On Extending the Quasilinearization Method to Higher Order Convergent Hybrid Schemes Using the Spectral Homotopy Analysis Method

We propose a sequence of highly accurate higher order convergent iterative schemes by embedding the quasilinearization algorithm within a spectral collocation method. The iterative schemes are simple to use and significantly reduce the time and number of iterations required to find solutions of highly nonlinear boundary value problems to any arbitrary level of accuracy. The accuracy and convergence properties of the proposed algorithms are tested numerically by solving three Falkner-Skan type boundary layer flow problems and comparing the results to the most accurate results currently available in the literature. We show, for instance, that precision of up to 29 significant figures can be attained with no more than 5 iterations of each algorithm.


Introduction
The quasilinearization method (QLM) was originally developed by Bellman and Kalaba [1] as a generalization of the Newton-Raphson method to provide lower and upper bound solutions of nonlinear differential equations.The attraction of quasilinearization is that the algorithm is easy to understand and the method generally converges rapidly if the initial guess is close to the true solution.
Bellman and Kalaba [1] established that the method converges quadratically.However, the original proof of quadratic convergence was subject to restrictive conditions of small step size and convexity or concavity of nonlinear functions, Maleknejad and Najafi [2].These conditions were subsequently relaxed and the method generalized to be applicable to a wider class of problems; see, for instance, papers by Mandelzweig and his coworkers [3][4][5][6] and Lakshmikantham [7,8].Parand et al. [9] used the quasilinearization method to solve Volterra's model for population growth in a closed system.Other uses of the quasilinearization method include application to reaction diffusion equations, Jiang and Vatsala [10], and to Volterra integro-differential equations, Ahmad [11], Pandit [12], and Ramos [13].
An often noted disadvantage of quasilinearization is the instability of the method whenever a poor initial guess is chosen, Tuffuor and Labadie [14].To improve the accuracy and convergence of the quasilinearization method for all initial guesses, we propose in this paper to embed the QLM algorithm within the spectral homotopy analysis method (SHAM) to obtain a sequence of integration schemes with arbitrary higher order convergence.
The spectral homotopy analysis method was introduced by Motsa et al. [15,16] to address some limitations of the standard homotopy analysis method of Liao [17,18] by, for example, improving the rate of convergence and extending the region of validity of solutions.The SHAM has been used to solve nonlinear equations that arise in the study of fluid flow problems and other areas of science and engineering, Sibanda et al. [19] and Motsa and Sibanda [20].
Abbasbandy [21] and Chun [22] proposed and studied several methods for nonlinear equations with higher order convergence by using the Adomian decomposition technique [23][24][25].Higher order Newton-like iteration formulae for the computation of the solutions of nonlinear equations were also derived in Chun [26] using the homotopy analysis method and by [27] using the homotopy perturbation method.In this paper we extend the ideas used for the solution of nonlinear equations to obtain higher order iteration schemes for solving nonlinear boundary value problems.We propose an extension of the quasilinearization method by using the spectral homotopy analysis method within the QLM algorithm to obtain a sequence of highly accurate and convergent higher order iterative schemes for solving boundary value problems.
For illustration purposes we have presented three QLM-SHAM hybrid iteration schemes that are used to solve the Blasius and Falkner-Skan equations.The results are compared to the most accurate skin friction coefficients currently available in the literature by Boyd [28] and Ganapol [29].Ganapol [29] used an algorithm based on a Maclaurin series with Wynnepsilon convergence acceleration and analytical continuation to obtain highly accurate skin friction coefficients for the Blasius and Falkner-Skan boundary layer flows.The schemes derived in this paper however require neither convergence acceleration nor analytical continuation to remain steady and accurate for up to 29 digits of precision.In addition, the present schemes are highly efficient with 29-digit precision achieved with five or fewer iterations as compared with at least 104 iterations of Ganapol's algorithm.
The structure of this paper is as follows.Section 2 gives a general framework for the derivation of the hybrid quasilinearization-SHAM schemes for the solution of nonlinear differential equations.Section 3 illustrates the application of the three schemes derived in this paper to the solution of Blasius and Falkner-Skan equations.In Section 4, the results are presented and comparison made with the most accurate skin friction results for Blasius and Falkner-Skan equations currently available in the literature.
Here G 1 is a nonlinear function that is decomposed using the spectral homotopy analysis method [15,16].We define the following zeroth-order deformation equations: where  ∈ [0, 1] is an embedding parameter, (; ) are unknown functions, ℎ is the convergence controlling parameter, and the "bar" has been introduced for convenience to denote the associated function and its  derivatives.For example,  ≡ (,  (1) ,  (2) , . . .,  () ) .
The nonlinear operator N 1 is defined by By differentiating the zeroth-order equations ( 11)  times with respect to , setting  = 0, and finally dividing the resulting equations by ! (see, e.g., [17,[30][31][32]), we obtain the following th order deformation equations: where After obtaining solutions for (12), the approximate solution for () is determined as the series solution The SHAM solution is said to be of order  if the previous series is truncated at  = , that is, if The initial approximation  0 required for solving the sequence of linear higher order deformation equations ( 12) is chosen as the solution that results from solving the linear part of (5) subject to the given boundary conditions (2).That is, we solve We note that with L 1 as defined in ( 6), ( 16) cannot be solved exactly by means of analytical techniques.Numerical methods such as finite differences, finite element method, and spectral method can be used to solve equations of the form (16). Thus, if only the initial approximation is used to approximate the solution () of the governing nonlinear differential equation (1), that is, if () ≈  0 (), the ( + 1)th approximation of ( 1) is a solution of which, on using the definitions ( 6) and (7), can be written as () (  ,  (1)   , . . .,  ()  ) +  [  ,  (1)   , . . .,  ()  ] = 0. ( We note that the iterative scheme ( 18) is, in fact, the quasilinearization method of Bellman and Kalaba [1].For  = 1, we have where  1 is obtained as a solution of This produces the iteration scheme where and  0,+1 is the solution of  [ 0,+1 ,  (1)  0,+1 , . . .,  () 0,+1 ] + () (  ,  (1)   , . . .,  ()  ) = Φ (  ,  (1)   , . . .,  ()  ] .
For  = 2, we have where  2 is obtained as a solution of This produces the iteration scheme where  1,+1 is obtained as the solution of In general, for any  > 1, we have where   () is obtained as a solution of Thus, a general scheme when the SHAM is truncated at order  (where  ≥ 1), hereinafter referred to as scheme-, can be obtained as where each  ,+1 is obtained as the solution of (1)   , . . .,  ()  ] when  = 0.

Solution of the Falkner-Skan Equation
In this section we demonstrate how the numerical schemes derived in the previous section may be used to solve the Falkner-Skan equation: subject to the boundary conditions It is convenient to first define so that where Using ( 35)-(37) the first three iterative schemes corresponding to  = 0, 1, 2 may now be defined as follows.
3.1.Scheme-0.In this scheme we set subject to It is worth noting that Scheme-0 is, in fact, equivalent to the original QLM algorithm; see Mandelzweig and Tabakin [5] and Mandelzweig [6].
3.2.Scheme-1.For this scheme we set subject to where and  0,+1 is the solution of 3.3.Scheme-2.The complexity of the defining equations increases with the order of the scheme.For Scheme-2 we have subject to where and  1,+1 is the solution of with Equations ( 39), (41), and (46) describing the three solution schemes can be solved numerically using standard methods such as finite difference, finite elements, and spline collocation methods.In this study we use the Chebyshev spectral collocation method to solve the iteration schemes, (see [33][34][35][36]).To allow for numerical implementation of the pseudospectral method, the physical region [0, ∞) is truncated to [0, ] where  is chosen to be sufficiently large.The truncated region is further transformed to the space [−1, 1] using the transformation As with any other numerical approximation method, some sort of discretization is introduced in the interval [−1, 1].We choose the Gauss-Lobatto collocation points to define the nodes in [−1, 1] as where ( + 1) is the number of collocation points.The essence of the Chebyshev spectral collocation method is the idea of introducing a differentiation matrix .The differentiation matrix maps a vector of the function values Y = [( 0 ), . . ., (  )]  at the collocation points to a vector Y  defined as In general, a derivative of order  for the function () can be expressed as where D = 2/ ∞ .The matrix  is of size ( + 1) × ( + 1) and its entries are defined as ,  = 1, 2, . . .,  − 1, with Thus, applying the spectral method to the iteration Scheme-0 (39) and the corresponding boundary conditions gives the following matrix system: with boundary conditions where where Φ  corresponds to the function Φ(, ,   ) when evaluated at the collocation points and a , ( = 0, 1, 2) is a diagonal matrix corresponding to the vector of  , .The boundary conditions (58) are imposed on the first, th, and ( + 1)th rows of   and Φ  to obtain a system of the form  Starting from a suitable initial guess  0 (), the iteration scheme (60) can be used to iteratively give approximate solutions of the governing equation (32) for Scheme-0.The application of the pseudospectral method for Scheme-1 and Scheme-2 can be done in a similar manner.The initial approximation used in all the algorithms is The number of collocation points used in all the results presented here is  = 200 with  ∞ = 20.

Results and Discussion
In this section we present solutions of the Falkner-Skan equation (32) using the QLM-SHAM hybrid iteration schemes.Numerical simulations were conducted for the following special classes of the F-S equations: To assess the accuracy and performance of our schemes, the numerical results were compared to the recently reported results of Ganapol [29].To date, these results are the most accurate results for the Blasius and Falkner-Skan class of equations.Ganapol [29] reported highly accurate results between 10 and 30 decimal places using a robust algorithm based on Maclaurin series with convergence acceleration and analytical continuation techniques.
The comparison between the present findings and the results in the literature is made for the skin friction which is proportional to   (0).Table 1 shows a comparison between the computed skin friction values of the Blasius equation using the three QLM-SHAM iteration schemes.The results are compared with the results reported in Ganapol [29] which are accurate to 29 decimal places.We observe that all the iteration schemes rapidly converge to the results of [29] to all 29 decimal places.Full convergence is achieved after 6 iterations when using Scheme-0, 4 iterations when using Scheme-1, and after 3 iterations when using Scheme-2.It is worth noting that the results of [29] were achieved after 104 decimal places.Prior to Ganapol [29], the most accurate Blasius skin friction results had been published to 17 decimal places by Boyd [28] as   (0) = 0.33205733621519630.This result was obtained after 5 iterations using Scheme-0 and 3 iterations for both Schemes-1 and -2.The value reported after 52 iterations in [29] is   (0) = 0.3320573362151965.It is clear that the proposed iteration schemes converge significantly faster than the method of [29].That the results of Boyd [28] and Ganapol [29] were obtained only after a few iterations validates both the higher order convergence and the accuracy of the present solution methods.
Table 2 shows the computed values of   (0) using Schemes-0 and -1 for the Pohlhausen flow ( = 0,  1 = 1).For this particular flow, the exact value of   (0) is known to be 2/ √ 3. The iteration Scheme-0 matches the exact result after only 5 iterations and Scheme-1 converges after only 3 iterations.The method used in Ganapol [29] converged to the exact result after 104 iterations.This again demonstrates the superior convergence of the present method.Table 3 gives the numerical simulations of the skin friction results for the Homann flow.We observe that the 29-digit results reported in [29] are achieved in 5 iterations, 4 iterations, and 3 iterations for Scheme-0, -1, and -2, respectively.This result indicates that adding an additional level in the QLM-SHAM scheme would further significantly increase the convergence of the iteration scheme.We further note from Tables 1-3 that all three schemes converge significantly much faster than the spectral homotopy analysis method on its own.

Conclusion
In this study we presented three hybrid QLM-SHAM iteration schemes for the solution of Falkner-Skan type boundary layer equations.We have shown through numerical experimentation that the proposed numerical schemes significantly enhance the convergence rate of the quasilinearization method.By comparison with the most accurate solutions of the Falkner-Skan equations currently available in the literature, we have shown that the schemes are highly accurate and efficient in terms of the number of iterations required to determine the solution to the required level of accuracy.The schemes presented provide robust tools for the efficient solution of nonlinear equations by offering superior accuracy to many existing methods.In addition, the approach used in deriving these schemes provides a suitable framework for extension to higher level schemes by adding more terms of the SHAM component of the method.