MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

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1 MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section Outline nd Introduction ecturer: Dr. Willim J. Prnell (room 2.238; Willim.Prnell@mnchester.c.uk Exercise sheets, solutions, lecture notes nd ll other mteril relevnt to this course re vilble (t the pproprite time t wprnell/teching.html Summry: Green s functions re enormously importnt in mny res of pplied mthemtics, physics nd engineering. Consequently there re mny good books rnging from those which re firly informl/chtty (for scientists without too much mthemtics to those which re imed squrely t pure mthemticins. This course shll introduce Green s functions in firly rigorous fshion s sometimes lck of cre in the mthemtics cn led to misleding conclusions, or even incorrect solutions! Green s functions were devised essentilly s tool to solve boundry nd initil vlue problems (henceforth referred to s BVPs nd IVPs. Their use nturlly leds to integrl forms for the solutions of such BVPs nd IVPs which cn then be solved in number of wys. In prticulr they hve been the bsis of numericl solution method clled the boundry integrl method. Often, the ppliction of Green s functions to such problems leds to integrl equtions of specific type; therefore, this course will offer n introduction to this interesting nd importnt topic. The finl prt of the course will consider the relted nd eqully importnt topic of vritionl clculus. The rough syllbus for the course is s presented on the webpge: Green s functions: Definition nd bsic properties. [3 lectures] Appliction to the solution of ordinry differentil equtions. [4] Fredholm nd Volterr equtions of the 1st nd 2nd kinds. [2] Fredholm equtions with seprble kernels. [3] Neumnn series. [3] Euler-grnge eqution(s. [3] 1st nd 2nd functionl derivtives. [2] grnge multipliers. [2] Note tht this course is under review, so the topics covered, their order delivered, nd the time llocted to ech, my vry somewht. However, I intend (brodly to cover the course in the following sections: ( Introductory lecture(s (1 Function spces nd Opertors Some proofs but minly discussion of some importnt nottion nd issues ssocited with doing things properly Abstrct nottion - g(xf(xdx f, g

2 MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions 2 Opertors: differentil nd integrl (2 Green s Functions Method of solving differentil equtions (3 iner Integrl Equtions Unknown function ppers under n integrl sign - lterntive to differentil equtions (DEs Sometimes preferble - differentition roughens function, integrtion smooths it (4 Clculus of Vritions Concerns integrls contining objects tht depend on functions - functionls. The generl im is to extremize (mximise/minimise the vlue of the integrl Relted problems: shortest distnce between two points, mximl re enclosed by given perimeter, curve giving shortest time of descent Pre-requisites: (i Bsic ordinry differentil equtions (ODEs (ii Bsic prtil differentil equtions (ODEs - Het, wve nd plce equtions References for Green s Functions Texts whose primry focus is Green s functions Brton, G., Elements of Green s Functions nd Propgtion, Oxford University Press, New York, USA, Chester, C.R., Techniques in Prtil Differentil Equtions, McGrw-Hill, New York, USA, Kellog, O. D., Foundtions of Potentil Theory, Dover Science Publictions, New York, USA, Melnikov, Y., Green s Functions in Applied Mechnics, Computtionl Mechnics Publictions, Southmpton, Gret Britin, Roch, G. F., Green s Functions, 2nd Edition, Cmbridge University Press, Cmbridge, Gret Britin, Stkgold, I., Green s Functions nd Boundry Vlue Problems, Wiley-Interscience Publictions, New York, USA, Texts with n emphsis on mthemticl physics with substntil sections on Green s functions Crslw, H. S. nd Jeger, J. C., Conduction of Het in Solids, 2nd Edition, Clrendon Press, Oxford, Gret Britin, Cournt, D., nd Hilbert, D., Methods of Mthemticl Physics, Wiley-Interscience Publictions, New York, USA, Hbermn, R., Elementry Applied Prtil Differentil Equtions, Prentice-Hll Interntionl, New Jersey, 1987 Morse, P. M. nd Feshbch, H., Methods of Theoreticl Physics, McGrw-Hill, New York, USA, 1953.

3 MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions 3 Green s functions: An overview of wht they re nd wht they do! The bsic ide of Green s function will become very fmilir to students of mthemticl physics. However, most mthemticins nd engineers re not typiclly exposed to the concept of Green s function until possibly t the upper-grdute level. In Mnchester we re unusul in offering such course in the third yer. Green s functions ply n importnt role in the solution of liner ordinry nd prtil differentil equtions, nd re key component to the development of boundry integrl eqution methods. Consider liner ordinry differentil eqution written in the generl form (xu(x = f(x (.1 for x [, b] where (x is liner opertor, u(x is the unknown function, f(x is known forcing term nd we shll ssume homogeneous boundry conditions, i.e. u( = u(b =. We shll tighten up exctly wht we men by (x soon. Inhomogeneous boundry conditions cn be formulted s well (see lter. Formlly, we cn write solution to eqution (.1 s u(x = 1 f(x (.2 where 1 is the inverse of the differentil opertor. Since is differentil opertor, it is resonble to expect its inverse to be n integrl opertor. We expect the usul properties of inverses to hold, 1 = 1 = I, (.3 where I is the identity. More specificlly, we shll see tht we cn define the inverse opertor s 1 f = G(x, yf(ydy (.4 where the kernel G(x, y is the Green s Function ssocited with the ordinry differentil opertor. Note tht G(x, y is function of the two independent vribles x nd y where y is the integrtion or dummy vrible. To complete the ide of the inverse opertor, we now need to introduce new function. We shll cll this the Dirc delt function δ(x which we define s: { b f(x, x [, b], δ(x yf(ydy = (.5, x / [, b], the property bove is often known s the sifting property in tht the delt function picks out the vlue of the function where x y =. Tking nd b, we get δ(x yf(ydy = f(x, (.6 δ(ydy = 1 (.7 where the lst eqution is prt of the definition of the function ( normliztion condition. This peculir function δ(x tkes the vlue zero everywhere except t x = where it is infinite. It is not relly function - it is wht we shll cll generlized function nd we shll mke the definition of the Dirc delt precise lter in the course. The Green s function G(x, y then stisfies the differentil eqution x G(x, y = δ(x y (.8

4 MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions 4 with ppropritely defined boundry conditions on nd b (see shortly. The subscript x on the liner opertor indictes tht it is operting with respect to x. Thus y cts merely s prmeter in the problem - the position of the forcing on the RHS. It trnspires tht the solution to eqution (.1 cn then be written directly in terms of the Green s function s u(x = G(x, yf(ydy. (.9 Note tht this is nice form becuse it is vlid for ny source distribution f(x. To show tht eqution (.9 is indeed solution to eqution (.1, simply substitute s follows: u(x = = G(x, yf(ydy = G(x, yf(ydy δ(x yf(ydy = f(x. (.1 Note tht we hve used the linerity of the differentil nd inverse opertors in ddition to equtions (.4, (.6, nd (.8 to rrive t the finl nswer, noting tht x [, b] for the lst equlity. Exmple.1: A simple first order ODE et s try this on very simple differentil eqution which DOESN T quite fit the bove theory but gives the ide of how it works: subject to y(c =. du dt Au = f(t (.11 This is liner first order ordinry differentil eqution with constnt coefficients rising in mny res including the subject of popultion dynmics. It hs given function on the right hnd side so is n inhomogeneous ODE. Written s eqution (.1, the opertor (t is d A, nd we cn solve this simple ODE by the use of n integrting fctor. In dt this cse multiply both sides of (.11 by e At : so tht the left hnd side simplifies to give the eqution e Atdu dt e At u = e At f(t (.12 d dt (e At u = e At f(t. (.13 Therefore, this my be integrted immeditely with the condition y(c =, to yield u = e At e Aτ f(τdτ = c c e A(t τ f(τdτ (.14 where τ is the dummy integrtion vrible nd c is constnt, nd we cn immeditely see tht the inverse opertor to (t is 1 f(t = c G(t, τf(τdτ (.15

5 MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions 5 in which G(t, τ = e A(t τ. Note tht this does not fit into the bove wy of thinking becuse the upper limit of the integrl is t (i.e. the independent vrible wheres ll limits so fr hve been constnts. This is specil type of integrl eqution nd will be discussed in lter lectures. Exmple.2: Stedy stte het eqution in 1D (ODE Consider heting long thin br of length with prescribed (equl temperture boundry conditions t ech end of T = T t x =, nd sources of het distributed vi source term f(x. We cn define u = T T s reltive temperture. Show tht the solution to the stedy stte het eqution with u( = = u( stisfies d 2 u = f(x (.16 dx2 u(x = where the Green s function stisfies with G(, y = = G(, y. f(yg(x, ydy (.17 2 G(x, y x 2 = δ(x y (.18 To show this, multiply (.18 by u(x nd subtrct from this (.16 multiplied by G(x, y to get: u(x d2 G(x, y G(x, y d2 u = δ(x yu(x G(x, yf(x dx 2 dx2 Then integrte both sides (with respect to x between x = nd x = : ( u(x d2 G(x, y G(x, y d2 u dx = (δ(x yu(x G(x, yf(xdx dx 2 dx 2 Re-write left hnd side (HS: d dx ( dg(x, y u(x G(x, y du dx dx nd cn then integrte HS immeditely dx = (δ(x yu(x G(x, yf(xdx ( dg(, y u( G(, y du ( dx dx ( dg(, y u( G(, y du dx dx ( = HS is zero due to boundry conditions so tht (lmost there! δ(x yu(xdx = (δ(x yu(x G(x, yf(xdx (.19 G(x, yf(xdx.

6 MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions 6 By the definition of the Dirc delt, for y [, ], nd finlly interchnge y nd x u(x = u(y = G(y, xf(ydy = G(x, yf(xdx G(x, yf(ydy which is the required form where we used G(x, y = G(y, x in this instnce. The Green s function is not lwys symmetric but in tht cse we use slightly modified rgument to tht bove (see end of Section 2. Wht is the Green s function here? Well see Section 2 for this but it trnspires tht it is: { x ( y, x < y, G(x, y = y ( x, x > y. Note lso tht: G(x, y is continuous but its derivtive is discontinuous - this is dictted by the governing eqution. G(x, y = G(y, x Wht hppens if we hve different boundry conditions on u(x? Choose different boundry conditions for the Green s function so tht the boundry terms in (.19 cn be set to zero. This then modifies the Green s function - see Q. 2 on Exmple Sheet 1. Exmple.3: Time-dependent het eqution in 1D (PDE Consider the het eqution governing some reltive temperture u(x, t subject to the boundry conditions u t = u k 2 x 2 nd initil condition u(, t =, u(, t = (.2 u(x, = U (x In pst course you will hve (probbly! shown, by seprtion of vribles tht the solution to this problem cn be written in the form u(x, t = n sin ( nπx e k(nπ/2 t

7 MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions 7 where the initil condition thus gives nd orthogonlity thus gives U (x = n = 2 n sin U (y sin ( nπx ( nπy dy. Put this bck into the solution form, interchnge integrtion nd summtion to obtin the form ( 2 ( nπy ( nπx u(x, t = U (y sin sin e k(nπ/2 t dy This eqution represents the fct tht the temperture t the loction x t time t is due to the initil temperture t the loction y nd to obtin the ctul temperture we hve to integrte the initil condition over ll such loctions y with specific weighting - which corresponds to the function in brckets. We could write this solution in the form u(x, t = U (yg(x, t; y, τ = dy where G(x, t; y, τ = is Green s function. Furthermore, it cn be shown tht for the corresponding problem (sme boundry conditions nd initil condition u t = u k 2 + F(x, t x2 then the solution cn be written where u(x, t = U (yg(x, t; y, τ = dx + G(x, t; y, τ = 2 ( nπy sin sin F(y, τg(x, t; y, τdτdy (.21 ( nπx e k(nπ/2 (t τ. The first term in (.21 represents the influence of the initil condition, wheres the second term represents the influence of the source term. Note. Even though we hve net nlyticl form for the solution, it does not men tht it will be esy to evlute - note tht it is n infinite sum, so for smll times in prticulr, this my converge very slowly. This is common problem, i.e. it is nice to hve nlyticl solutions but tht does not men we cn lwys evlute them esily. Don t worry if this quick summry seems confusing/complicted t present - we will be developing it slowly over the coming lectures. The best wy to understnd the ides is to try out the problems on the exmple sheets, nd if necessry to look over other exmples in books.