1B Methods 1. 1B METHODS ECTURE NOTES Richrd Jozs, DAMTP Cmbridge rj31@cm.c.uk October 213 PART I: Fourier series, Self djoint ODEs
1B Methods 2 PREFACE These notes (in four prts cover the essentil content of the 1B Methods course s it will be presented in lectures. They re intended be self-contined but they should not be seen s full substitute for other good textbooks, which will contin further explntions nd more worked exmples. I m grteful to the previous lecturer of this course, Dr Colm- Cille Culfield, for mking his notes vilble to me; they were very useful in forming bsis for the notes given here. The term mthemticl methods is often understood to imply kind of prgmtic ttitude to the mthemtics involved, in which our principl im is to develop ctul explicit methods for solving rel problems (mostly solving ODEs nd PDEs in this course, rther thn crefully justified development of n ssocited mthemticl theory. With this focus on pplictions, we will not give proofs of some of the theorems on which our techniques re bsed (being stisfied with just resonbly ccurte sttements. Indeed in some cses these proofs would involve formidble fory into subjects such s functionl nlysis nd opertor theory. This mthemticl methods ttitude is sometimes frowned upon by pure-minded mthemticins but in its defence I would mke two points: (i developing n bility to pply the techniques effectively, provides relly excellent bsis for lter ppreciting the subtleties of the pure mthemticl proofs, whose considerble bstrctions nd complexities if tken by themselves, cn sometimes obfuscte our understnding; (ii much of our gretest mthemtics rose in just this cretively plyful wy of cvlierly pplying not-yet-fully-rigorous techniques to obtin nswers, nd only lter, guided by gined insights, developing n ssocited rigorous mthemticl theory. Exmples include mnipultion of infinite series (without worrying too much bout exct convergence criteri, use of infinitesimls in the erly development of clculus, even the notion of rel number itself, the use of the Dirc delt function (llowing infinity s vlue, but in controlled fshion nd mny more. Thus I hope you will pproch nd enjoy the content of this course in the spirit tht is intended. Richrd Jozs October 213
1B Methods 3 1 FOURIER SERIES The subject of Fourier series is concerned with functions on R tht re periodic, or equivlently, re defined on bounded intervl [, b which my then be extended periodicllly to ll of R. A function f on R is periodic with period T if f(t + T = f(t for ll t (nd conventionlly we tke the smllest such T. Thus f is fully specified if we give its vlues only on [, T or ny other intervl of length t lest T. Bsic exmple: S(t = A sin wt nd C(t = A cos wt. A is the mplitude. Interpreting the vrible t s time, we hve: period T = 2π/w = time intervl of single wve, frequency f = w/2π = 1/T = number of wves per unit time, ngulr frequency f ng = 2π/T = w = number of wves in 2π intervl (useful if viewing t s n ngle in rdins. Sometimes the independent vrible is spce x e.g. f(x = A sin kx nd we hve: wvelength λ = 2π/k = sptil extent of one wve, wvenumber 1/λ = k/2π = number of wves in unit length, ngulr wvenumber k = 2π/λ = number of wves in 2π distnce. Bewre: lthough w resp. k re ngulr frequency resp. wvenumber, they re often referred to simply s frequency resp. wvenumber, nd the terminology should be cler from the context. In contrst to the infinitely differentible trig functions bove, in pplictions we often encounter periodic functions tht re not continuous (especilly t, T, 2T,... but which re mde up of continuous pieces e.g. the swtooth f(x = x for x < 1 with period 1; or the squre wve f(x = 1 for x < 1 nd f(x = for 1 x < 2 with period 2. 1.1 Orthogonlity of functions Recll tht for vectors, sy 2-dimensionl rel vectors u = i + bj v = ci + dj we hve the notion of orthonorml bsis i, j nd inner product u v which we ll write using brcket nottion (u, v = c + bd (not to be confused with n open intervl! - the mening should lwys be cler from the context!. For complex vectors we use (u, v = c+b d where the str denotes complex conjugtion. u is normlised if (u, u = 1 nd u nd v re orthogonl if (u, v =. Consider now the set of ll (generlly complex-vlued functions on n intervl [, b]. These re like vectors in the sense tht we cn dd them (pointwise nd multiply them by sclrs. Introduce the inner product of two functions f, g : [, b] C s follows: (f, g = f (xg(xdx. (1 Note tht this even looks rther like n infinite dimensionl version of the stndrd inner product formul, if we think of the function vlues s vector components prmeterised
1B Methods 4 by x i.e. we multiply corresponding vlues nd sum (integrte them up. (f, f is clled the squred norm of f nd two functions re orthogonl if (f, g =. Note tht (f, g = (g, f so if either is zero then the other is too. Sometimes we ll restrict the clss of functions by imposing further boundry conditions (BCs tht re required to hve the following property: if f nd g stisfy the conditions then so does c 1 f + c 2 g for ny constnts c 1, c 2. Such BCs re clled homogeneous BCs nd the resulting clss of functions will lwys still form vector spce (i.e. be closed under liner combintions. Exmple. For functions on [, 1] the following BCs re homogeneous BCs: ( f( =, (bf( = f(1, (c f( + 2f (1 =. The following BCs re not homogeneous: ( f( = 3, (b f( + f (1 = 1. Importnt exmple of orthogonlity: On the intervl [, 2] consider S m (x = sin mπx C n (x = cos nπx m, n =, 1, 2,... Note tht S m (for m comprises m full sine wves in the intervl [, 2] (nd similrly for C n. To clculte their inner products we cn use the stndrd trig identities cos A cos B = 1 [cos(a B + cos(a + B] 2 sin A sin B = 1 [cos(a B cos(a + B] 2 sin A cos B = 1 [sin(a + B + sin(a B]. 2 We get for exmple, for m, n 2 (S m, S n = sin mπx nπx sin dx = 1 2 (m nπx cos dx 1 2 2 { if m n. = if m = n. More concisely, using the Kronecker delt 2 cos (m + nπx dx (S m, S n = δ mn m, n (2 so the rescled set of functions 1 S n (x for n = 1, 2,... is n orthonorml set. Similrly you cn esily derive tht for ll m, n = 1, 2,... we hve (C m, C n = δ mn (C m, S n = (S m, C n =. (3 Finlly consider the cses with m = or n = : S is identiclly zero so we exclude it wheres C (x = 1 nd it is esy to see (C, C = 2 (C, C m = (C, S m = for ll m = 1, 2,...
1B Methods 5 Putting ll this together we see tht the infinite set of functions B = { 1 2 C, S 1, C 1, S 2, C 2,...} is n orthogonl set with ech function hving norm. Indeed it my be shown (but not proved in this methods course... tht these functions constitute complete orthogonl set, or n orthogonl bsis for the spce of ll functions on [, 2] (or functions on R with period 2 in the sme sense tht i, j is complete orthogonl set for 2-dimensionl vectors it is possible to represent ny (suitbly well behved function s (generlly infinite series of functions from B. Such series is clled Fourier series. 1.2 Definition of Fourier series We cn express ny suitbly well-behved (cf lter for wht this mens periodic function f(x with period 2 s Fourier series: f(x = 1 2 + [ n cos + b n sin ], (4 where n nd b n re constnts known s the Fourier coefficients of f, (This expression pplies lso if f is complex-vlued function in which cse the coefficients n nd b n re themselves complex numbers we cn just tret the rel nd imginry prts of ny such complex-vlued f s seprte rel functions. The Fourier series expnsion will lso pply to (suitbly well behved discontinuous functions: if f is discontinuous t x then the HS of eq. (4 is replced by f(x + + f(x 2 where f(x + = lim ξ x f(ξ nd f(x = lim ξ x f(ξ re the right nd left limits of f s we pproch x from bove nd below respectively. Thus the Fourier series will converge to the verge vlue of f cross jump discontinuity. Indeed in our present context this provides convenient wy to re-define the vlue of function t bounded discontinuity e.g. we would replce the step function f(x = if x nd f(x = 1 if x > by the function g(x = if x <, g( = 1 nd g(x = 1 if x >. Often, this subtlety will be 2 glossed over, nd the left hnd side will just be written s f(x (s in eq. (4, with the behviour t bounded discontinuity being understood. Determining the n nd b n is esy by exploiting the orthogonlity of the sines nd cosines. In terms of our previous nottion of S m nd C n we cn write eq. (4 s f(x = ( 1 2 C + n C n (x + b n S n (x. (5 Consider now (S m, f = 2 f(x sin mπx dx.
1B Methods 6 Substituting the RHS of eq. (5 nd ssuming it is oky to swp the order of summtion nd integrtion, we get (S m, f = (S m, 1 2 C + n (S m, C n + b n (S m, S n. According to the orthogonlity reltions eqs. (2,3 ll inner products on RHS re zero except for (S m, S m which is. Hence we get (S m, f = b m i.e. b m = 1 2 Similrly by tking inner products of eq.(5 with C m we get m = 1 2 f(x sin mπx dx m = 1, 2,... (6 f(x cos mπx dx m =, 1, 2,... (7 The fctor of 1 in the 2 term of eq. (4 conveniently mkes the m formul eq. (7 vlid for m both zero nd nonzero (reclling tht (C, C = 2 but (C m, C m = for m. Remrks 2 f(x dx of f over its (i The constnt term /2 equls the verge vlue f = 1 2 period nd then subsequent sine nd cosine terms build up the function by dding in terms of higher nd higher frequency. Thus the Fourier series my be thought of s the decomposition of ny signl (or function into n infinite sum of wves with different but discrete wvelengths, with the Fourier coefficients defining the mplitude of ech of these countbly-mny different wves. (ii The rnge of integrtion in the bove formuls cn be tken to be over ny single period. Often it s more convenient to use the symmetricl rnge. (iii Wrning if we strt with function hving period T, be creful to replce in the bove formuls by T/2 (since bove, we wrote the period s 2! Dirichlet conditions So, wht is ment by well-behved function in the definition of Fourier series? Here it is defined by the Dirichlet conditions: periodic function f(x with period T is sid to stisfy the Dirichlet conditions if f is bounded nd hs finite number of mxim, minim nd discontinuities on [, T (nd hence lso T f(x dx is well-defined. Then we hve the theorem (not proved in this course: Bsic theorem: If f stisfies the Dirichlet conditions then f hs unique Fourier series s in eq. (4 with coefficients given by eqs. (7,6. This series converges to f(x t ll points where f is continuous, nd converges to the verge of the left nd right hnd limits t ll points where f is discontinuous. Smoothness nd order of Fourier coefficients According to the Dirichlet conditions, it is possible to estblish Fourier series representtion of certin kind of discontinuous function. More generlly it cn be shown tht the mount of non-smoothness is reflected by the rte t which the Fourier coefficients decy with n, s follows. Theorem. Suppose tht the p th derivtive of f is the lowest derivtive tht is discontinuous somewhere (including the endpoints of the intervl. Then the Fourier coefficients
1B Methods 7 for f fll off s O[n (p+1 ], s n. Thus smoother functions (i.e. lrger p hve coefficients flling off fster, nd hence better convergence properties of the Fourier series. Exmple Consider the swtooth function hving period 2, given by: f(x = x for x <, nd repeting periodiclly outside [,. Since the function is odd, we immeditely get n = for ll n. Integrtion by prts shows (s you cn check, tht b m = 2 mπ ( 1m+1, f(x = 2 [ ( πx sin π 1 ( 2πx 2 sin + 1 ( ] 3πx 3 sin +.... (8 This series is ctully very slowly convergent the smoothness prmeter p bove is zero in this exmple nd indeed the coefficients fll off only s O(1/n, s expected. In the figure we plot few of its prtil sums f N (x: f N (x N b n sin. Note tht the series converges to t x = ± i.e. to the verge vlue cross these jump discontinuities. The Gibbs phenomenon ooking t the prtil sums f N (x for the discontinuous swtooth function s plotted in figure 1, we cn see tht there is persistent overshoot t the discontinuity x = ±. This is ctully generl feture of Fourier series convergence ner ny discontinuity nd is clled the Gibbs-Wilbrhm phenomenon. It is illustrted even more clerly in figure 2, showing prtil sums for the squre wve function. These re pictoril illustrtions nd on exmple sheet 1 (question 5 you cn work through derivtion of the Gibbs phenomenon. Although the sequence of prtil sums f N, N = 1, 2,... of the Fourier series of function f (stisfying the Dirichlet conditions lwys converges pointwise to f, the Gibbs phenomenon implies tht the convergence is not uniform in region round discontinuity. [Recll tht f N converges to f pointwise if for ech x nd for ech ɛ > there is n integer N (which cn depend on x s well s on ɛ such tht f N (x f(x < ɛ for ll N > N. If for ech ɛ >, N cn be chosen to be independent of x, then the convergence is sid to be uniform.] Exmple/Exercise The integrl of the swtooth function: f(x = x 2 /2, x As n exercise, show tht the Fourier series representtion of this function is [ x 2 2 = 1 2 6 + 2 ( 1 n ] (nπ cos. (9 2 Note tht the coefficients fll off s O(1/n 2, consistent with the fct tht f is continuous but hs discontinuous first derivtive.
1B Methods 8 1.5 1.5.5 1 1.5 1.8.6.4.2.2.4.6.8 1 Figure 1: Plots (with = 1 of the swtooth function f(x = x (thin solid line nd the prtil sums f 1 (x (dots; f 5 (x (dot-dshed; f 1 (x (dshed; nd f 2 (x (solid.
1B Methods 9 If we substitute x = nd = 1 into the series we get π 2 12 = 1 1 4 + 1 9 1 16 + 1 25... Mny such interesting formule cn be constructed using Fourier series (cf. more lter! Finlly, notice the coincidence of the term-by-term derivtive of this Fourier series eq. (9 nd the series in eq. (8. We now look into this property more crefully. 1.3 Integrtion nd differentition of Fourier series Integrtion is lwys ok Fourier series cn lwys be integrted term-by-term. Suppose f(x is periodic with period 2 nd stisfies the Dirichlet conditions so it hs Fourier series for x : f(x = 2 + [ n cos + b n sin ]. It is then lwys vlid to integrte term by term to get (here x nd F is extended to ll of R by periodicity F (x x where we hve used eq. (8. f(udu = (x + + 2 b n + nπ n nπ sin [ ( 1 n cos = 2 + ( 1 n b n nπ b ( n nπx nπ cos ( n ( 1 n + nπ ], sin, Note tht the first infinite series on RHS of the lst equlity bove, forms prt of the constnt term in the Fourier series for F (x. This infinite series is lwys gurnteed to converge since b n comes from Fourier series we know tht b n is t worst O(1/n so ( 1 n b n /n converges by comprison test with M/n 2 for suitble constnt M. It is to be expected tht the convergence of the Fourier series for F (x will be fster (i.e. fewer terms will give certin level of pproximtion thn for f(x due to the extr fctor of 1/n mking the coefficients decrese fster. This is unsurprising since integrtion is nturlly smoothing opertion. Recll lso tht the Dirichlet conditions
1B Methods 1 1.5 1.5.5 1 1.5 1.8.6.4.2.2.4.6.8 1 Figure 2: Plots of f(x = 1 for < x < 1 nd f(x = 1 for 1 < x < (thin solid line nd the prtil sums f 1 (x (dots; f 5 (x (dot-dshed; f 1 (x (dshed; nd f 2 (x (solid. llow for finite jump discontinuities in the underlying function. But integrtion cross such jump leds to continuous function, nd F (x will lwys stisfy the Dirichlet conditions if f(x does. Differentition doesn t lwys work! On the other hnd, term-by-term differentition of the Fourier series of function is not gurnteed to yield convergent Fourier series for the derivtive! Consider this counterexmple. et f(x be periodic function with period 2 such tht f(x = 1 for < x < 1 nd f(x = 1 for 1 < x <, s shown in the figure. You cn redily clculte (exercise! its Fourier series to be f(x = 4 π sin([2n 1]πx 2n 1 (1 nd formlly differentiting term by term we get f (x =? 4 cos([2n 1]πx (11
1B Methods 11 which is clerly divergent even though for our ctul function we hve f (x = for ll x! The ltter my look like rther hrmless function, but wht bout f (? f is not defined t x = so f does not stisfy the Dirichlet conditions. Why not just put in some vlue f ( = c t the single point x =? e.g. the verge of left nd right limits, c =? Well, then consider the desired reltionship f(x = f( 1 + x f (tdt. 1 For ny finite c, f(x will remin t f( 1 s x crosses x = from below. To get the jump in f(x t x =, intuitively we d need f ( to introduce finite re under the grph of f, but only over x = with zero horizontl extent! i.e. we d need f ( = with = 1! Thus the opertion of differentition behves bdly (or rther most interestingly, cf lter when we discuss the so-clled Dirc delt function! when we try to differentite over jump discontinuity, even if we hve nice differentible pieces on both sides. So, when cn we legitimtely differentite the Fourier series of function term by term? Clerly it is not enough for f to stisfy the Dirichlet conditions (merely gurnteeing Fourier series for f itself. It suffices for f to lso not hve ny jump discontinuities (on R nd we hve the following result. Theorem: Suppose f is continuous on R nd hs period 2 nd stisfies the Dirichlet conditions on (,. Suppose further tht f stisfies the Dirichlet conditions. Then the Fourier series for f cn be obtined by term-by-term differentition of the Fourier series for f. To see this, note tht the conditions imply tht both f nd f hve Fourier series: f(x = 1 [ ] 2 + n cos + b n sin, nd so f (x = 1 2 A + A = 1 A n = 1 = 1 2 2 [ f(x cos [ A n cos + B n sin ]. f f(2 f( (x dx = = by periodicity, f (x cos dx, ] 2 + nπ 2 f(x sin dx, 2 = + nπb n. (12 where we hve gin used periodicity nd eqs. (7,6 for the Fourier coefficients. Similrly B n = nπ n so the series for f is obtined by term-by-term differentition of the series for f. Note tht the differentition of f hs been reduced to just simple multipliction of the Fourier coefficients by nπ (together with cross-relting the roles of n nd b n nd dding in minus sign for the B n s.
1B Methods 12 1.4 Complex form of Fourier series When deling with sines nd cosines it is often esier nd more elegnt to use complex exponentils vi de Moivre s theorem so cos sin nd our Fourier series becomes f(x + + f(x = 2 2 + [ n 2 where = e iθ = cos θ + i sin θ = 1 2 = 1 2i ( ( e inπx e inπx ( e inπx + e inπx, e inπx + e inπx + b ( n e inπx 2i ] e inπx, c n e inπx, (13 c n = ( n ib n /2 n > ; c n = ( n + ib n /2 n > ; c = /2. This is neter (though completely equivlent formultion. (These formuls ll remin vlid even if f is complex-vlued, in which cse the n s nd b n s re themselves complex. We cn work directly in this complex formultion by noting tht the relevnt complex exponentils re orthogonl functions: (e imπx inπx, e = 2 (14 e inπx imπx e dx = 2δ nm for m, n Z. (15 Note the signs (i.e. complex conjugtion in the integrl here! in ccordnce with our definition of inner products for complex vlued functions. Using orthogonlity, in the by now fmilir wy, we get from eq. (13: c m = 1 2 2 f(xe imπx dx m Z. For rel-vlued functions f (most functions in this course we immeditely get c m = c m so we need only compute c (which is rel nd c m for m >. Exmple. (Differentition rule revisited. Assuming we cn differentite the Fourier series term by term, in the complex representtion we write f(x = c n e inπx, n= df dx = n= C n e inπx
1B Methods 13 nd the differentition rule then gives the single simple formul: 1.5 Hlf-rnge series C n = inπ c n holding for ll n Z. Consider function f(x defined only on the hlf rnge x. It is possible to extend this function to the full rnge x (nd then to 2-periodic function in two nturl different wys, with different symmetries. Fourier sine series: odd functions The function f(x cn be extended to be n odd function f( x = f(x on x, nd then extended s 2-periodic function. In this cse, from eq. (7, n = for ll n nd we get Fourier sine series for f (note the rnge of integrtion: f(x + + f(x 2 = b n = 2 b n sin f(x sin ; dx. (16 i.e. f(x on [, ] hs been represented s Fourier series with only sine terms. Fourier cosine series: even functions Alterntively, the function f(x cn be extended to be n even function f( x = f(x on x, nd then extended s 2-periodic function. In this cse, from eq. (6, b n = for ll n nd we get Fourier cosine series (note gin the rnge of integrtion: f(x + + f(x 2 = 2 + n cos n = 2 f(x cos ; dx. (17 which gin represents f on [, ] but now s Fourier series with only cosine terms. 1.6 Prsevl s theorem for Fourier series The integrl of squred periodic function (or squred modulus for complex functions is often of interest in pplictions, e.g. representing the energy of periodic signl E = 2 Substituting the complex form of the Fourier series f(x = c n e inπx f(x 2 dx = (f, f. (18 n=
1B Methods 14 nd using the orthogonlity property eq. (15 of the complex exponentils we immeditely get 2 f(x 2 dx = 2 c n 2. (19 n= This result is clled Prsevl s theorem. Equivlently this cn be expressed in terms of the n nd b n using eq. (14 s [ ] 2 [f(x] 2 2 dx = + ( n 2 + b n 2 (2 2 i.e. the energy is obtined by dding together contributions from seprte sinusoidl hrmonics whose energies re proportionl to their squred mplitudes. If f is relvlued function, we cn remove ll the modulus signs in the bove formul. Remrk: Prsevl s formul cn be interpreted s kind of infinite dimensionl version of Pythgors theorem (tht the squred length of vector is the sum of the squred components in ny orthonorml bsis. Indeed on [, ] the following functions form n orthonorml set (i.e. pirwise orthogonl nd ech hving norm 1: c = 1/ 2 f n (x = 1 sin nπx for n = 1, 2,... g n (x = 1 cos nπx for n = 1, 2,... The Fourier series eq. (4 with these slightly rescled bsic functions becomes f(x = ( 2 c + n f n (x + b n g n (x n 1 nd then Prsevl s theorem eq. (2 is formlly just Pythgors theorem in this infinite dimensionl setting. For second interprettion of Prsevl s formul, we strt by viewing the Fourier series for f s mpping M from functions f to doubly infinite sequences {c n : n Z} of Fourier coefficients. Then viewing the ltter s components of n infinite dimensionl vector, Prsevl s theorem eq. (19 sttes tht the mpping M (up to n overll constnt 2 is n isometry (i.e. length-preserving, ccording to nturl notions of length on both sides. Exmple. Consider gin the swtooth function f(x = x for x. If we substitute eq. (8 into Prsevl s formul eq. (2 we get x 2 dx = 23 3 = 4 2 m 2 π 2 m=1 giving the nice formul π 2 6 = 1 + 1 4 + 1 9 + 1 16 +...
1B Methods 15 Prsevl s theorem is indeed commonly used to construct such tntlising equlities. As nother exmple (exercise Prsevl s formul cn be pplied to eq. (9 to obtin m=1 1 m 4 = π4 9. (21
1B Methods 16 2 STURM-IOUVIE THEORY Sturm-iouville (S theory is bout the properties of prticulr clss of second order liner ODEs tht rise very commonly in physicl pplictions (s we ll see more lter. Recll tht in our study of Fourier series we intuitively viewed (complex-vlued functions on [, b] s vectors in n infinite dimensionl vector spce equipped with n inner product defined by (f, g = f (xg(xdx. (22 A fundmentlly importnt feture ws tht the bsic Fourier (trig or complex exponentil functions were orthogonl reltive to this inner product nd the set of them ws complete in the sense tht ny (suitbly well behved function could be expressed s n infinite series in terms of them. In finite dimensionl liner lgebr of vectors with n inner product we hve very nice theory of self-djoint or Hermitin mtrices (tht you sw in first yer! viz. their eigenvlues re rel, eigenvectors belonging to different eigenvlues re orthogonl nd we lwys hve complete set (i.e. full bsis of orthonorml eigenvectors. S theory cn be viewed s lifting of these ides to the infinite dimensionl setting, with vectors being replced by functions (s before, mtrices (i.e. liner mps on vectors by liner second order differentil opertors, nd we ll hve notion of self-djointness for those opertors. The bsic formlism of Fourier series will repper s simple specil cse! 2.1 Revision of second order liner ODEs Consider the generl liner second-order differentil eqution y(x = α(x d2 dx y + β(x d y + γ(xy = f(x, (23 2 dx where α, β, γ re continuous, f(x is bounded, nd α is nonzero (except perhps t finite number of isolted points, nd x b (which my tend to or +. The homogeneous eqution y = hs two non-trivil linerly independent solutions y 1 (x nd y 2 (x nd its generl solution is clled the complementry function y c (x = Ay 1 (x + By 2 (x. Here A nd B re rbitrry constnts. For the inhomogeneous or forced eqution y = f (f(x describes the forcing it is usul to seek prticulr integrl solution y p which is just ny single solution of it. Then the generl solution of eq. (23 is y(x = y c (x + y p (x. Finding prticulr solutions cn sometimes involve some inspired guesswork e.g. substituting suitble guessed form for y p with some free prmeters which re then mtched
1B Methods 17 to mke y p stisfy the eqution. However there re some more systemtic wys of constructing prticulr integrls: ( using the theory of so-clled Green s functions tht we ll study in more detil lter, nd (b using S theory, which lso hs other importnt uses too lter we will see how it cn be used to construct solutions to homogeneous PDEs, especilly in conjunction with the method of seprtion of vribles, which reduces the PDE into set of inter-relted Sturm-iouville ODE problems. In physicl pplictions (modelled by second order liner ODEs where we wnt unique solution, the constnts A nd B in the complementry function re fixed by imposing suitble boundry conditions (BCs t one or both ends. Exmples of such conditions include: (i Dirichlet boundry vlue problems: we specify y on the two boundries e.g. y( = c nd y(b = d; (ii Homogeneous BCs e.g. y( = nd y(b = (homogeneous conditions hve the feture tht if y 1 nd y 2 stisfy them then so does c 1 y 1 + c 2 y 2 for ny c 1, c 2 R; (iii Initil vlue problems: y nd y re specified t x = ; (iv Asymptotic boundedness conditions e.g. y s x for infinite domins; etc. 2.2 Properties of self-djoint mtrices As prelude to S theory let s recll some properties of (complex N-dimensionl vectors nd mtrices. If u nd v re N-dimensionl complex vectors, represented s column vectors of complex numbers then their inner product is (u, v = u v where the dgger denotes complex conjugte trnspose (so u is row vector of the complex conjugted entries of u. If A is ny N N complex mtrix, its djoint (or Hermitin conjugte is A (i.e. complex conjugte trnsposed mtrix nd A is self-djoint or Hermitin if A = A. There is neter (more bstrct.. wy of defining djoints: B is the djoint of A if for ll vectors u nd v we hve: (u, Av = (Bu, v (24 (s you cn esily check using the property tht (Bu = u B. Note tht this chrcteristion of the djoint depends only on the notion of n inner product so we cn pply it in ny other sitution where we hve notion of inner product (nd you cn probbly imgine where this is leding!... Now let A be ny self-djoint mtrix. Its eigenvlues λ n nd corresponding eigenvectors v n re defined by Av n = λ n v n (25 nd you should recll the following fcts: If A is self-djoint then (1 the eigenvlues λ n re ll rel;
1B Methods 18 (2 if λ m λ n then corresponding eigenvectors re orthogonl (v m, v n = ; (3 by rescling the eigenvectors to hve unit length we cn lwys find n orthonorml bsis of eigenvectors {v 1,..., v N } so ny vector w in C N cn be written s liner combintion of eigenvectors. Note: it is possible for n eigenvlue λ to be degenerte i.e. hving more thn one linerly independent eigenvector belonging to it. For ny eigenvlue, the set of ll ssocited eigenvectors forms vector subspce nd for our orthogonl bsis we choose n orthonorml bsis of ech of these subspces. If λ is non-degenerte, the ssocited subspce is simply one-dimensionl. (4 A is non-singulr iff ll eigenvlues re nonzero. The bove fcts give net wy of solving the liner eqution Ax = b for unknown x C N, when A is nonsingulr nd self-djoint. et {v 1,..., v N } be n orthonorml bsis of eigenvectors belonging to eigenvlues λ 1,..., λ N respectively. Then we cn write b = β i v i x = ξ i v i where the β j = (v j, b (by orthonormlity of the v i re known nd ξ j re the unknowns. Then Ax = A ξ i v i = ξ i Av i = ξ i λ i v i = b = β i v i. (26 Forming the inner product with v j (for ny j gives ξ j λ j = β j so ξ j = β j /λ j nd we get our solution x = β j λ j v j. For this to work, we need tht no eigenvlue is zero. If we hve zero eigenvlue i.e. nontrivil solution of Ax = then A is singulr nd Ax = b either hs no solution or non-unique solution (depending on the choice of b. 2.3 Self-djoint differentil opertors Consider the generl second order liner differentil opertor : y = α(x d2 dx 2 y + β(x d dx y + γ(xy for x b (nd α, β, γ re ll rel vlued functions. In terms of the inner product eq. (22 of functions, we define to be self-djoint if i.e. (y 1, y 2 = (y 1, y 2 y 1(x y 2 (x dx = (y 1(x y 2 (x dx for ll functions y 1 nd y 2 tht stisfy some specified boundry conditions. It is importnt to note tht self-djointness is not property of lone but lso incorportes specifiction of boundry conditions restricting the clss of functions being considered i.e. we re lso ble to vry the underlying spce on which is being tken to ct. This feture rises nturlly in mny pplictions. Note tht (y = (y since we re tking to hve rel coefficient functions α, β, γ. Furthermore if we work with rel-vlued functions y then the complex conjugtions in eq. (27 cn be omitted ltogether. (27
1B Methods 19 Eigenfunctions of nd weight functions et w(x be rel-vlued non-negtive function on [, b] (with t most finite number of zeroes. A function y (stisfying the BCs being used is n eigenfunction for the self-djoint opertor with eigenvlue λ nd weight function w if y(x = λw(xy(x. (28 Note tht this is formlly similr to the mtrix eigenvector eqution eq. (25 but here we hve the extr ingredient of the weight function. Equtions of this form (with vrious choices of w occur frequently in pplictions. (In the theory developed below, the ppernce of w could be eliminted by mking the substitution ỹ = wy nd replcing y by 1 w ( ỹ w but it is generlly simpler to work with w in plce, s done in ll textbooks, nd express our results correspondingly. Eigenvlues nd eigenfunctions of self-djoint opertors enjoy series of properties tht prllel those of self-djoint mtrices. Property 1: the eigenvlues re lwys rel. Property 2: eigenfunctions y 1 nd y 2 belonging to different eigenvlues λ 1 λ 2 re lwys orthogonl reltive to the weight function w: w(xy 1(xy 2 (x dx =. (29 Thus by rescling the eigenfunctions we cn form n orthonorml set Y (x = y(x/ w y 2 dx. Remrk: Note tht the inner product eq. (27, used to define self-djointness, hs no weight function (i.e. w = 1 there wheres the eigenfunctions re orthogonl only if we incorporte the weight w from the eigenvlue eqution eq. (28 into the inner product. Alterntively we my think of the functions wy i s being orthogonl reltive to the unweighted inner product. Remrk: We my lwys tke our eigenfunctions to be rel-vlued functions. This is becuse in eq. (28, λ, w nd the coefficient functions of re ll rel. Hence by tking the complex conjugte of this eqution we see tht if y is n eigenfunction belonging to λ then so is y. Hence if the eigenvlue is nondegenerte, y must be rel (i.e. y = y. For degenerte eigenvlues we cn lwys tke the two rel functions (y + y nd (y y /i s our eigenfunctions, with the sme spn. In this course we will lwys ssume tht our eigenfunctions re rel-vlued, so we cn omit the complex conjugtion in the weighted inner product expressions such s eq. (3. Property 3: There is lwys countble infinity of eigenvlues λ 1, λ 2, λ 3,... nd the corresponding set of (normlised eigenfunctions Y 1 (x, Y 2 (x,... forms complete bsis for functions on [, b] stisfying the BCs being used i.e. ny such function f cn be
1B Methods 2 expressed s nd property 2 gives A n = f(x = A n Y n (x w(xf(xy n (x dx. (Don t forget here to insert the weight function into the inner product integrl! Remrk: the discreteness of the series of eigenvlues is remrkble feture here. The eigenvlue eqution itself ppers to involve no element of discreteness, nd this cn be intuitively ttributed to the imposition of boundry conditions, s illustrted in the next exmple below. Demonstrtion of the completeness property 3 is beyond the scope of this course, but properties 1 nd 2 cn be seen using rguments similr to those used in the finite dimensionl cse, for self-djoint mtrices. Introduce the nottion (f, g = f g dx (f, g w = wf g dx so (since w is rel (wf, g = (f, wg = (f, g w. (3 Now since is self-djoint we hve (y 1, y 2 = (y 1, y 2 for ny y 1, y 2 stisfying the BCs. (31 If lso y 1, y 2 re eigenfunctions belonging to eigenvlues λ 1, λ 2 respectively i.e. y i = λ i wy i, then eq. (31 gives (λ 1 wy 1, y 2 = (y 1, λ 2 wy 2 nd pplying eq. (3 we get λ 1(y 1, y 2 w = λ 2 (y 1, y 2 w i.e. (λ 1 λ 2 (y 1, y 2 w =. (32 Now tking y 1 = y 2 with λ 1 = λ 2 = λ, eq.(32 gives λ λ = (s (y, y w for y i.e. ny eigenvlue λ must be rel. Finlly tking λ 1 λ 2 we hve λ 1 λ 2 = λ 1 λ 2 so eq. (32 gives (y 1, y 2 w =, completing the proof of properties 1 nd 2. et s now illustrte these ides with the simplest exmple. Exmple (Fourier series gin! Consider = d2 dx 2 on x i.e. the coefficient functions re α(x = 1, β(x = γ(x =. We impose the homogeneous boundry conditions: y( = y( = nd we tke the weight function to be simply w(x = 1.
1B Methods 21 We will work only with rel-vlued functions (nd hence omit ll complex conjugtions. Is self-djoint? Well, we just need to clculte (y 1, y 2 = Integrting by prts twice we get so y 2 y 1 dx = [y 2 y 1] y 1 y 2 dx nd (y 1, y 2 = y 2y 1 dx = [y 2 y 1 y 2y 1 ] + (y 1, y 2 (y 1, y 2 = [y 2 y 1 y 2y 1 ]. y 1y 2 dx y 2y 1 dx With our BCs we see tht RHS = so with this choice of BCs is self-djoint. et s clculte its eigenvlues nd eigenfunctions: y = λwy = λy (the minus sign on RHS being for convenience, just relbelling of the λ vlues i.e. y = λy with y( = y( =. For λ the BCs give y(x =. For λ > the solutions re well known to be y n = sin nπx λ n = n2 π 2 2. Properties 2 nd 3 then reproduce the theory of hlf-rnge Fourier sine series. You cn esily check tht if we hd insted tken the sme but on [, ] with periodic boundry conditions y( = y( nd y ( = y ( (nd weight function w(x = 1 gin then sin nπx/ nd cos nπx/ would be (rel eigenfunctions belonging to the (now degenerte eigenvlues n 2 π 2 / 2 nd properties 2 nd 3 give the formlism of full rnge Fourier series (t lest s pplicble to suitbly differentible functions. 2.4 Sturm-iouvlle theory The bove exmple is the simplest cse of so-clled Sturm-iouville eqution. Consider gin the generl second order liner differentil opertor (with new nmes for the coefficient functions, s often used in texts y p(xy + r(xy + q(xy (33 where p, q, r re rel-vlued functions on x b. How cn we choose the functions p, q, r (nd lso ssocited BCs to mke self-djoint? An importnt wy is the following. We will require tht r(x = dp (34 dx
1B Methods 22 so we cn write s y = (py + qy. Cn be self-djoint? (reclling tht we still need to specify some BCs!. Well, integrting by prts twice (s in the bove exmple, nd tking ll functions to be rel you cn redily verify tht (y 1, y 2 (y 1, y 2 = y 1 [(py 2 + qy 2 ] y 2 [(py 1 + qy 1 ] dx = [p(y 1 y 2 y 2 y 1] b = [ ( y1 y p det 2 y 1 y 2 will be self-djoint if we impose BCs mking the bove boundry term combintion zero. The following re exmples of such BCs: (i y = t x =, b; (ii y = t x =, b; (iii y + ky = t x =, b (for ny constnts k which my differ t x = nd x = b; (iv periodic BCs: y( = y(b nd y ( = y (b; (v p = t x =, b (the endpoints of the intervl re then singulr points of the ODE. If w(x is ny weight function then the corresponding eigenfunction eqution y = (py + qy = λwy (with ny choice of BCs mking self-djoint is clled Sturm-iouville eqution. Such equtions rise nturlly in mny pplictions (we ll see some in section 2 nd the eigenfunctions/vlues re then gurnteed to stisfy ll the extremely useful properties 1,2,3 bove. The Fourier series exmple bove corresponds to the simplest non-trivil cse of n S eqution (with p(x = 1, q(x = nd w(x = 1. Reducing generl second order s to S form The S condition eq. (34 viz. tht r = p, on the coefficients of generl second order opertor ppers to be nontrivil restriction, but in fct, ny second order differentil opertor s in eq. (33 cn be re-cst into the S form s follows. Consider the generl eigenfunction eqution ] b py + ry + qy = λwy x b (35 (with weight w non-negtive but r not necessrily equl to p. Consider multiplying through by some function F (x. The new coefficients of y nd y re F p nd F r respectively nd we wnt to choose F (then clled the integrting fctor to hve so Then eq.(35 tkes the S form (F r = (F p i.e. pf = (r p F x ( r p F (x = exp dx. p [F (xp(xy ] + F (xq(xy = λf (xw(xy
1B Methods 23 with new weight function F (xw(x (which is still non-negtive since F (x is rel exponentil nd hence lwys positive nd new coefficient functions F p nd F q. Exmple. (An S eqution with integrting fctor nd non-trivil weight function Consider the eigenfunction/eigenvlue eqution on [, π]: with boundry conditions y = y + y + 1 4 y = λy y = t x = nd y 2y = t x = π. This is not in S form since p(x = 1 nd r(x = 1 p (x. But the integrting fctor is esily clculted: x r p F = exp dx = e x. p Multiplying through by this F gives the self-djoint form (noting lso the form of the given BCs!: ( d e x dy + ex dx dx 4 y = λex y (nd we cn view λ s the eigenvlue. To solve for the eigenfunctions it is esier here to use the originl form of the eqution (second order liner, constnt coefficients using stndrd methods (i.e. substitute y = e σx giving σ 2 + σ + 1 4 + λ = so σ = 1 2 ± i λ to obtin y(x = Ae x/2 cos µx + Be x/2 sin µx where we hve written µ = λ (with µ nd A, B re rbitrry constnts of integrtion. The first BC requires A = nd then the second BC gives the trnscendentl eqution (s you should check: tn µπ = µ. (36 Now to study the ltter condition, in the positive qudrnt of n xy plne imgine plotting the 45 line y = x nd the grph of y = tn πx. The line crosses ech brnch of the tn function once giving n infinite sequence µ 1, µ 2,... of incresing solutions of eq. (36. As n (i.e. lrge x nd y vlues in the plne these crossing points pproch the verticl symptotes of the tn πx function, which re t xπ = (2n + 1π/2 so we see tht µ n (2n + 1/2 s n i.e. the eigenvlues hve symptotic form λ n (2n+1 2 /4. The ssocited eigenfunctions re proportionl to y n (x = e x/2 sin µ n x. They re orthogonl if the correct weight function e x is used: π e x y m (xy n (x dx = if m n s you could verify by direct integrtion (nd you will need to use the specil property eq. (36 of the µ n vlues.
1B Methods 24 2.5 Prsevl s identity nd lest squre pproximtions ooking bck t Prsevl s theorem for Fourier series we see tht its derivtion depends only on the orthogonlity of the Fourier functions nd not their prticulr (e.g. trig form. Hence we cn obtin similr Prsevl formul ssocited to ny complete set of orthogonl functions, such s our S eigenfunctions. Indeed let {Y 1 (x, Y 2 (x,...} be complete orthonorml set of functions reltive to n inner product with weight function w nd suppose tht f(x = wy i Y j dx = δ ij (37 A n Y n (x A n = wy m f dx (38 (for simplicity we re ssuming here tht f nd ll Y n s re rel functions. Using the series for f nd the orthogonlity conditions we redily get wf 2 dx = w( A i Y i ( A j Y j dx = A 2 n. Finlly, it is possible to estblish tht the representtion of function in n eigenfunction expnsion is the best possible representtion in certin well-defined sense. Consider the prtil sum S N (x = N A n Y n (x, The men squre error involved in pproximting f(x by S N (x is ɛ N = w[f S N (x] 2 dx. How does this error depend on the coefficients A m? Viewing the A m s s vribles we hve A m ɛ N = 2 = 2 w[f N A n Y n ]Y m dx, wfy m dx + 2 = 2A m + 2A m =, N A n wy m Y n dx, once gin using eqs. (37,38. Therefore the ctul S coefficients extremize the error in men squre sense (in fct minimise it since 2 ɛ N = wy A 2 m my m dx >, nd so give the best prtil sum representtion of function is in terms of ny (prtil eigenfunction expnsion. This property is importnt computtionlly, where we wnt the best pproximtion within given resources.
1B Methods 25 2.6 S theory nd inhomogeneous problems For systems of liner equtions Ax = b whose coefficient mtrix ws self-djoint, we described method of solution tht utilised the eigenvectors of A look bck t eq. (26 et. seq. The sme ides my be pplied in the context of self-djoint differentil opertors nd their eigenfunctions. Consider generl inhomogeneous (or forced eqution y(x = f(x = wf (x (39 where on [, b] (with specified BCs is self-djoint, nd we hve lso introduced weight function w. Mimicking the development of eq. (26, let {Y n (x} be complete set of eigenfunctions of tht re orthonorml reltive to the weight function w: Y n = λ n wy n wy m Y n dx = δ mn nd (ssuming tht F cn be expnded in the eigenfunctions write F (x = A n Y n y(x = B n Y n where A n = wy nf dx re known nd B n re the unknowns. Substituting these into eq. (39 gives y = B n λ n wy n = w A n Y n. Multiplying by Y m nd integrting from to b immeditely gives B m λ m = A m (by orthonormlity of the Y n s nd so we obtin the solution y(x = Here we must ssume tht ll eigenvlues λ n re non-zero. A n λ n Y n (x. (4 Exmple. In some pplictions, when system modelled by homogeneous eqution y = (py + qy = is subjected to forcing, the function q develops weighted liner term nd we get y = (py + (q + µwy = f where w is weight function nd µ is given fixed constnt. This occurs for exmple in the nlysis of vibrting non-uniform elstic string with fixed endpoints; p(x is the mss density long the string nd µ, f depend on the pplied driving force. The eigenfunction eqution for (with weight function w, eigenvlues λ is y = y + µwy = λwy.
1B Methods 26 Hence we esily see tht the eigenfunctions Y n of re those of but with eigenvlues (λ n µ where λ n re the eigenvlues of, nd our formul eq. (4 bove gives y(x = A n (λ n µ Y n(x. This derivtion is vlid only if µ does not coincide with ny eigenvlue λ n. If µ does coincide with n eigenvlue then this method fils nd (s in the finite dimensionl mtrix cse the solution becomes either non-unique or non-existent, depending on the choice of RHS function f. 2.7 The notion of Green s function et us look little more closely t the structure of the solution formul eq. (4. Substituting A n = w(ξy n(ξf (ξ dx nd interchnging the order of summtion nd integrtion we get Y n (xy n (ξ y(x = w(ξf (ξ dξ (41 λ n i.e. y(x = G(x; ξf(ξ dξ (42 where we hve reinstted f = F w nd introduced the Green s function G defined by G(x; ξ = Y n (xy n (ξ λ n. (43 Note tht the Green s function depends only on (i.e. its eigenfunctions nd eigenvlues nd not the forcing function f. It lso depends the boundry conditions, tht re needed to mke self-djoint nd used in the construction of the eigenfunctions. Vi eq. (42, it provides the solution of y = f for ny forcing term. By nlogy with Ax = b x = A 1 b we cn think of integrtion ginst G in eq. (42 s n expression of forml inverse 1 of the liner differentil opertor : y = f y = G(x; ξf(ξ dξ. This notion of Green s function nd its ssocited integrl opertor inverting differentil opertor, is very importnt construct nd we ll encounter it gin lter in more generl contexts, especilly in solving inhomogeneous boundry vlue problems for PDEs.