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.
The quasilinearization method (QLM) was originally developed by Bellman and Kalaba [
Bellman and Kalaba [
An often noted disadvantage of quasilinearization is the instability of the method whenever a poor initial guess is chosen, Tuffuor and Labadie [
The spectral homotopy analysis method was introduced by Motsa et al. [
Abbasbandy [
The structure of this paper is as follows. Section
In this section we present a framework for the derivation of general QLM-SHAM iterative schemes for solving one-dimensional nonlinear differential equations. We consider a general
Following [
Here
The nonlinear operator
By differentiating the zeroth-order equations (
After obtaining solutions for (
The SHAM solution is said to be of order
The initial approximation
We note that with
We note that the iterative scheme (
This produces the iteration scheme
For
This produces the iteration scheme
In general, for any
Thus, a general scheme when the SHAM is truncated at order
In this section we demonstrate how the numerical schemes derived in the previous section may be used to solve the Falkner-Skan equation:
In this scheme we set
For this scheme we set
The complexity of the defining equations increases with the order of the scheme. For Scheme-2 we have
The boundary conditions (
In this section we present solutions of the Falkner-Skan equation ( Blasius flow: Pohlhausen flow: Homann flow:
To assess the accuracy and performance of our schemes, the numerical results were compared to the recently reported results of Ganapol [
The comparison between the present findings and the results in the literature is made for the skin friction which is proportional to
Comparison between the computed values of the skin friction
iter. |
|
Error |
---|---|---|
SHAM | ||
| ||
1 | 0.361245275076317576714175486031 | 0.02918793886112127777699542402 |
10 | 0.332060294018222201221920675183 | 0.00000295780302590228474061317 |
20 | 0.332057337274670714006530763206 | 0.00000000105947441506935070120 |
30 | 0.332057336215760109582560249181 | 0.00000000000056381064538018717 |
40 | 0.332057336215196653344046405966 | 0.00000000000000035440686634396 |
60 | 0.332057336215196298937262729762 | 0.00000000000000000000008266775 |
| ||
Scheme-0 | ||
| ||
1 | 0.36124527510805664031836423508 | 0.02918793889286034138118417307 |
2 | 0.33293906079206190667160822082 | 0.00088172457686560773442815881 |
3 | 0.33205878995514977263006366166 | 0.00000145373995347369288359965 |
4 | 0.33205733621994973222724960736 | 0.00000000000475343329006954535 |
5 | 0.33205733621519629893723540415 | 0.00000000000000000000005534214 |
6 | 0.33205733621519629893718006201 | 0.00000000000000000000000000000 |
| ||
Scheme-1 | ||
| ||
1 | 0.33849743020925601396026175681 | 0.00644009399405971502308169480 |
2 | 0.33205889444389263880627992358 | 0.00000155822869633986909986157 |
3 | 0.33205733621519633877777093517 | 0.00000000000000003984059087316 |
4 | 0.33205733621519629893718006201 | 0.00000000000000000000000000000 |
| ||
Scheme-2 | ||
| ||
1 | 0.33398877527020321822942828158 | 0.00193143905500691929224821957 |
2 | 0.33205733679309573625968418056 | 0.00000000057789943732250411855 |
3 | 0.33205733621519629893718006201 | 0.00000000000000000000000000000 |
| ||
[ |
Table
Comparison between the computed values of the skin friction
iter. |
|
Error |
---|---|---|
SHAM | ||
| ||
1 | 1.15819390472196206795661617787 | 0.00349336634271053893831861687 |
5 | 1.15470068226816961126259064868 | 0.00000014388891808224429308768 |
10 | 1.15470053837716000145607336675 | 0.00000000000209152756222419425 |
20 | 1.15470053837925152901716901667 | 0.00000000000000000000112854434 |
25 | 1.15470053837925152901829759222 | 0.00000000000000000000000003121 |
30 | 1.15470053837925152901829756100 | 0.00000000000000000000000000000 |
| ||
Scheme-0 | ||
| ||
1 | 1.1581939047219620679566161779 | 0.0034933663427105389383186169 |
2 | 1.1547034510528929300844093465 | 0.0000029126736414010661117855 |
3 | 1.1547005383817023293095010719 | 0.0000000000024508002912035109 |
4 | 1.1547005383792515290182994735 | 0.0000000000000000000000019125 |
5 | 1.1547005383792515290182975610 | 0.0000000000000000000000000000 |
| ||
Scheme-1 | ||
| ||
1 | 1.1544901934962778016810840055 | 0.0002103448829737273372135555 |
2 | 1.1547005383778620432865956388 | 0.0000000000013894857317019222 |
3 | 1.1547005383792515290182975610 | 0.0000000000000000000000000000 |
| ||
EXACT: 1.1547005383792515290182975610 | ||
[ |
Table
Comparison between the computed values of the skin friction
iter. |
|
Error |
---|---|---|
SHAM | ||
| ||
1 | 1.335633919867798255673072885920 | 0.023696225987993120191426715220 |
10 | 1.311933330446726656235176983180 | 0.000004363433078479246469187519 |
20 | 1.311937690198936556741381203220 | 0.000000003680868578740264967481 |
30 | 1.311937693875056506843830283790 | 0.000000000004748628637815886910 |
40 | 1.311937693879797836016540925840 | 0.000000000000007299465105244860 |
50 | 1.311937693879805123127203300860 | 0.000000000000000012354442869842 |
60 | 1.311937693879805135459418974120 | 0.000000000000000000022227196583 |
70 | 1.311937693879805135481601938190 | 0.000000000000000000000044232511 |
| ||
Scheme-0 | ||
| ||
1 | 1.3356339198662404626942038769 | 0.0236962259864353272125577062 |
2 | 1.3121878643609977795027970207 | 0.0002501704811926440211508500 |
3 | 1.3119377351892323008775816635 | 0.0000000413094271653959354928 |
4 | 1.3119376938798066169255039256 | 0.0000000000000014814438577549 |
5 | 1.3119376938798051354816461707 | 0.0000000000000000000000000000 |
| ||
Scheme-1 | ||
| ||
1 | 1.3064680181439000004910779728 | 0.0054696757359051349905681979 |
2 | 1.3119375731135522096139489286 | 0.0000001207662529258676972421 |
3 | 1.3119376938798051354785690801 | 0.0000000000000000000030770906 |
4 | 1.3119376938798051354816461707 | 0.0000000000000000000000000000 |
| ||
Scheme-2 | ||
| ||
1 | 1.3136564701352183697646278650 | 0.0017187762554132342829816943 |
2 | 1.3119376938938730064725895040 | 0.0000000000140678709909433333 |
3 | 1.3119376938798051354816461707 | 0.0000000000000000000000000000 |
| ||
[ |
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.