Abstract
We propose an iterative method for solving the Falkner-Skan equation. The method provides approximate analytical solutions which consist of coefficients of the previous iterate solution. By some examples, we show that the presented method with a small number of iterations is competitive with the existing method such as Adomian decomposition method. Furthermore, to improve the accuracy of the proposed method, we suggest an efficient correction method. In practice, for some examples one can observe that the correction method results in highly improved approximate solutions.
1. Introduction
We consider the well-known Falkner-Skan equation subjected to boundary conditions This boundary value problem arises in the research of viscous flow past a wedge of angle . corresponds to flow toward the wedge and does to flow away from the wedge. The special case, , is called the Blasius equation where the wedge reduces to a flat plate. It is well known that a unique smooth solution of the Falkner-Skan equation (1.1) with the condition (1.2) exists for [1, 2]. For , there exists two solutions, that is, one with and the other one with , as shown in the literature [3]. For the solution is unique under the restriction [4, 5].
In order to obtain approximate solutions of the Falkner-Skan equation, analytical methods such as Adomian decomposition methods [6–10], variational iterative methods [11–15], and homotopy analysis methods [16–18] can be referred. The objective of this work is to present a new analytical method which provides a simple form of iterate solutions and can be a match for existing methods in accuracy.
In the next section we develop an iterative method based on a decomposition of the Falkner-Skan equation (1.1) which was recently introduced in [19] for a particular case, . In the result we derive an iterative formula producing approximate analytical solutions in the form of polynomial series without requiring any differentiations or integrations of the previous iterate solutions. The degree of the th iterate solution increases very rapidly, in fact, like as . In Section 3, for some values of we compare the presented solutions with the existing iterate solutions. In Section 4, to improve the accuracy of the presented solutions we suggest a correction method which is composed of the successive differences of the iterations.
2. Iteration for Approximate Analytical Solutions
To develop a new iterative method generating analytical series solutions, we consider the following one-point boundary conditions instead of the boundary conditions in (1.2). where the curvature of the solution is assumed to be known. Actually, the value of can be obtained by numerical evaluation [8, 20, 21]. In this section we derive iterate solutions based on the method which was introduced in [19] for a special case, .
First, for the Falkner-Skan equation (1.1) becomes From the boundary conditions in (2.1), it follows that and thus In the result we have
Denoting by the th iterate solution and substituting it into the right-hand side of (2.5), with a given initial solution , we have an iteration formula where
Let denote a degree of the th iterate solution . Then, from the formulas (2.6) and (2.7), it follows that for all . Solving this recurrence equation, we have
For an initial solution satisfying the conditions in (2.1), we may set the th iterate solution as for any integer . Then, by performing integrations in (2.6) and (2.7) directly, we have Thus, by integrating , we have
On the other hand, referring to the boundary conditions in (2.1) and (1.1), we may take an initial solution as
3. Examples
3.1. A Case of (Blasius Problem)
A case of is called the Blasius problem, and a well-known power series for the solution is where the coefficients can be computed from the recurrence [1]: In fact, and the series becomes The Blasius series, however, converges only for [16, 22, 23]. In this paper we denote by a truncated Blasius series to the th term of .
For the Blasius problem the first four iterate solutions generated by the formula (2.11), with , are as follows: It should be noted that the degree of the presented solution increases like as shown in (2.8) while that of the truncated Blasius series is , or it increases like as .
For comparison between and , Figure 1 includes graphs of the approximations to the numerical solution , which is taken for an exact solution, and their errors over an interval . We set which is close to the radius of convergence, of the Blasius series solution mentioned above. Figure 1 shows that the presented solutions approximate the exact solution better than the truncated Blasius series .
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3.2. A Case of
We refer to another analytical solution obtained by the Adomian decomposition method as follows: with , and Adomian polynomial generated by the recursive formula [8]: where is an inverse operator of and is a nonlinear operator defined as .
Presented solutions , , given in (2.11) and their errors are compared with those of Adomian's solutions for the Falkner-Skan equation with and in Figures 2 and 3, respectively. We chose in (2.12) as an initial solution and took for and for , as given in [1]. One can see that the presented solutions are competitive with the Adomian solutions. Moreover, it should be noticed that unlike the Adomian solutions, the presented solutions do not require any integration or differentiation.
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For the so-called decelerated flow (or ), it is well known that a unique solution with exists when [4]. So we take an example of with as given in [8]. Comparison between the presented solutions and the Adomian solutions for this case is given in Figure 4. Therein we can find superiority of the presented solutions in accuracy.
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4. Improvement of the Approximate Solution
In Figure 1, for the case of , one can see that both the proposed solutions and the truncated Blasius series overshoot and undershoot, alternately, as increases. This tendency, in the proposed solutions, is continued to the cases of other values of as can be observed in Figures 2–4. Considering this feature, we propose a correction method as follows.
Let , , be three successive iterate solutions obtained by the formula (2.11). To improve the accuracy of the iterate solutions, we suggest two corrections of the form for a weight function such that . Setting a constraint , we can determine as where is a ratio of the successive differences defined as We may surmise that because, as mentioned above, the proposed solutions alternately overshoot and undershoot as increases. In practice, for the Blasius problem (), Figure 5(a) depicts the graphs of on the interval for , which shows for all . Thus from (4.2) we have so that the correction or in (4.1) with the weight in (4.2) is reasonable. The graph of is given in Figure 5(b).
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For , for example, Figure 6 shows the approximations of , , and their errors on an extended interval . Furthermore, Table 1 includes -norm errors of , , and with respect to the numerical solution on the interval . The -norm error is defined as From Figure 6 and Table 1, compared with the results of in Figure 1, we can see that the correction highly improves the accuracy of the original iterate solution .
In addition, absolute errors of and , , for each are depicted in Figure 7. Similar to Table 1, Table 2 includes -norm errors of , , and on the interval . We can also observe that the correction highly improves the accuracy of the original iterate solution .
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5. Conclusion
We have developed a new iterative method generating analytical solutions to the Falkner-Skan equation (1.1) with the boundary conditions (2.1). In practice, for several cases of , it was shown that the presented method with a small number of iterations is available and efficient, compared with a well-known existing method such as the Adomian decomposition method. Moreover, we have proposed a simple correction method which improves the accuracy and the rate of convergence of the presented method.
Although in this paper we considered with the boundary conditions , the proposed method may be extended to the general case of with boundary conditions, and , as long as the existence of the solution is assured under some additional restrictions. In addition, we leave theoretical analysis of the error and the radius of convergence for a future work.
Acknowledgments
The author would like to show his sincere gratitude to Professor Anthony Peirce who invited the author to work as a visiting scholar at the University of British Columbia. This research was supported by Basic Science Research program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0004716).