Simplified Application of Lighthill's Uniformization Technique Using Lagrange Expansion Formulas

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1 /. Inst. Maths Applies (1967) 3, Simplified Application of Lighthill's Uniformization Technique Using Lagrange Expansion Formulas E. DALE MARTIN National Aeronautics and Space Administration, Ames Research Center, Moffett Field, California, U.S.A. [Received 8 September, 1966] Application of Lighthill's uniformization technique in a class of singular perturbation problems is simplified by using formulas derived from Lagrange's expansion. Uniformly valid solutions are given directly in terms of the singular perturbation solutions. 1. Introduction IN one class of singular perturbation problems, expanding the unknown solution in powers of a small parameter artificially shifts a singularity to a location within the defined domain of the independent variables, the perturbation solution is then not a valid approximation. The essential mechanism of the method developed by M. J. Lighthill (Lighthill, 1949, 1961; Tsien, 1956; Van Dyke, 1964) for rendering approximate solutions to physical problems uniformly valid, is a transformation of variables that moves the singularity back to the proper location of its occurrence in the exact solution function, and out of the desired physical domain. A typical physical problem this type of non-uniformity occurs is in the small-perturbation solution of thin-airfoil theory near a rounded leading edge (Lighthill, 1951; Van Dyke, 1964). The purpose of this communication is to show that part of the rather tedious procedure normally followed in applying Lighthill's technique can be eliminated and, at least in many problems, the Lighthill uniformization applied directly with use of formulas derived from Lagrange's expansion. 2. Lagrange's Expansion Lagrange's expansion is a generalization of Taylor's expansion (e.g., Whittaker & Watson, 1927; Sack, 1965) the independent variable is defined by an implicit equation. The usual form for Lagrange's expansion of a function/(x) is x = xtfd = Z+egQc). (lb) In an obvious generalization of (1), we may include a parameter a: (2a)

2 APPLICATION OF LIGHTHUX'S UNIFORMIZATION TECHNIQUE 17 x = x(z,e,<x) = +eg(x,a). (2b) [For example, starting with (2b), one can derive (2a) from Taylor's series in a manner closely following Laplace's derivation of (la), outlined briefly by Sack (1965).] Equations (2) may now be specialized to the case a = e, so that Equation (3b) has the expansion If now and equations (3a) and (3c) become f(x,e) = ^T ^ { ^ j (3a) x = x^,e,e) = Z+eg(x,e). * = ^t^[^' E)] - (3b) Ax,e) = h{x)+emx)+e2mx) +... (4a) (3C) g(x,e) = ftw+^w+e^w + -, (4b) x = Z+sg^+B^g^+gMg'm+OiE 3 ). ), ' (5a) (5b) At this point, equations (5) are valid (at least asymptotically as e-*0) for arbitrary functions/and g that can be expanded as in (4). Equations (4) and (5) will be seen to be useful for a direct application of LighthilTs uniformization. 3. Lighthill's Uniformization In solving a differential equation with boundary conditions for f(x,e), one may assume a perturbation expansion in the form of (4a). One substitutes the assumed expansion into the differential equation and boundary conditions and collects the coefficients of various powers of e to solve for the respective/ n (;c). In a class of problems to which Lighthill's technique applies, the f n (x) becomes infinite at some x (without loss of generality, say, at x = 0). The Lighthill's uniformization procedure is then to expand both/and x in powers of e and in terms of a common independent variable : f(x,e) = h 1 (,0+Bh 2 (O+e 2 h^)+..., (6a) x = Z+ex 1 tf)+e2x 2 (.Z)+..., and to choose x\{^), X2(<0> etc., by requiring that: (6b) the h n (Q be no more singular at = 0 than Ai(«J). (7) The standard procedure, after observing the singular behaviour in the f n (x), is to substitute equations (6) back into the problem, collect powers of e, solve the respective

3 18 E. DALE MARTIN differential equations with boundary conditions for the respective terms hj ), and then apply rule (7) to determine xj ). Instead, one can observe now, from comparison of (5) with (6), that etc., and (8a) (8b) ), (8c) (9a) (9b) etc. ( the/, are already determined), and simply specify the g n ( ) in equations (8) so that (7) is satisfied. By this approach the usual tedious procedure of substituting (6a) and (6b) into the differential equation and boundary conditions and again solving the problem in the new form to determine the uniformly valid solution has been circumvented, but Lighthill's uniformization has been accomplished. 4. A Completely Specified, Uniformly Valid Solution The specification of # ( ) in (8) to satisfy (7) is not unique (e.g., see Van Dyke, 1964, p. 103). However, by making certain predetermined specifications of g n ( ) that satisfy Lighthill's condition, (7), one can establish formulas for a completely determined, uniformly valid solution, once and for all, for problems in the class of interest. One such specification can be made by requiring that all the hjj;) vanish identically for n ^ 2. From equations (3a), (3c), (4a) and (4b), one obtains f(x,e)=m0+h, (10a) and.?ihl?/hl oo i fin i CV oo "Tn oo "\ <1Ob) Specification of h n {^) = 0 for n'z. 2 then determines explicitly g^). The final result for the complete, uniformly valid solution to replace the singular perturbation expansion f(x,e)= f et-'ux), (11) A*,<0=/i > (12) x is given by (10b) and the # (< ) are evaluated explicitly by setting the coefficients h n in (10c) equal to zero. For convenience, the first few terms of the complete explicit solution are written out: (13a)

4 APPLICATION OF LIGHTHIIX'S UN1FORM1ZATION TECHNIQUE 19 and -, (13b) etc. More specifically to order B 2 : (13e) /(*, ) = m), (14a) and the/, are the functions in the original singular perturbation expansion (11). The specification of hjg) = 0 for n^ 2, which results in (12), (13) and (14), may not be the most advantageous for all problems in the class of interest, for example, because of slow convergence. However, it does have several advantages: (a) The resulting uniformly valid solution consists of only one expansion, (14b), to which an Euler's transformation could be applied to improve convergence, if desired (e.g., Meksyn, 1961, pp ). (b) If/i( ) is simple enough, one can solve (14a) for as a function of/, then find x as a function of <f; from (14b), and, hence, obtain x explicitly as a function of/for the uniformly valid solution. These same advantages would be realized if, instead of requiring h n ( ) = 0 for n i> 2, one specifies /* ( ) = hi( ) = /i({) for all n, since one would then obtain f(x,e)=mq/(l-e). The corresponding appropriate expansion for x(,e) is given by (10b) or (13b), with the gj ) determined by equations like (13c), (13d), and (13e) with/2,/3 and/, (but not their derivatives) in the latter three equations replaced by/ 2 -/i,/3-/i and/ 4 -/i. 5. A Simple Example As a simple illustrative example, consider (x+ef)f'+f= 1, (0 x l); (15a) x=l, /=2. (15b)

5 20 E. DALE MARTIN The straightforward expansion solution has the form of (11) with (16) / 3 (x)= -l-5x x" 3 + l-5x" x" s ; etc. J Each of the fj^x) is more singular at x = 0 than the preceding term. With application of equations (14), which incorporate Lighthill's technique, one obtains directly the uniformly valid solution: /(X,E) = l+{-i, x = 5+s(l-5$ 1 O-55" 1 )- ( 17b ) In this example problem, it happens that this uniformly valid solution obtained from (14a) and two terms of (14b) is the exact solution, which may be put into the form (cf. Van Dyke, 1964, pp ): 6. Concluding Remark [GMTHS Lighthill's uniformization has been accomplished directly by substituting the singular perturbation solution, having the form of equation (11), into formulas (14), derived herein from LagTange's expansion. REFERENCES LiGHTHni, M. J Phil. Mag. 7, 40, LIGHTHIIX, M. J Aeronaut. Quart. 3, LIGHTHILL, M. J Z. Flugwiss. 9, MEKSYN, D New Methods in Laminar Boundary-Layer Theory. New York: Pergamon Press. SACK, R. A /. Soc. ind. appl. Math. 13, TSIEN, H. S In Advances in Applied Mechanics, Vol. IV, New York: Academic Press. VAN DYKE, M Perturbation Methods in Fluid Mechanics. New York: Academic Press. WHTTTAKER, E. T. & WATSON, G. N A Course of Modern Analysis. 4th edn., Cambridge University Press. (17a)