LECTURE 1 Introduction 1. Rough Clssiction of Prtil Dierentil Equtions A prtil dierentil eqution is eqution relting function of n vribles x 1 ;::: ;x n, its prtil derivtives, nd the coordintes x =èx 1 ;:::;x n è; i.e., n eqution of the form è1.1è F ëx; ; @ i ; @ i @ j ;::: ;@ i @ j @ k ëèxè =0 The order of the highest derivtive ppering in è1.1è is the order of the prtil dierentil eqution è1.1è. If the dependence of the functionl F on nd its prtil derivtives is liner, then the PDE è1.1è is sid to be liner ènote, however, tht liner PDE is llowed to hve nonliner dependence on xè. Exmple 1.1. is nonliner PDE of degree 1. Exmple 1.. is liner PDE of degree. @ @ =3 @x 1 @x @ @x 1, @ @x x 1 x =0 By solution of the PDE è1.1è in region R R n,we men n explicit function : R n! R such tht F ëx; ;@ i ;@ i @ j ;::: ;@ i @ j @ k ë èxè vnishes identiclly t ech point x R. Note tht if è1.1è hs degree d then must be of clss C d èi.e., nd ech of prtil derivtives up to order d must be continuous throughout Rè.. Three Fundmentl Exmples of nd Order Liner PDEs:.1. Generic nd Stndrd Forms of nd Order Liner PDEs. The generic form of second order liner PDE in two vribles is è1.è Aèx; yè @ @x Bèx; yè @ @x@y @ Cèx; yè @ @ Dèx; yè Eèx; yè F èx; yè = Gèx; yè @y @x @y We shll see ltter tht by suitble chnge of coordintes x; y! èx; yè;èx; yè we cn cst ny PDE of the form è1.è into one of the following three èstndrdè forms. è1.3è 1. Prbolic Equtions: @ f 1è; è @ @ f è; è @ @ f 3è; è = gè; è 1
. Elliptic Equtions: 3. BOUNDARY CONDITIONS @ è1.4è @ @ f 1è; è @ @ f è; è @ @ f 3è; è = gè; è 3. Hyperbolic Equtions: è1.5è @@ f 1è; è @ @ f è; è @ @ f 3è; è = gè; è Associted to ech of these stndrd forms re prototypicl exmples, ech of which, remrkbly, corresponds to fundmentl PDE occuring in physicl pplictions. During the next few weeks we shll discuss the solutions or ech of these equtions extensively. è1.6è.. The Het Eqution. @, @ @t =0 @x This eqution rises in studies of het ow. For exmple, if 1-dimensionl wire is heted t one end, then the function èx; tè describing the temperture of the wire t position x nd time t will stisfy è1.6è. The het eqution is the prototypicl exmple of prbolic PDE. è1.7è.3. Lplce's Eqution. @ @ =0 @x @y This eqution rises in vriety of physicl situtions: the function èx; yè might be interpretble s the electric potentil t point èx; yè in the plne, or the stedy stte temperture of point in the plne. Lplce's eqution is the prototypicl exmple of n elliptic PDE. è1.8è.4. The Wve Eqution. @ @t, @ =0 @x This eqution governs the propgtion of wves in medium, such s the vibrtions of tunt string, pressure uctutions in compressible uid, or electromgnetic wves. The wve eqution is the prototypicl exmple of hyperbolic PDE. The coordinte trnsformtion tht csts è1.8è into the form è1.5è is = x, ct = x ct 3. Boundry Conditions In strk constrst to the theory of ordinry dierentil equtions where boundry conditions ply reltively innocuous role in the construction of solutions, the nture of the boundry conditions imposed on prtil dierentil eqution cn hve drmtic eect on whether or not the PDEèBVP èprtil dierentil eqution è boundry vlue problemè is solvble. 3.1. Cuchy Conditions. The speciction of the function nd its norml derivtive long the boundry curve. Cuchy boundry conditions re commonly pplicble in dynmicl situtions èwhere the system is interpreted s evolving with respect to time prmeter t:
4. SIMPLE SOLUTIONS OF THE HEAT EQUATION - SEPARATION OF VARIABLES 3 3.. Dirichlet Conditions. The speciction of the function on the boundry curve. As n exmple of PDEèBVP with Dirichlet boundry conditions, consider the problem of nding the equilibrium temperture distribution of rectngulr sheet whose edges re mintined t some prescribed èbut non-constntè temperture. 3.3. Neumnn Conditions. The speciction of the norml derivtive of the function long the boundry curve. As n exmple of PDEèBVP with Neumnn boundry conditions, consider the problem of determining the electric potentil inside superconducting cylinder. 4. Simple Solutions of the Het Eqution - Seprtion of Vribles In order to get feel for the generl nture of prtil dierentil equtions, we shll now look for simple solutions for the het eqution è1.9è @ = @ @t @x : We shll construct solutions of this eqution by presuming the existence of solutions of prticulrly simple èbut suciently generlè form. Our initil ssumptions will be justied by the fct tht we obtin in this mnner lots of solutions. Let us then suppose tht there exist solutions of è1.9è of the form è1.10è èx; tè =Gètè where F is function of x lone nd G is function of t lone. Substituting this nstz for into è1.9è yields G 0 ètè = GètèF 00 èxè or è1.11è G 0 ètè Gètè = F 00 èxè : Now this eqution should hold for ll x nd t. However, the left hnd side depends only on t while the right hnd side depends only on x. Consequently, ifwevry t but keep x xed, we must hve G0 ètè Gètè equl to the xed number F 00 èxè ètè Gètè equls some constnt; sy C. Similrly,byvrying x nd keeping t xed we cn conclude tht F 00 èxè is constnt swell; sy D. Eqution è1.11è then becomes C = D : Thus, when we presume the existence of solutions of the form è1.10è, the diusion eqution è1.9è is equivlent to the following pir of ordinry dierentil equtions è1.1è è1.13è G 0 ètè Gètè F 00 èxè = C = C : Therefore, if we cn construct solutions G nd F of the ordinry dierentil equtions è1.1è nd è1.13è, then è1.10è will be solution of the prtil dierentil eqution è1.9è. Rewriting è1.1è nd è1.13, respectively, s è1.14è G 0 ètè, CGètè =0
4. SIMPLE SOLUTIONS OF THE HEAT EQUATION - SEPARATION OF VARIABLES 4 è1.15è F 00 èxè, C =0 We see tht both of these ordinry dierentil equtions re liner with constnt coecients. The generl solution of è1.14è will be nd the generl solution of è1.15è will hve the form Thus, ny function of the generl form Gètè =G o e Ct =F 1 e x F e,x ; = C èx; tè = G o e Ct, F1 e x F e,x c 1 e Ctx c e Ct,x r C : will be solutions of è1.9è. Note tht there re 3 undetermined prmeters here; C, c 1 nd c.for xed vlues of 6= 0,we obtin two dimensionl spce of solutions, since C;1 èx; tè =e Ctx nd C; èx; tè =e Ct,x re linerly independent. However, if C 0 6= C, then the functions f C;1 ; C; ; C 0 ;1; C 0 ;g re ll linerly independent. If we tke the seprtion constnt C = k, with k rel, we obtin b 1 e k x b e, k x k èx; tè =e k t Vrying c we thus obtin two 1-prmeter fmilies of liner independent solutions whose mgnitudes grow exponentilly in time: k;1 èxè = e kt e kx k; èxè = If we tke C =,, with rel constnt, we hve nd so nd èx; tè = c 1 e, t e i = e, t = e, t e kt e,kx r, = = i ;1 èx; tè = e,t e i x ; èx; tè = e,t e,i x x c e,,i t e x x c 1 cos 1 cos x In the second step we hve used Euler's formul ic 1 sin sin x x c cos e i = cosèèi sinèè to replce the exponentil functions e i x by sine nd cosine functions: 1 = c 1 c = ic 1, ic : : x, ic sin Vrying we obtin two more 1-prmeter fmilies of linerly independent solutions tht decy exponentilly s t!1, nd oscillte sinusoidlly s one vries, x. ;1 èxè = e,k t kx è1.16è cos, R ; èxè = e,k t,kx è1.17è sin R x
4. SIMPLE SOLUTIONS OF THE HEAT EQUATION - SEPARATION OF VARIABLES 5 In summry, the method of seprtion of vribles èi.e., the nstz èx; tè = Gètèè produces four 1-prmeter sets of linerly independent, rel-vlued solutions k;1 èxè = e k kx t è1.18è e è1.19è è1.0è è1.1è k; èxè = e k,kx t e ;1 èxè = e,kt cos kx ; èxè = e,kt sin,kx R R Given this plethor of linerly independent solutions, it is pproprite to sk under wht dditionl conditions cn we expect to nd unique solution. Clerly, specifying the vlue of t single point will be insucient. We shll see ltter tht in order to obtin unique solution we will hve to specify the vlues of nd its prtil derivtives t every point long some curve in order to completely determine solution.