An efficient analytic approach for solving two-point nonlinear boundary value problems by homotopy analysis method

Abstract

In this paper, we use homotopy analysis method (HAM) to solve two-point nonlinear boundary value problems that have at least one solution. The new approach provides the solution in the form of a rapidly convergent series with easily computable components using symbolic computation software. The scheme shows importance of choice of convergencecontrol parameter h to guarantee the convergence of the solutions of nonlinear differential equations. This scheme is tested on three nonlinear exactly solvable differential equations. Two of the examples are practical in science and engineering. The results demonstrate reliability, simplicity and efficiency of the algorithm developed.

Analytic approximate solution for the Bratu's problem by optimal homotopy analysis method

“…The absolute residual | ( )|(27) has maximum magnitude 0.00003 in lower solution case and 0.12 in the upper solution case, which show the accuracy of the approximate solution and the efficiency of the proposed method. The Bratu's problem (28) has been resolved in previous papers [7] and [20] at only, but in this paper successfully solved the problem with high accuracy for different values of . as shown in Fig.4, fig.5 and fig.6 show the problem solutions at respectively, the sets lower and upper solutions ℏ can be summarized by the Table 6: Maximum magnitude of | ( )| in both solutions for different values of …”

“…N. Hassan et all; [7] and Abbasbandy et all; [20], applied successfully the HAM to obtain multiple branches of solutions of this nonlinear problem for only. In …”

“…In recent years, this method has been successfully employed to solve many types of nonlinear problems in science and engineering such as Troesch's problem [7], the Fitzhugh-Nagumo equation [8], heat radiation equations [9], MHD viscoelastic fluid flow [10], the Coupled nonlinear schrödinger equations [11], the Continuous population models for single and interacting species [12], the Boussinesq problem [13,14], the Sturm-Liouville problems [15], the wave propagation problems [16], Laplace equation with Dirichlet and Neumann boundary conditions [17], differential-difference equation [18], fractional equations [19]. Many authors trying to solve equation (1) by homotopy analysis method [7,20] but for λ=3 only. The basic motivation of this paper is to obtain analytical approximate solution of the Bratu's problem (1) by using homotopy analysis method for different values of λ and determined the optimal value of convergence-controller parameter ℏ using the averaged residual error [21] .…”

Nonlinearities distribution Laplace transform-homotopy perturbation method

“…As it is widely known, the importance of research on nonlinear differential equations is that many phenomena, practical or theoretical, are of nonlinear nature. In recent years, several methods focused to find approximate solutions to nonlinear differential equations, as an alternative to classical methods, have been reported, such those based on: variational approaches (Assas 2007; He 2007; Kazemnia et al 2008; Noorzad et al 2008), tanh method (Evans & Raslan 2005), exp-function (Xu 2007; Mahmoudi et al 2008), Adomian’s Decomposition Method (ADM) (Adomian 1988; Babolian & Biazar 2002; Kooch & Abadyan 2012; Kooch & Abadyan 2011; Vanani et al 2011; Chowdhury 2011; Elias et al 2000), parameter expansion (Zhang & Xu 2007), HPM (Aminikhan & Hemmatnezhad 2012; Aminikhah 2012; Filobello-Nino et al 2013; Aminikhah 2011; Khan et al 2011; Marinca & Herisanu 2011; He 1998; He 1999; He 2006a; Vazquez-Leal et al 2014; Belendez et al 2009; He 2000; El-Shaed 2005; He 2006b; Vazquez-Leal et al 2012a; Ganji et al 2008; Fereidon et al 2010; Sharma & Methi 2011; Biazar & Ghanbari 2012; Biazar & Eslami 2012; Araghi & Sotoodeh 2012; Araghi & Rezapour 2011; Bayat et al 2014; Bayat et al 2013; Vazquez-Leal et al 2012b; Vazquez-Leal et al 2012c; Filobello-Niño et al 2012; Biazar & Aminikhan 2009; Biazar & Ghazvini 2009; Filobello-Nino et al 2012; Khan & Qingbiao 2011; Madani et al 2011; Ji Huan 2006; Feng et al 2007; Mirmoradia et al 2009; Vazquez-Leal et al 2012; Vazquez-Leal et al 2013), Homotopy Analysis Method (HAM) (Rashidi et al 2012a; Rashidi et al 2012b; Patel et al 2012; Hassana & El-Tawil 2011), and perturbation method (Filobello-Nino et al 2013; Holmes 1995; Filobello-Niño et al 2013b; Filobello-Nino et al 2014) among many others. Also, a ...…”

“…The case of Boundary Value Problems (BVPs) for nonlinear ODES includes, Michaelis Menten equation (Murray 2002; Filobello-Nino et al 2014), that describes the kinetics of enzyme-catalyzed reactions, Gelfand’s differential equation (Filobello-Nino et al 2013; Filobello-Niño et al 2013b) which governing combustible gas dynamics, Troesch’s equation (Elias et al 2000; Feng et al 2007; Mirmoradia et al 2009; Vazquez-Leal et al 2012; Hassana & El-Tawil 2011; Erdogan & Ozis 2011), arising in the investigation of confinement of a plasma column by a radiation pressure, among many others.…”