Using Laplace Equation PDE 40(a) Solve for u(x,y): x.x (b) From your solution, evaluate u(3, 1), let's say correct to two decimal places 40(a) Solve for u(x,y): x.x (b) From your solution, e...
14 points Consider the following equation : PDE: u+ 0 ,0<x <1, 0<y <1 BCs: u(0, y)= 0, u (1, y ) = 0 ,0<y <1 ICs: u (x,0)=0, u (x,1)=2 ,0<x <1 a) Using the PDE and the boundary conditions write the form of the solution u (x ,t) b) Now apply the initial condition to solve for the unknown coefficients in the solution from part (a)
14 points Consider the following equation : PDE: u+ 0 ,0
Using Laplace Equation PDE
42.(a) Solve for u(r, e): That is, the region is an annulus betweenr 1 andr 2. HINT: First draw a picture of it, to get a look at the problem. Now, you should be able to readily get Then, see that you have 27-periodicity, so K n (n-1, 2, ) and D-0, so u (r, θ) A' + B' In r + an infinite series with r's and θ's in it. But look at your picture:...
3. Solve the following integral equations using Laplace transforms. (a) (t)= te! - 2e x(u)e"du (b) y(t) 1 - sinht +(1+T)y(t - T)dT. netions
3. Solve the following integral equations using Laplace transforms. (a) (t)= te! - 2e x(u)e"du (b) y(t) 1 - sinht +(1+T)y(t - T)dT. netions
Fourier transform:
3. Consider the equation a(x, 0) = f(x) u(x,t) lim 0 Using a Fourier transform, solve this equation. Evaluate your solution in the case when f(x)-δ(x).
3. Consider the equation a(x, 0) = f(x) u(x,t) lim 0 Using a Fourier transform, solve this equation. Evaluate your solution in the case when f(x)-δ(x).
Use the finite difference approach to solve the following differential equation with Δ,-2 and y 0-5 and answer to five decimal places) )(20) 8, (Round the final d y The solution of the equation at x= 12 is 40 points Skipped
Use the finite difference approach to solve the following differential equation with Δ,-2 and y 0-5 and answer to five decimal places) )(20) 8, (Round the final d y The solution of the equation at x= 12 is 40...
7.4 Solve the Laplace equation Δ11-0 in the square 0 < x, y < π, subject to the bound- ary condition 11(0, y) u(T, y) = 0. 11(x, 0) = 11(x, π) = 1, = 1/(π, y) =
7.4 Solve the Laplace equation Δ11-0 in the square 0
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 nx · sin ny. [Suggested Solution Steps for Problem 3] (1) Apply the method of separation of variables as w(x,y) = X(x) · Y(y); (2) substitute into the...
2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F is given as 1; x< 10 x > 10 u(x, 0) = F(x) = use part (a) to write the solution u(x, t) c) Sketch u(x,0) and u(x,1) on the same u-versus-x graph d) Explain in your own...
2. In lectures we solved the heat PDE in 1 +1 dimensions with constant-temperature boundary conditions u(0,t)u(L,t) -0. If these boundary conditions change from zero temperature, we need to do a little bit more work. Consider the following initial/boundary-value problem (IBVP) 2 (PDE) (BCs) (IC) u(0,t) = a, u(x,00, u(L, t)=b, st. and let's take L = 1, a = 1, b = 2 throughout for simplicity. Solve this problem using the following tricks b and A"(x)-0 (a) Find a...
3. Using Laplace transform, solve the differential equation y" +2y' +y=te* given that y(0) = 1, y'(0)= -2.