This article presents an improved spectral-homotopy analysis method (ISHAM) for solving nonlinear differential equations. The implementation of this new technique is shown by solving the Falkner-Skan and magnetohydrodynamic boundary layer problems. The results obtained are compared to numerical solutions in the literature and MATLAB's bvp4c solver. The results show that the ISHAM converges faster and gives accurate results.
Keywords:Falkner-Skan flow; MHD flow; improved spectral-homotopy analysis method
Boundary layer flow problems have wide applications in fluid mechanics. In this article, we propose an improved spectral-homotopy analysis method (ISHAM) for solving general boundary layer problems. Three boundary layer problems are considered and solved in this study using the novel technique. The first problem considered is the classical two-point nonlinear boundary value Blasius problem which models viscous fluid flow over a semi-infinite flat plate. Although solutions for this problem had been obtained as far back as 1908 by Blasius , the problem is still of great interest to many researchers as can be seen from the several recent studies [2-5].
The second problem considered in this article is the third-order nonlinear Falkner-Skan equation. The Falkner-Skan boundary layer equation has been studied by several researchers from as early as 1931 . More recent studies of the solutions of the The Falkner-Skan equation include those of Harries et al. , Pade  and Pantokratoras . The third problem considered is magnetohy-drodynamic (MHD) boundary layer flow. Such boundary layer problems arise in the study of the flow of electrically conducting fluids such as liquid metal. Owing to its many applications such as power generators, flow meters, and the cooling of reactors, MHD flow has been studied by many researchers, for example [10,11].
Owing to the nonlinearity of equations that describe most engineering and science phenomena, many authors traditionally resort to numerical methods such as finite difference methods , Runge-Kutta methods , finite element methods  and spectral methods  to solve the governing equations. However, in recent years, several analytical or semi-analytical methods have been proposed and used to find solutions to most nonlinear equations. These methods include the Adomian decomposition method [15-17], differential transform method , variational iteration method , homotopy analysis method (HAM) [20-23], and the spectral-homotopy analysis (SHAM) (see Motsa et al. [24,25]) which sought to remove some of the perceived limitations of the HAM. More recently, successive linearization method [26-28], has been used successfully to solve nonlinear equations that govern the flow of fluids in bounded domains.
In this article, boundary layer equations are solved using the ISHAM. The ISHAM is a modified version of the SHAM [24,25]. One strength of the SHAM is that it removes restrictions of the HAM such as the requirement for the solution to conform to the so-called rule of solution expression and the rule of coefficient ergodicity. Also, the SHAM inherits the strengths of the HAM, for example, it does not depend on the existence of a small parameter in the equation to be solved, it avoids discretization, and the solution obtained is in terms of an auxiliary parameter ħ which can conveniently be chosen to determine the convergence rate of the solution.
We consider the general nonlinear third-order boundary value problem
subject to the boundary conditions
where ci, bj (i = 1, ..., 4 j = 1, 2, 3) are constants.
Equation 2.1 can be solved easily using methods such as the HAM and the SHAM. In each of these methods, an initial approximation f0(η) is sought, which satisfies the boundary conditions. The speed of convergence of the method depends on whether f0(η) is a good approximation of f (η) or not. The approach proposed here seeks to find an optimal initial approximation f0 that would lead to faster convergence of the method to the true solution. We thus first seek to improve the initial approximation that is used later in the SHAM to solve the governing nonlinear equation.
We assume that the solution f(η) may be expanded as an infinite sum:
where fi's are unknown functions whose solutions are obtained using the SHAM at the ith iteration and fn, (n ≥ 1) are known from previous iterations. The algorithm starts with the initial approximation f0(η) which is chosen to satisfy the boundary conditions (2.2). An appropriate initial guess is
Substituting (2.3) in the governing equation (2.1-2.2) gives
subject to the boundary conditions
where the coefficient parameters ak,i-1, (k = 1, ..., 3) and ri-1 are defined as
Starting from the initial approximation (2.4), the subsequent solutions fi (i ≥ 1) are obtained by recursively solving Equation 2.5 using the SHAM, [24,25]. To find the solutions of Equation 2.5, we begin by defining the following linear operator:
where q ∈ 0 is the embedding parameter, and Fi(η; q) is an unknown function.
The zeroth-order deformation equation is given by
where ħ is the non-zero convergence controlling auxiliary parameter and is a nonlinear operator given by
Differentiating (2.10) m times with respect to q and then setting q = 0, and finally dividing the resulting equations by m! yield the mth-order deformation equations:
subject to the boundary conditions
The initial approximation fi,0 that is used in the higher-order equations (2.12) is obtained on solving the linear part of Equation 2.5 which is given by
subject to the boundary conditions:
Since the coefficient parameters and the right-hand side of Equation 2.15 for i = 1, 2, 3, ... are known (from previous iterations), the equation can easily be solved using numerical methods such as finite differences, finite elements, Runge-Kutta-based shooting methods or collocation methods. In this article, Equation 2.15 are solved using the Chebyshev spectral collocation method. The method (see, for example, [29-31]), is based on the Chebyshev polynomials defined on the interval [-1, 1] by
To implement the method, the physical region [0, ∞) is transformed into the region [-1, 1] using the domain truncation technique whereby the problem is solved in the interval [0, L] instead of [0, ∞). This leads to the mapping
where N is the number of collocation points. The variable fi,0 is approximated by the interpolating polynomial in terms of its values at each of the collocation points by employing the truncated Chebyshev series of the form:
where Tk is the kth Chebyshev polynomial. Derivatives of the variables at the collocation points may be represented by
Substituting Equations 2.20-2.21 in 2.15-2.16 gives
In the above definitions, T stands for transpose and ak,i-1(k = 1, 2, 3) denotes a diagonal matrix of size (N + 1) × (N + 1). The boundary condition fi (ξN ) = 0 is implemented by deleting last row and last column of Ai-1, and deleting the last rows of Fi,0 and Ri-1. The derivative boundary conditions in (2.24) are then imposed on the resulting first row and last row of Ai-1 and setting the first and last rows of Fi,0 and Ri-1 to be zero. The solutions for fi.0(ξ) are then obtained from soloving
In a similar manner, applying the Chebyshev spectral transformation on the higher order deformation equations (2.12)-(2.13) gives
subject to the boundary conditions
where Ai-1 and Ri-1, are as defined in (2.25) and (2.27), respectively, and
To implement the boundary condition fi,m(ξN ) = 0, we delete the last rows of Pi,m-1 and Ri-1 and delete the last row and the last column of Ai-1 in (2.29). The other boundary conditions in (2.30) are imposed on the first and the last rows of the modified Ai-1 matrix on the left side of the equal sign in (2.29). The first and the last rows of the modified Ai-1 matrix on the right side of the equal sign in (2.29) are then set to be zero. This results in the following recursive formula for m ≥ 1:
where Ãi-1 is the modified matrix Ai-1 after incorporating the boundary conditions (2.30). Thus, starting from the initial approximation, which is obtained from (2.28), higher-order approximations fi,m(ξ) for m ≥ 1, can be obtained through the recursive formula (2.33).
The solutions for fi are then generated using the solutions for fi, m as follows:
The [i, m] approximate solution for f (η) is then obtained by substituting fi (obtained from 2.34) in equation 2.3.
Results and discussion
Table 1 shows the values of f" (0) at different orders [i, m] of the ISHAM approximation for the Blasius boundary layer flow when L = 30, ħ = -1 and N = 80. It is worth noting here that the numerical solution given by Howarth  is f" (0) = 0.332057, while the numerical result by the Matlab bvp4c routine is f" (0) = 0.33205734. Asaithambi  found this number correct to nine decimal positions as 0.332057336. It is evident that the ISHAM converges to the numerical result at orders [3,1] and [2,2]. Moreover, Table 1 shows that the ISHAM solution converges to the accurate solution of Howarth and the bvp4c result faster than the original SHAM results of which are those given in the first row of Table 1 (for the case when i = 1).
Table 1. Order [i, m] ISHAM approximate results for f" (0) of the Blasius boundary layer flow (Example 1) using L = 30, ħ = -1 and N = 80
In general, at order [i, m], i is the number of improvements of the initial approximation f0(η) for f(η), and m is the number of improvements of the initial guess fq,0(η); q = 1, 2, ..., i, for each application of the ISHAM. Table 2 gives a sense of the convergence rate of the ISHAM when compared with the numerical method for the Blasius problem at different values of η. In all the instances, convergence of the ISHAM is achieved at the second order.
Table 2. Comparison between the [m, m] ISHAM results and the bvp4c numerical results for the velocity pro le f' (η) at selected values of η for the Blasius boundary layer flow (Example 1) using L = 30, ħ = -1 and N = 200
Table 3 gives the values of f" (0) obtained used the ISHAM and the numerical method for various values of β for the Falkner-Skan boundary layer problem. Full convergence is again achieved at order [2,2] for all the parameter values.
Table 3. Order [m, m] ISHAM approximate results for f" (0) of the Falkner-Skan boundary layer flow (Example 2) using L = 30, ħ = -1 and N = 80
For the MHD boundary layer problem, Tables 4 and 5 illustrate the exact and approximate values of f' (η) and f" (0) at different values of η and the magnetic parameter M, respectively. The absolute errors in the approximations are also given. The tables show that the ISHAM converges rapidly with marginal or no errors after order [2,2].
Table 4. Order [m, m] ISHAM approximate results for the velocity profile f' (η) of the MHD boundary layer flow (Example 3) when M = 10 using L = 10, ħ = -1 and N = 200
Table 5. Order [m, m] ISHAM approximate results for f" (η) of the MHD boundary layer flow (Example 3) for different values of M using L = 10, ħ = -1 and N = 200
In this article, we have proposed an ISHAM for solving general nonlinear differential equations. This novel technique was compared against both numerical approximations and the MATLAB bvp4c routine for solving Falkner-Skan and MHD boundary layer problems. The results demonstrate the relatively more rapid convergence of the ISHAM, and they show that the ISHAM is highly accurate.
HAM: homotopy analysis method; ISHAM: improved spectral-homotopy analysis method; MHD: magnetohydrodynamic; SHAM: spectral-homotopy analysis.
The authors declare that they have no competing interests.
SSM developed the Matlab codes and generated the results. GTM and PS conceived of the study and formulated the problem. SS participated in the analysis of the results and manuscript coordination. All authors typed, read and approved the final manuscript.
The authors wish to acknowledge financial support from the University of Swaziland, University of KwaZulu-Natal, University of Venda, and the National Research Foundation (NRF).
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