1. Solve the Laplace equation in the disk of radius a with the boundary condition uIr=a în3(8). V...
(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 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 the diffusion equation in the disk of radius a, with u = B on the boundary and u = 0 when t = 0, where B is constant Solve the diffusion equation in the ball of radius a, with u = B on the boundary and u = C when t 0, where B and C are constants. Solve the diffusion equation in the disk of radius a, with u = B on the boundary and u = 0...
PDE problem diffusion equation Solve the diffusion equation in the disk of radius a, with u = t 0, where B is constant B on the boundary and u = 0 when Solve the diffusion equation in the disk of radius a, with u = t 0, where B is constant B on the boundary and u = 0 when
Consider the Laplace equation on a circle of radius a around the origin of the xy-plane: p?u=0, Osr<a, -Isosa. The boundary condition is u(a,0)= p cos?o, with p a positive constant. Find the solution u(r,o) by separation of variables. Require that the solution is finite at r = 0, and that the solution is continuous with a continuous derivative at 0 = Ín. To check your solution, set r = a and 0 = 0. You should get u(a,0) =...
(1 point) Solve the boundary value problem by using the Laplace transform 22 w ²w + sin(6ax) sin(16t) = 0 < x < 1, t> 0 дх2 dt2 w(0,t) = 0, w(1,t) = 0, t> 0, w(x,0) = 0, dw -(x,0) = 0, 0 < x < 1. dt First take the Laplace transform of the partial differential equation. Let W be the Laplace transform of w. Then W satisfies the ordinary differential equation W" = subject to W(0) =...
Let a >0 Solve the following Laplace's equation in the disk: with the boundary conditions Assume that is a given periodic function with satisfying f (0) = f (2π) and Moreover, u(r,0 is bounded for r s a Which of the following is the (general) solution Select one: A. where for B. where )cos(n)de and for C. where and 2m for n- 1,2,3, D. where Co E R f(0) cos(n0)de and for Let a >0 Solve the following Laplace's equation...
5. Solve the Laplace equation 0 inside the annocular domain R1 < r < R2 with boundary conditions or Ri 5. Solve the Laplace equation 0 inside the annocular domain R1
3. Consider another Volterra integral equation (a) Solve the integral equation (4) by using the Laplace transform. (b) Convert the integral equation (4) into an initial value problem, as in Problem 2. (c) Solve the initial value problem in part (b), and verify the solution is the same as the one in part (a)
the below is the previous question solution: 1. Recall the following boundary-value problem on the interval [0, 1] from Homework 2: f" =-Xf, f'(1) =-f(1). f(0) = 0, Show that if (Anh) and to this boundary-value problem, λι, λ2 〉 0, λιメÂn then fi and f2 are orthogonal with respect to the standard inner product (.9)J( gr)dr. (You may use the solution posted on the course website, or work directly from the equation and boundary conditions above.) (λ2'J2) are two...