Question

(1 point) Use eigenvalues and elgenfunction expansion expansion to solve the mixed Dirichlet- Neumann problem for the Laplace equation Au(x, y) = 0 on the rectangle {(x,y) : 0<x<1, 0<y<1} satisfying the BCS ux(0,y) = 0, ux(1, y) = 0, 0 < y < 1 u(x,0) = x, u(x, 1) = 0, 0<x<1 The solution can be written as The u(x, y) = Covo(y)+(x) + .(x).(y) where on is a normalized eigenfunction for "(x) = 10(x) with x(0) = 0 and 4x(1) = 0. Find u(x, y) u(x, y) = IM:

Answer 1

use eigenvalues and eigenfunction expansion to solve the mixed Dirchlet Neumann problem for the Laplace equation

use cigan value and Cigen fonction expansion expansion to solve the mixed Dixich let - Neumam pooblem por the Laplace . Buation su (8,91:0 on teoroctangle $28,91:0CHC1,0cycis. Satisfying the BCS wolow:0, U8 (14) :o, o cgel 4(8,0) = &, 4(8,1) =0,04841. the son can be written as u(x,y) = Cololu I coon (8) Unly) nel Given that PDE - Du(2,4)=0,0CM<!, O CY41. → Usede (8,4) + Ulyy (8,) 0,04&21, ocy<& s0 BC.: 418 (0,4)20. 48 (1,4)=0 ; OcYC! 10 4(2,0) = 4, 468,1)=0 , 02&cl > ☺ Step - 1,- Assume that the solution of the diffe - rental on as of the form.

418,9) = X() Yey) → Sub a onto de" (a) y Cy) + (x)}"() - 0 x"(x) syy) Y(x) yy) x' (a) - KX(X) +0 → Ý (Y) - KY (Y) =O 6 Step 2 i- Caye (7) in when r so, set k = x The solution of the ODE's are => X() - Ae**Bem © => Yly) - Coos (day) + Osan (19) the solution of PDE U(ay) is gillen by 4(x,y) - Che Mat Be AR] [ccos (149)+ Osen ().

Coje (il;- when kao The Soletion of ODES QUE =X(X) - A+Bd © >> Yy) = CHOy The solution of the PDE o (8,4) as Given by 0 (38,9) - (A488] [C+04] case tooni) - when kao set ka The solution of the ODES are © => X(X) = A COS (Ay) + B son (ay) 6 y(y) = cely Doty : The solution of PDES U(8,4) is given by 168.) - Cacos (1x) + BS90 (18) ] [center pony

Step - Apply the boundary conditions us (0,4)=0 and us (14) = 0 Case (ali 4(8,) [te " t Be PJ Froscay) + OS9n(ay) On differentiating we get. curcle, y} - (nment 10€ ""][ccoschy ) + o'smca us (0,4)=8 =) A-B=0 & Us (1,4)=0 as ponsable. =) Ach-Be =o. which has a trival solution A & B .O. the trival Solution ulay) =0 "S This only possiable Cage (09) - ucse, y) = (A+B&] [C+Oy on diffentia - ting we get

Us (8,4) = (B) (CADY) IR up (0,4) = 0 => B=0 De ux(0,4)=6 » B=0 IR 4(8,0) = . = c(A+BX) = X =) CA + CBX = X → CA:0 -C=0 (0) A -O If A o this leads to o trival Solution u(x, y) =0 if not than c=0, Applying a (8,1)=0 – (A+BX)(C+01:0 (C+D) = 0 D=0 also leads to a trivial solution This u(8,4)=0

Case (0) - 478,4) = (a cos(4x) + Bsen(x2)+fee met de nyy diffentiate both sides we get 48 (8.4) - (-RASAD (18) + BACOS CA] [ce toe"] usco,y) to 13:01 Ux(1,) =0 >> (Asyn (ne)] [ce4970e") -o for a non trival Solution A0 => San(x) = 0 s : Saillo) 1 = n/ . given as The sequence of non- Torival Solution by un (se, y) = cos(ne) (che tone y 19 anny

③ u18, 1) =0 un (8,1) = cos Childe] (enettone") -o a che + Dne no © à non ty) → un (8,4) - cos (one) [cnecne 'xes = cos(ne (con certes, como conveyan, cany sa le os crimes (saucern cering unting) - (San) as com asian nuray] con los comentan cos (come use " (wewn) set en aan lent

Super position parinciple we obtain 4162,Ž un cosy). t's always lehes crane) a su camera). no