c) Solve the equation for A2 > 0, First impose the boundary conditions φ(0) = 0...
solve the PDE
+u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
3. Consider the damped wave equation with boundary conditions where 0 < β < 21tc/ L. (i) Explain the physical meaning of the term-8ut. Why is β > 0? (ii) Explain the physical meaning of the boundary conditions. ii) Using separation of variables and superposition, solve the initial value problem (iv) What is the long-time behavior of the solution?
9. Solve the wave equation subject to the boundary and initial conditions u(0,t) = 0, u(x,0) = 0, U(TT, t) = 0, t> 0 $ (3,0) = sin(x), 0<x<a
Solve the heat equation Ut = Uxx
+ Uyy on a square 0 <= x <= 2, 0<= y<= 2 with the
following boundary and initial conditions
2. Solve the heat equation boundary conditions uvw on a square O S r s 2, 0 S vS 2 with the (note the mix of u and tu) and with initial condition 0 otherwise Present your answer as a double trigonometric sum.
2. Solve the heat equation boundary conditions uvw on a...
3.24 Solve the differential equation in Example 3.4.1 for the mixed boundary conditions u(0) = 0, (d) = 1 dx/x=1 Use the uniform mesh of three linear elements. The exact solution is mm)_ 2 cos(1 – 2) - sin 2 - + x2 – 2 cos(1) Answer: U2 = 0.4134, Uz = 0.7958, U4 = 1.1420, (Q1)def = -1.2402. Example 3.4.1 Use the finite element method to solve the problem described by the following differential equation and boundary conditions (see...
(1 point) Solve the following differential equation with the given boundary conditions -If there are infinitely many solutions, use c for any undetermined constants - If there are no solutions, write No Solution - Write answers as functions of 2 (ie.y=y(2)). y" +9y=0 • A) Boundary conditions: y(0) = 2 • B) Boundary conditions: y(0) = 2 y= No Solution • C) Boundary conditions: y(0) = 2 No Solution
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...
3. Find all critical points of dt dt with the constraint PP = 8 0 (c and boundary conditions x(0) - 0, x(1)- 3. Hint: Write the Euler Lagrange equation (there is no dependence on t), and then use the boundary conditions and the constraint to reach a system of 2 equations (with quadratic terms) of two unknown constants a, b Solve it by first finding a quadratic equation for a/b
3. Find all critical points of dt dt with...
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
PDE Problem: homogenous diffusion equation with non-homogenous
boundary conditions
27. Solve the nonhomogeneous initial boundary value problem | Ut = kuzz, 0 < x < 1, t > 0, u(0, t) = T1, u(1,t) = T2, t> 0, | u(x,0) = 4(x), 0 < x < 1. for the following data: (c) T1 = 100, T2 = 50, 4(x) = 1 = , k = 1. 33x, 33(1 – 2), 0 < x <a/2, /2 < x < TT, [u(x,...