by introducing a semi-contraction operator and completed the proof like the proof of the Banach's fixed point theorem. On the other hand, the PIM was utilized for solving various kind of differential equations like Abel equation [8], nonlinear chaotic Genesio system [9], boundary value problems [10], linear optimal control problems [11] and etc. Convergence theorem for some particular cases was discussed in some of these literature (e.g.…”

The Numerical Solution of Linear FredholmIntegro-Differential Equations via Parametric Iteration Method

“…The various numerical methods exist for solving LFIDEs for example variation iteration method [4], Adomian decomposition method [5], Chebyshev Polynomials [6], Bernstein's approximation [7]. PIM was applied successfully for solving boundary value problems [8]. We consider linear integro-differential equations as the following: …”

On the coupling of the homotopy perturbation method and Laplace transformation

“…He [8][9][10][11][12] developed the homotopy perturbation method for solving linear, nonlinear, initial and boundary value problems by merging two techniques, the standard homotopy and the perturbation. The homotopy perturbation method was formulated by taking the full advantage of the standard homotopy and perturbation methods and has been modified later by some scientists to obtain more accurate results, rapid convergence, and to reduce the amount of computation [13][14][15][16]. The Homotopy Perturbation Method (HPM) has been applied to a wide class of functional equations; see [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] and the references therein.…”