Question

For the following boundary-value problem (BVP):

d²u/dx²= δ(x− 1/3) for 0≤x≤1 subjected to the boundary conditions u(0) = 0, du/dx(1) = 0 Obtain the solution for u and plot it.

Answer 1

To solve the given boundary value problem (BVP), we need to integrate the equation twice, determine the constants of integration using the boundary conditions, and obtain the solution for u. The solution for u is (-1/6)x + (1/2)x². We can plot this function for better visualization.

Step-by-step explanation:

To solve the boundary value problem (BVP) d²u/dx²= δ(x− 1/3) for 0≤x≤1, subjected to the boundary conditions u(0) = 0 and du/dx(1) = 0, we can start by integrating the given equation twice with respect to x.

Integrating the equation d²u/dx²= δ(x− 1/3) twice will give us du/dx = C1 + x and u = C1x + (1/2)x² + C2, where C1 and C2 are constants of integration.

To determine C1 and C2, we can use the given boundary conditions. Substituting u(0) = 0 and du/dx(1) = 0 into the equations, we get C2 = 0 and C1 = -1/6. Therefore, the solution for u is u(x) = (-1/6)x + (1/2)x². We can plot this function to visualize the solution.