Question

mechanical engineering analysis help, please show all work, thanks.

Problem 3. Show that the solution of the partial differential equation (Laplace equation), Wxx(x,y) + wyy(x, y) = 0, with the four boundary conditions: w(x,0) = 0, w(x,1) = 0, w(0,y) = 0 and w(1, y) = 24 sin ny, can be obtained as w(x,y) = 2 sinh Tx. sin ny.

Answer 1

The given equation is, War (x, y) + Wzy (pf)=0.-1) W(140)=0, w(x, 1) = 0 W10,y) = 0 and W1,7) = 24 sin try. het W(ny)= XC40 Ylz) be a solution of (1). Then the equation (1) becomes, go" vpn + y como till yery" LI or, 11 X u х Y 01 = Let th Y Y u ea Constants (says - (2) X-MX0. y" thy zo (3) So W(npol=0 X (0) 410)=0 and w(X, 1) = x(x1Y (1)-0 :: Y10) 20 Y(I) -O Now we where X(+) fo , since of herwise W=0 which does not imply Werry) = 24 Sinity, are to solve the equation (3). ret h=0. the solution of (2) is YOF Ay+ B. Where ApB are Castants. Case 1: Then

Y10)= 0 =) Ba Yll)=0 Ar B=0 :: A=0 W=0 which does not so, y = 0 satisfy = (18) بيا 24 sinity Care 2! het M= -x (negative). (negative). Here , #o. Then the solution of (3) is, YA 411) = A + Be +Beryx Y(o)=0 = A+B=0 Y (1) A Cat Bed=0 ert_ex to. So, Y=0 A =B20 trivial Solvhu Theo W(2,1)=0 wrich does not satisfy wlly) 224 =Lasinny Cose 3: Here fo. Then let M = x (positics. the solution of (3) is, Y(Y) = A los xy of Bsindy. 07 A = 0 Y(o)=0 Y11)=0 B sind all

sinx=0 siree B fo if B=0 ñ Then Y=0 and W(x,y)=0, which does not imply will y)=24sinly. sinx=0 2) Да ти n=1, 2, 3, .. Hence YoY) = Bn sinnity Also, then M=-X = -nity, then the equation (2) reduces to xl-max=0. n=1,43 so the solution is, Xn(n)= en el nox -nou tone 24). Nou, w(0,4)=0 = Xorly)=0. :: X(0) = 0 So An(0) to sorth Xncol=0 gives en + An = 0, Pnarch En=2&non Xn(n)= encenum – e nx) aan sinnnita. Walny) = xnlu) Yn (Y) = Ep on Therefore sinh nix. Sinnay.

= WEIH). n디 Ewn (1,4)=24sinity sin hnit sinnity atsinaty n=1,2,3,. E En for na Ei sinhit sinty = /24 Sinity Ei sinni = 24 터 ( 24 sinnt En sin nnot 2/ 24 sinity sin niny dy 25 En a 2 sinnnit 24sinity Sinning ff . 0 Si wn (y) = En Sinh non sinnity. W(w, y) = ElEn sinh nux) Sinnity ny 2 Sin hatx Sinity (proveof).