Question

Solve the following Dirichlet boundary value problem for diffusion equation. u t​ =ku xx

​ , for all 00.
u(0,t)=u(10,t)=0 for all t>0.
u(x,0)=10sin(πx)+20sin(2πx)+20sin(3πx).
​

Answer 1

To solve the given Dirichlet boundary value problem for the diffusion equation, we need to use the method of separation of variables to find the solutions for X(x) and T(t). Applying the given boundary conditions will help us determine the constants and obtain the complete solution.

Step-by-step explanation:

The given problem is a Dirichlet boundary value problem for the diffusion equation, which is represented as ut = kxx. The problem requires finding the solution u(x,t) subject to the given boundary conditions and initial condition.

1. First, we need to solve the diffusion equation ut = kxx using the method of separation of variables. Assume the solution to be u(x,t) = X(x)T(t).
2. Substitute the separation of variables into the diffusion equation and rearrange the equation to obtain two separate ordinary differential equations: X''(x)/X(x) = -T'(t)/kT(t).
3. Solve each differential equation separately to obtain the solutions X(x) and T(t).
4. Apply the boundary conditions u(0,t) = u(10,t) = 0 to find the values of X(0) and X(10).
5. Finally, substitute the solutions X(x) and T(t) back into the assumed form u(x,t) = X(x)T(t) to obtain the complete solution.