perturbation - 23 Perturbation theory From this book and...

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Unformatted text preview: 23 Perturbation theory From this book and most other books on mathematical physics you may have ob tained the impression that most. equations in the physical sciences can be solved. This is actually not true; most textbooks (including this book) give an unrepresenw tative state of affairs by only showing the problems that can be solved in closed form. It is an interesting paradox that our theories or the physical world become more accurate. the resulting equations become more difficult to solve. In classical mechanics the problem of two particles that interact with a central force can be solved in closed form. but the threewbody problem in which three particles inter— act has no analytical solution. In quantum mechanics. the one—body problem of a particle that moves in a potential can be solved for a limited number of situations only: for the free particle, the particle in a box, the harmonic oscillator. and the hydrogen atom. In this sense the oae~body problem in quantum mechanics has no general solution. This shows that as a theory becomes more accurate. the re sulting complexity of the equations makes it often more. difficult to actually find solutions. One way to proceed is to compute numerical solutions of the equations. Comput— ers are a powerful tool and can be extremely useful in solving physical problems. Another approach is to find approximate solutions to the equations. in Chapter l2. scale analysis was used to drop from the equations terms that appear to be irrel- evant. In this chapter, a systematic method is introduced to account for terms in the equations that are small but that make the equations difficult to solve. The idea is that a complex problem is compared to a simpler problem that can be solved in closed form. and to consider these small terms as a perturbation to the original equation. The theory of this chapter then makes it possible to determine how the solution is perturbed by the perturbation in the original equation; this technique is called perrurbarirm theory. A classic reference on perturbation theory has been written by Nayfeh £747]. The book by Bender and Orszag [ 14} gives a useful and illustrative overview of a wide variety of perturbation methods. 412 i. . 23.] Regular perturbation theory 41 3 The central idea of perturbation theory is introduced for an algebraic eqoation a Section 231. Sections 23.2. 23.3, and 23.5 contain important applications of erturbation theory to differential equations. As shown in Section 23.4, perturbation theory has a limited domain of applicability. and this may depend on the way the 'erttirbation pi‘obtem is formulated. Finally, it is shown in Section 23.? that not Every perturbation problem is well behaved; this Eeads to singular perturbation heory. Chapter 24 is devoted to the asymptotic evaluation of integrals. 23.1 Regular perturbation theory an introduction to perturbation theory let us consider the following equation — 4x2 + 4x = 0.01. (23.1) gfleet us for the moment assume that we do not know how to find the roots of a third ijgorder polynomial, so we cannot solve this equation. The problem is the small term liliim on the rightwhand side. Ifthis tern: were equal to zero, the resetting equation can jibe solved; x3 M 4x2 ~l~ 4x = O is equivaient to 1:063 ~ 4x +4) : x(x — 2)2 : 0. ijeiwhieh has the solutions )5 m 0 and x m 2. in Figure 23.1 the polynomial of (23. l) Esis shown by the thick solid line; it is indeed equal to zero for :r a l) and x m 2. The problem that we face is that the right—hand side of ('23. E) is no! equal to ifiero. ln perturbation theory one studies the perturbation of the solution under a :igperturbation of the original equation. in order to do this, we replace the original iiequation {23.1) by the more general equation — 4x2 + 4x = g. (23.2) grill/hen e = 0.01 this equation is identical to the original problem. while for a 2 0 it educes to the unperturbed problem that. we can solve in closed form. It may appear Fig. 23.1 The polynomial X3 - 4x2 ~l~ 4); (thick solid line) and the lines a m 0.15 (dotted line) and r: 2 MO. 15 (dashed line). 414 Perturbation them-"r that we have made the problem more complex because we stilt need to solve the same equation as our original equation. but. it now contains a new variable 8 as well! However, this is also the strength of this approach. The solution of (23.2) is a function oils so that .r = .t'(t~?). (23.3) In Section 3.1 the Taylor series was used to approximate a function fit} by a power series in the variable- .r: I ‘ .1“2 If ‘ rm 2 for + x 51o :01+ —‘ i (x : 0) - - -. (3.11) (1.x 2! (hr When the sotutton .t‘ of (23.2) depends in a regular way on 8. this solution can also be written as a similar power series by making the substitutions ,r —> s and f --+ .r in (3.1 l): ’P ’) (Li- 5* (111' its“) = “0) + ‘9 at“? Z 0) + i (it-:3 (e=0)+---. (23.4) This expression is not very asefu] because we need the derivative (1.x Me and hi gher derivatives (Pit/dis” as well: 111 order to compute these derivatives we need to find the solution .rte) first. but this is just what we aretrying to do. There is. however. another way to determine the series (23.4}. Let us write the soiution rte) as a power series in e .t‘(e_) 2411+ ex; + 821'} + - - -. (23.5) The coefficients 1,, are not known at this point. but once we snow them the solution 1' can be found by inserting the numerical value 8 = 0.01 . In practice one truncates the series (23.5); it is this truncation that makes perturbation theory an approximation. When the series (23.5) is inserted into (23.2) one. needs to compute. x: and X3 when .r is given by (23.5). Let us first consider the .rZ-term. The square of a sum of terms is given by (cr+b+c‘+---)2=nl+bz+c3+~- 23.. miw 2a!) wt 2w? +21%: + - - - . ( 6) Let us apply this to the series (23.5) and retain only the terms up to order 52. this gives 7 "1 2 ~ 2 2 4 2 (.m—l—sx; +8 .rg+~-) :xo—t—e .ri +8.1“: +--- + 2mm: % 283x01: + 283,111.11: + - - A . (23.7) If we are only are interested in retaining the terms tip to order 92. the terms 54x3 and 28’.rl.r3 in this expression can be ignored. Coltecting terms oi’equai powers of 23.] Regular pei'rtu'bmion theory 41 5 e ther: gives 3 '3 (X0 + 8X5 ‘l— 8-x; 7L ' ‘ ')- : X5 ‘i” 2€XOX§ ‘i” 82 (X? "i‘ 2XOXg) “i” 0(53). A similar expansion in powers of 8 can be used for the term x3. This expansion is based on the identity (_a—§—b+('-l—---_)3 mrtg+b3+rj+s~ + 3612b er 3th)”2 + 3(126‘ + 36M?" + 31926 + 33963 + A — -. (23.9) Problem 21 Apply this identity to the series (23.5), collect together all the terms with equal powers of s and show that up to order 83 the result is given by a, _’ (I0 + “I + 52x2 + . . .) w x3 + Bangs"; + (W; + x5e) + 0(53). (23.10) Problem b At this point we can express all the terms in (23.2) in a power series of Insert (23.5), (23.82), and (23.10) into the original equation (23.2) and coliect together terms of equal powers of e to derive that m 41:; «t» 4A1; we 5 (3.)c5xl w 8mm + 4x1 w l) + 82(31‘01‘12 + .xfixz - 41“? — 8mm + 4le + ‘ " m 0 - (23-11) ln this and subsequent expressions the dots denote terms of order 0(83). The term —E in the term that multiplies 8 comes from the righbhand side of (23.2). At this point we use that .9 does not have a fixed value, butthat it can take any value within certain bounds. This means that expression (23.1 1) must be satisfied for a range of values of This can only be the case when the coefficients that multiply the different powers 8" are equal to zero. This means that 23.E l) is equivalent to the following system of equations which consists of the terms that multiply the 0 terms 5 . 5' and 82 respectively: 0(E)~lermst x3 — 4x5 + 4x0 = 0r O(E}—terms: 3x5,“ e 8,159,171 +4xE e E m e, (23.12) 0(53)—terms: 3xoxl'3 + .rgxg — 4x13 —— 81:92:; + 4.36; m (l . You may wonder whether we have not made the problem more complex. We started with a single equation for a singEe variable x, and now we have. a system ot‘eoupled equations for many variables. Howevera we could not solve (_ 23.2) for the single variable x, while it is not difficult to soEve (23.0). 41 6 Pertirrbnrion r/iem'_\.' Problem c Show that (23.12) can be rewritten in the following form: The first equation is simply the unperturbed problem. this has the solutions .ru : i3 and in m 2. For reasons that will become ciearin Section 23.7 we focus here on (in solution If) : 0 only. Given .rn. the parameter .l‘] 't'ollows from the second equation because this is a linear equation in rt. The last equation is a linear equation in the unknown A“; which can easily be soiyed once to and M are known. Problem d Sol\re(23.13) in this way to show that the solution near .r z 0 is giycn by 1 i 1- r is - ( ls.) W L In K It : Now we are close to the final solution of our problem. The coefficients of the previous expression can be. inserted into the perturbation series (23.5) so that the solution as a function of r: is given by 1 1 .. . .' :0+3 7.9+ =5fi+ O . 23.15% ‘ 4 16“ < J ( At this point we can revert to the original equation (23.1) by inserting the nurnericai value 5‘ = 0.01. which gives: 1 1 1 _ .\' = i x 10—“ A x 10—4 + O( 10—“) = 0.002506. {23.16: 4 16 it should be noted that this is an approximate solution because the terms ol‘ order 5:3 and higher have been ignored. This is indicated by the term ONO—b) in (23.16}. Assuming that the error made by truncating the perturbation series is of the same order as the first term that is truncated. the error in the solution (23.16) is ol‘ the order 10"". For this reason the number on the right-hand side of (23.16) is given to six decimals; the last decimal is of the same order as the truncation error. 11' this result is not sufficiently accurate for the application that one has in mind. then one can easily extend the analysis to higher powers 5-)” in order to reduce the truncation error ol‘thc truncated perturbation series. Although the algebra resulting: from doing this can be tedious. there is no reason why this analysis cannot be extended to higher orders. A truly formal anal ysis ol‘ perturbation problems can be difficult. For example. the perturbation series (23.5) converges only for sufficiently small values of‘rs. It is 23. 2 Born approximation 4 l 7 often not ciear whether the employed vaiue of s (in this case a : 0.01 ) is sut‘ticientiy small to ensure convergence. Even when a perturbation series does not couverge for a given vaiue of one can often obtain a use-fat approximation to the solution by truncating the perturbation series at a suitabiy chosen order {£41. In this case one speaks of an nsyrnptotirr series. When one has obtained an approximate solution of a perturbation prohtem, one can sometimes substitute it back into the original equation to verify whether this solution indeed satisfies the equation with an acceptable accuracy. For example, inserting the numerical value x = 0.002 506 in (23. E) gives 3'3 7 4x? +4x m 0.009 9989 : 0.01 m 0.00000i i . (23.17) This means that the approximate solution satisfies (23. i") with a relative error that is given by 0.000001 1 /0.0E = 10""4. This is a very accurate resttit given the fact that only three terms were retained in the perturbation anaiysis of this section. 23.2 Born approximation in many scattering problems one wants to account for the scattering of waves by the heterogeneities in the medium. Usually these prohietns are so complex that they cannot be solved in ciosed form. Suppose one has a background medium in which scatterers are embedded. When the background medium is sufficientiy simple, one can soive the wave propagation problem for this background medium. For example, in Section 19.3 we computed the Green’s function for the Helmholtz equation in a homogeneous medium. in this section we consider the Helmhoitz equation with a variable velocity ("(1‘) as an exampie of the appiication of perturbation theory to scattering problems. This means we consider the wave lield p(r. w) in the frequency domain that satisfies the following equation: ’7 (t)— Vipfl". to) + [)(I‘. 0)) m Str. to} . (23.l8) (:3 r) in this expression S(r. w) denotes the source that generates the wave field. In order to facilitate a systematic perturbation analysis we decompose l /c*3(r) into a term l [65 that accounts for a homogeneous reference model and a perturbation: l 1 a = “; it + 8mm}. (23.t9) ("(1‘) ('6 in this expression 8 is a smali parameter which measures the strength of the het— erogeneity. The function 118‘) gives the spatial distribution of the heterogeneity. {Tombining the previous expressions it follows that the wave fieid satisfies the 23.7 Singular perturbation theoifiv 433 how [3? that for elastic waves aiso the disptacernent is inverser proportional to 1/ where c is the propagation vetocity of the elastic wave under consideration.) _ The fact that the gronnd motion is inversely proportional to the square—root of the impedance is one of the factors that made the i985 earthquake along the west coast of Mexico cause so much damage in Mexico City. This city is constructed on Soft sediments which have fitted the swamp onto which the city is built. The smail value of the associated eiastic impedance was one of the causes of the extensive damage in Mexico City after this earthquake. 23.7 Singular perturbation theory Section 231 we anaiyzed the behavior of the root of the equation x3 -— 4x2 ~t~ {ix m 8 that was located near x = {3. As shown in that section, the unperturbed emblem also has a root x x 2. The roots .r m 0 and x m 2 can be seen graphicaily in Figure 23.3. because for these vatues of x the polynomiai shown by the thick lid line is equal to zero. in Figure 23.1 the value 8 = +0.15 is shewn by a dotted line whiie the value a 2: m 0.15 is indicated by the dashed line. There is a profound difference between the two roots when the parameter s is nonzero. The too: near .‘r m 0 depends in a continuous way on e, and (23.2) has for the root near .r 2 0 a soiution regardless of whether 8 is positive or negative. This situation is compieteiy different for the root near .1" m 2. When 8 is positive (the dotted iine). the poiynornial has two intersections with the dotted tine. whereas when 8 is negative the polynomial does not intersect the dashed tine at all. This means that depending on whether a is positive or negative, the solution has two or zero _ otutions, respectively. This behavior cannot be described by a regatta pettorbation Series of the form {23.5) because this expansion assigns one soiation to each value of the perturbation parameter I Let us first diagnose where the treatment of Section 23.] breaks down when we pply it to the root near x x 2. " roblem a lnsert the unperturbed solution .rg m 2 into the second fine of (23.13) and show that the resulting equation for x] is 0 - x1 x l. (23.83) his equation obviously has no finite solution. This is related to the fact that the ingent of the poiynornial at x = 2 is horizontat. First—order perturbation theory ffectiveiy repiaees the polynomiai by the straight line that is tangent to the poly— orniai. When this tangent line is horizontai, itcan never have a vatue that is nonzero. 434 Pertio'brirroa theory Fig. 23.4 Graph of the l‘unetion Vile. This means that the regular perturbation series (23.5) is not the appropriate wa; to study the hehayitn‘ of the root near .t x 2. In order to lind out how this row behaves. let as set Problem b Show that under the substitution (23.84) the original problem {23.23: transforms to y" +0 2}'“' z g. (23.8%: We will not yet carry out a systematic perturbation analysis. but we will first deter mine the dependence of the solution _\_‘ on the parameter 5:. For small values ol' 5.: this parameter r is also small. This means that the term y} can be ignored with respeeg to the term yi. L'nder this assumption (23.85) is approximately equai to 2y: % 5-: so that. y E This means that the solution does not depend on integer powers oi e as in the perturbation series (23.5}. but that it does depend on the square-root of The square—toot ol‘ 8 is shown in Figure 23.4. Note that for to : 0 the tangent of this carve is vertical and that for a < O the t'unetion is not defined for reai values oi' a This reflects the fact that the roots near ,r : 2 depend in a very different way on 2 than the root. near x : 0. We know now that a regular perturbatitm series (23.5) is not the correct tool to ase to analyze the root near .r : 2. How-even we do not yet know what type ot perturbation series we should use for the root near .t‘ z 2; we only know that. the perturbation depends to leading order on That is. let us make the following: ' When one allons a ernnples solution .rtea ol' the equation there are always tuo roots near 4' t 2. HU\\C\'L‘E. these complex solutions also display a l'undamental Change in their behavior \\ hen it 0. \i hieh is Characterize- by a bifurcation. 23. 7 Singular perturbation theory 4.35 substittttion: x 2 2 + (23.86) obiem c Insert this solution into (23.2) and show that 5 satisfies the following equation: ~%— 2:3 = 1. (23.87) iii-\Iow we have a new perturbation problem with a small parameter. However, this its-mail parameter is not. the originai perturbation parameter a, but it is the square—root The perturbation problem in this section is a Singular perturbation problem. 3:51:71 a singular perturbation problem the solution is not a weiiwbehaved function of tithe perturbation parameter. This has the resuit that the corresponding perturbation if'series cannot be expressed in powers 8“, where n is a positive reai integer. instead. iiiiegative or fractional powers of r: are present in the pertnrbation series of a singular Eiipertorbation problem. a 'oblem (1 Since the small parameter in {23.87) is it rnakes sense to seek an expansion of r; in this parameter: : w tn + 81/33: + at; + v - -. (23-88} Coltect together the coefficients of equal powers of a when this series is in— serted into (23.87} and show that this leads to the foilowing equations for the coefficients in and :1: 0(l)—tertns: — 'l = 0, t, 2?. E 0(81"*)Mter.rn.s: + 4:051 m (l. ( j 8)) Problem e The first equation of {23.89) obviously has the solution at; m :‘e i / Show that for both the plus and the minus signs 2:; m m i /'2.I";Use these restiEts to derive that the roots near )5 x 2 are given by i 1 m x z 2 :e —fi m we + 0am. (23.90) It is illustrative to compute the. numericai values of these roots for the originai “emblem (23.1) where E.‘ 3 0.0L: this gives for the two roots: ,1: x 1.924 and A: = 2.065. (23.91) these numbers oniy three decimals are shown. The reason is that the error in the ncated perturbation series is of the order of the first truncated term, hence the 436 Perturbation theory error is of the order (0.0] )‘l’ll : 0.001. When these solutions are compared n at: perturbation soiution (23.16) for the root near .r : (1. it is striking that the shift: perturbation series for the root near X = 2 converges mach less rapidly the: reguiar perturbation series (23.16) for the root near .v : 0. This is a Coast-ins .- of the fact that the solution near .r : 2 is a perturbation series in fit: [1.1 i than at: 0.1M). When the roots (23.91) are inserted into the polynomial (2}- t s :' following solutions are obtained for the two roots: X 1.924: x3 A 4;? + 4x m 0.0: 11 z; 0.0i +0.0011. 2.005 ; .55 a 4x3 + 4x : 0.0087 2 0.01 41.0032. X Note that these resnits are much less accurate than the corresponding result ifs: for the root near X x 0. Again this is a consequence ofthe singular behavior oi roots near .v = 2. The singular behavior of the roots of the polynomial {23.1) near .r 2: L? a responds to the fact that the solution changes in a discontinuous way when perturbation parameter 8 goes to zero. it follows from Figure 23.1 that for the tie turbation problem in this section the problem has one root near .r m 2 when r‘ t; s has no roots when a < 0 and there are two roots when 8 > 0. Such a (.llSCDl'Hllitti change in the character of the solution also occurs in fluid mechanics in which it:- eqaation of motion is given by E) t p v} a t «1 V - (pvv‘) : avg" + F. (i i.fi'\~ In this expression the viscosity ofthe fluid gives a contribution ,ttVBV. where [t i viscosity. This viscous term contains the highest spatial derivatives of the velour that are present in the. equation. When the viscosity ,0: goes to zero. the eqiiziiiw for fluid [low becomes a first order differential equation rather than a second (this: differential equation. This changes the number of boundary conditions that needed for the soiution. and hence it drastically affects the mathematical stt‘Ltcitt:= of the solution. This has the effect that boundary-layer probiems are. in generaé singular perturbation problems | 1 1 1 I. When waves propagate. through an inhomogeneous medium they may be focusm: onto focaé points or focal surfaces [16]. These regions in space where the wax; amplitude is large are called caustics. The formation of caustics depends on r! where E is a measure of the variations in the wave veiocity [58. 102]. The non integer power of 5: indicates that the formation of caustics constitutes a singui..;; perturbation problem. ...
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