Question

Thank you.

5. Find the solution u(x, y) of Laplaces equation in the rectangle 0<<a, 0<y<b that satisfies the boundary conditions u(0, y

0 0
Add a comment Improve this question Transcribed image text
Answer #1

Ry oCり26case 皿 ehiuen utoh:40 ) x,o)0 using tas a solukon oth Jorm Suppos ucxy) ) Y) Subrrikmng thi valu of in => r Y Sirc xCase Phen on using bocndary conditen A-aトod nd9ニo => n1,2,3 もnl non-2ero A0le,ten Yn ) ar gveng Xn X Bn Sin(nn 272 wtore geneO 30) bub y o) XAOn thy as xx)0 yco RO thot Case nTTY/A nTr Dn TT Cn NTE Yn (o) o= Dn Dn- Ch Cn Teducesed + e Cn Ynly) CosCosh nITy .siny 3b) buHiny Cn a bove eh y b ao.sh 90) En sin n介ス Sin nT En cosろりかム wr hawe 9/2 Sin nnx dx a 9/2 9/2 Cos n dxPn cos わりTム 2 Sinnt sinDT 2 nET 49 n27 Cos/ク776 49 Bn= o dd ih1,3 2 nuodd En ard O whn nijeren Soley tion Laplace eqn n S}n D

Add a comment
Know the answer?
Add Answer to:
Thank you. 5. Find the solution u(x, y) of Laplace's equation in the rectangle 0<<a, 0<y<b...
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for? Ask your own homework help question. Our experts will answer your question WITHIN MINUTES for Free.
Similar Homework Help Questions
  • 1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x...

    1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x <a, 0 <y <b, that satisfies the boundary conditions u(0,y) = 0, u(a, y) = 0, u(x,0) = 0, u(x,b) = g(x), 0 0 0 < a. (b) Find the solution if a = 4, b = 2, and g(x) = 0 <r <a/2, a-r, a/2 < x <a.

  • We can expect the solution u(x,y) to be in the form X(x)Y(y). or I believe that...

    We can expect the solution u(x,y) to be in the form X(x)Y(y). or I believe that these are the correct forms of X(x) and Y(y). 2. Laplace's equation Consider Laplace's equation on the rectangle with 0 < x < L and 0 < < H: PDE BC BC BC u(x,0) 0, u(z, H) = g(z). (10) where a mixture of Dirichlet and Neumann boundary conditions is specified, and only one of the sides has a boundary condition that is nonhomogeneous...

  • (a) Find the solution u(x, y) of Laplace's equation in the semi-infinite strip 0 0, and the addit...

    (a) Find the solution u(x, y) of Laplace's equation in the semi-infinite strip 0<x<a, y>0, that satisfies the boundary conditions u(0, y)-0 u(a, y)-0, y > 0, and the additional condition that u(x, y) -0 as yoo, etnyla sin nTX where Cn X where Cn- NTX) where Cn = u(x, y) - -Ttny/a sin(where Cn u(x, y) n=1 u(x, y) - (b) Find the solution if f(x) = x(a-x) V(x)- (c) Let a9. Find the smallest value of yo for...

  • 2. Solve for the bounded solution of Laplace's equation v2T=0 in the UHP: [2] < 0,...

    2. Solve for the bounded solution of Laplace's equation v2T=0 in the UHP: [2] < 0, y > 0 with the following boundary conditions given on y = 0: T(x,0) = {A on x < l1, B on li < x < l2,C on x > la} A, B, C are real constants.

  • Problem 4. (25 points) Find the solution to the 2-dimensional Laplace's equation OLY + = 0...

    Problem 4. (25 points) Find the solution to the 2-dimensional Laplace's equation OLY + = 0 inside the square 0<x<1 0 <y <1 subject to the boundary conditions V(x,0) = 0 = V(x, 1) V(0,y) = 0 V(1,y) = 2 sin (31 y)

  • (a) Find the solution to Laplace's equation on a disk with boundary condition u(1,0) = 5...

    (a) Find the solution to Laplace's equation on a disk with boundary condition u(1,0) = 5 + sin(40). (You do not need to derive the general solution to the polar Laplace's equation.) (b) Verify that the solution to (a) satisfies the mean value property. (Hint: compare the average value of u(r, ) on the boundary r=1 to the value of u(r,() at r = 0.) (c) Find the minimum and maximum of the solution to (a) and verify they occur...

  • (Laplace's equation in polar coordinates) (a) Find the solution to Laplace's equation on a disk with...

    (Laplace's equation in polar coordinates) (a) Find the solution to Laplace's equation on a disk with boundary condition u(1,0) = 5 + sin(40). (You do not need to derive the general solution to the polar Laplace's equation.) (b) Verify that the solution to (a) satisfies the mean value property. (Hint: compare the average value of u(r, 0) on the boundary r = 1 to the value of u(r,) at r=0.) (c) Find the minimum and maximum of the solution to...

  • Solve heat equation in a rectangle du = k ( ou + dou), 0<x<t, 0<y< 1,...

    Solve heat equation in a rectangle du = k ( ou + dou), 0<x<t, 0<y< 1, t> 0 u(x, 0, 1) = 0, uy(x,1,1) = 0, with boundary conditions u(O, y,t) = 0, u(r, y, t) = 0, and initial condition u(x, y,0) = (y – į v?) sin(2x).

  • 3. This question is about non-homogeneous boundary conditions (a) Consider Laplace's equation on a rectangle, with...

    3. This question is about non-homogeneous boundary conditions (a) Consider Laplace's equation on a rectangle, with fully inhomogeneous boundary conditions =0 0 a, 0< y <b u(x, 0) fi() u(, b) f2(a) u(0, y)g (x) ua, y) = 92(r) 0 ra Homogenise the boundary conditions to convert the problem to one of the form 2 F(x, y) 0 xa,0 y < b + (x, 0)= fi() b(x, b) f2(x) b(0, y)0 (a, y) = 0 0y b 0 y sb...

  • In spherical polar coordinates (r, 0, ¢), the general solution of Laplace's equation which has cylindrical...

    In spherical polar coordinates (r, 0, ¢), the general solution of Laplace's equation which has cylindrical symmetry about the polar axis is bounded on the polar axis can be expressed as u(r, 0) = Rm(r)P,(cos 0), (A) where P is the Legendre polyomial of degree n, and R(r) is the general solution of the differential equation *() - n(n + 1)R = 0, (r > 0), dr dr where n is a non-negative integer. (You are not asked to show...

ADVERTISEMENT
Free Homework Help App
Download From Google Play
Scan Your Homework
to Get Instant Free Answers
Need Online Homework Help?
Ask a Question
Get Answers For Free
Most questions answered within 3 hours.
ADVERTISEMENT
ADVERTISEMENT