Diola Bagayoko (0 PARTAL DFFERENTAL EQUATONS SEPARATON OF ARABLES. troductio As discussed i previous lectures, partial differetial equatios arise whe the depedet variale, i.e., the fuctio, varies with ore tha oe idepedet variale. such cases, partial ψ dψ derivatives ( - as opposed to total derivatives ( - appear i the differetial equatios. d Please refer to the lecture o Maxwell s E & M equatios ad the wave, Poisso, ad Laplace equatios we derived fro the for further illustratio. For exaple, the Laplace equatio for the scalar potetial of E & M is y, z or y, z y, z z The separatio of variales that follows is differet fro the oe we used for solvig Newto s type equatios, d x( t d = f ( t, = f ( t, or d = f ( t dt dt dt t is also differet fro the oe we used for solvig hoogeeous first order differetial equatios (that are liear, y P( x y y = P( x y = P( x dx y oth the aove exaples, we separated the depedet ( or ad the idepedet (t or x variales. We gave the ae of first separatio to this for of separatio of variales. The oe discussed elow cosists of separatig the idepedet variales ( x, y, z, or t as i the Laplace equatio aove. t is essetial to ote that the geeral separatio of idepedet variales is oly the first step i solvig partial differetial equatios. This separatio leads to ordiary differetial equatios that are solved (a y usig the first separatio followed y itegratio, ( y utilizig e pt or e px sustitutio ethod for liear, d order, ordiary differetial equatios with costat coefficiets. (c y eployig the Froeius series ethod for ordiary differetial equatios (O.D.E for which the coefficiets are ot costat, ad
Diola Bagayoko (0 (d y usig the solutios otaied (as i a.,., ad c. aove to satisfy oudary or iitial coditios. As show through the exaple elow, satisfyig oudary coditios ofte leads to ifiite series!. The Separatio of ariales Through a exaple. a. Preliiaries A plate of legth a ad width is coected i a circuit as show. Solve the Laplace equatio for this plate. ( at y ad at y =, it is ad at x ad x = a y Solutio a x Due to the geoetry of the plate, we select the (x, rectagular Cartesia coordiate syste, as show aove, to express ad solve the prole. Due to this choice, the scalar potetial is = (x, ad the Laplace equatio is everywhere iside the plate. At the edges of the plate, the oudary coditios (.c. are: at y ad at y = (x, 0 ad (x, = at x (0, = = at x = a (a, = Key: Boudary coditios (.c. for which the depedet variale (the fuctio is zero are called hoogeous.c. Exaples are (x, 0 ad (x,. Boudary coditios that are ot hoogeeous are said to e ihoogeeous or o-hoogeeous. Exaples aove are (0, = ad (a, =. (cidetally, see your E & M lessos to uderstad that the groudig of the upper ad lower edges of the plate eas that the potetial is zero at those edges!. Also ote -- icidetally -- that coditios at extreal values of space variales like (x, y, z or (r, θ, ϕ are referred to as oudary coditios. Coditios for t or t = t 0 are called iitial coditios. Now that we have selected a coordiate syste, idetified the oudary coditios, let us proceed with the separatio of variales.
Diola Bagayoko (0.. Separatio of variales: asic steps [.0] a Assue (x, = ( x ( [.], i.e., is assued to e a product of fuctios ad each of which depeds oly o oe variale. Note well that this assuptio is ot always correct. Hece, whe it is ot correct, the the separatio of variales is ot possile. Get ( x = [ ] ( ad ( = ( x[ ] c Sustitute the i the D.E. ad divide y ( x ( = ( * ( x *. The first ter depeds oly o x ad the secod o y oly. So, the separatio of variale is possile ad we ca cotiue with the search of solutio y this ethod. d The last lie aove is possile oly if = k ad = k. [.] deed, if they are ot costat, the, for exaple, will take a value as x varies- - such that 0, give that x does ot appear i (ad does ot chage. Give that each ter depeds oly o oe variale, we ca switch to total derivatives: d dx d Further, we ust have k k. Hece, oe is positive ad the other oe is egative. We deterie, give the.c., which oe should e positive or egative. 3
Diola Bagayoko (0 Key: Select the sigs of the costats i such a way that you otai oscillatory (cos, si, e ix, e it fuctios i the directio of the hoogeeous.c. ad i the directio of ost cyclic variales [i.e., ϕ, of the π cycle, i ( ρ,ϕ, z cylidrical coordiate syste. our case aove, we eed oscillatios i the directio of y (the.c. for the extreal values of y are hoogeeous. To have oscillatios i y we eed the solutio of d = k to e oscillatory: " k = or " 0 k = px Fro the results of the lecture of e sustitutio, k ust e positive, or k ust e ixy egative, for to e e or cos( y, si( y where is a costat. Ay egative uer k ca e writte as ius a square (all squares are positive or zero. So, set k = -k. d d Hece, we ow have = k ad = k dx These two equatios are d order, ordiary differetial equatio with costat coefficiets. (We have leared how to solve the with e py or e px sustitutio ethod. Hece, these equatios give = Acos( k B si( k ad = ce De, with =, as their geeral solutios. The geeral solutio of the Laplace equatio is the = [ Acos( k B si( k]*[ Ce De ] We ust look for the particular solutio that satisfies the oudary coditios (.c.. Key: Always satisfy the hoogeeous.c. first. So we wat (x,0=0 ad (x,=0 or 0 = [ A* B * 0][ Ce De ] A or = B si ky[ Ce De ]. B si k[ Ce De ] si k or π si k k = π or k =. Note well the apparitio of! πx πx This value of k leads to (x,= B si( [ Ce De ]. Key: With the appearace of iteger, the questio ecoes the followig: Which value(s of do we take to write πx πx (x,= B si( [ Ce De ]? Ad the aswer is - - all of the satisfy the hoogeeous oudary coditios! So, we have to take the all! Hece, (x, is a ifiite series ad πx πx = = B si( [ Ce De ]. [.3] Notice the idices, eve for the coefficiets B, C ad D. 4
Diola Bagayoko (0 Let B C = ad B D = (We are just reaig costats, the πx πx = = ( e e si( Let us ow cosider the ihoogeeous oudary coditios: ( At x = = = si(. Usig the properties of orthogoal fuctios (i.e., i this case si( si * ( 0 π w( ad itegrate fro 0 to : y with W(= over the iterval [0, ], we ultiply y πy πy si*( = si*( ( si( [.4] The itegral o the left had side of Equatio [.4] is πy cos( π 0 which is πy cos( π 0 = π π [ ( ] [ ] for for [.4.] The right had side of Equatio [.4] is ( N δ = ( [.4.]. See the chapter o Fourier Series to otai N = (N is π as π is the rage, so it is if is the rage is i Equatio [.4]. Coiig [.4.] ad [.4.] leads to 4 ( = for odd ad 0 for eve. [.5] π At x = a We have πa πa ( a, = = ( e e si(. = Usig the orthogoal properties of si( ad proceedig as doe aove for x, we get odd eve 5
Diola Bagayoko (0 = a πa 4 for odd π e e [.6] 0 for eve Equatios [.5] ad [.6] for a syste of algeraic equatios with two ukow, ad. Solvig this syste for ad as = ( e odd πx, which we call or relael as e si( ad, gives potetial [You have to solve for ad ad rewrite the aove fial solutio usig their values i ters of,, a,, ad π] 6