Answer:
To find the solution of the heat equation with the given boundary and initial conditions, we can use the method of separation of variables. Let's solve it step by step:
Step 1: Assume a separation of variables solution:
u(x, t) = X(x)T(t)
Step 2: Substitute the assumed solution into the heat equation:
X(x)T'(t) = 9X'''(x)T(t)
Step 3: Divide both sides of the equation by X(x)T(t):
T'(t) / T(t) = 9X'''(x) / X(x)
Step 4: Set both sides of the equation equal to a constant:
(1/T(t)) * T'(t) = (9/X(x)) * X'''(x) = -λ^2
Step 5: Solve the time-dependent equation:
T'(t) / T(t) = -λ^2
The solution to this ordinary differential equation for T(t) is:
T(t) = Ae^(-λ^2t)
Step 6: Solve the space-dependent equation:
X'''(x) = -λ^2X(x)
The general solution to this ordinary differential equation for X(x) is:
X(x) = B1e^(λx) + B2e^(-λx) + B3cos(λx) + B4sin(λx)
Step 7: Apply the boundary condition u(0, t) = 0:
X(0)T(t) = 0
B1 + B2 + B3 = 0
Step 8: Apply the boundary condition u(3, t) = 0:
X(3)T(t) = 0
B1e^(3λ) + B2e^(-3λ) + B3cos(3λ) + B4sin(3λ) = 0
Step 9: Apply the initial condition u(x, 0) = 5sin(7πx/3):
X(x)T(0) = 5sin(7πx/3)
(B1 + B2 + B3) * T(0) = 5sin(7πx/3)
Step 10: Since the boundary conditions lead to B1 + B2 + B3 = 0, we have:
B3 * T(0) = 5sin(7πx/3)
Step 11: Solve for B3 using the initial condition:
B3 = (5sin(7πx/3)) / T(0)
Step 12: Substitute B3 into the general solution for X(x):
X(x) = B1e^(λx) + B2e^(-λx) + (5sin(7πx/3)) / T(0) * sin(λx)
Step 13: Apply the boundary condition u(0, t) = 0:
X(0)T(t) = 0
B1 + B2 = 0
B1 = -B2
Step 14: Substitute B1 = -B2 into the general solution for X(x):
X(x) = -B2e^(λx) + B2e^(-λx) + (5sin(7πx/3)) / T(0) * sin(λx)
Step 15: Substitute T(t) = Ae^(-λ^2t) and simplify the solution:
u(x, t) = X(x)T(t)
u(x, t) = (-B2e^(λx) + B2e^(-λx) + (5sin(7πx