Find the solution to the heat equation ∂u/∂t = α²∂²u/∂x² with boundary conditions u(0,t) = u(L,t) = 0 and initial condition u(x,0) = sin(πx/L).

Answer: $u(x,t) = \sin\left(\frac{\pi x}{L}\right) e^{-\alpha^2 (\pi/L)^2 t}$

Step-by-step solution

Step 1: Assume a Separable Solution

We begin by assuming a separable solution of the form $u(x,t) = X(x)T(t)$ , where $X(x)$ is a function of $x$ only and $T(t)$ is a function of $t$ only. This method is commonly used for solving partial differential equations like the heat equation.

Step 2: Substitute into the Heat Equation

Substituting $u(x,t) = X(x)T(t)$ into the heat equation $\frac{\partial u}{\partial t} = \alpha^2\frac{\partial^2 u}{\partial x^2}$ and separating variables leads to two ordinary differential equations. Since one side depends only on $t$ and the other only on $x$ , both must be equal to a constant, which we denote as $-\lambda^2$ for convenience in solving the resulting equations.

Step 3: Solve the Spatial Equation

The spatial equation $X''(x) + \lambda^2 X(x) = 0$ with boundary conditions $X(0) = 0$ and $X(L) = 0$ yields a general solution involving sines and cosines. Applying the boundary conditions forces $C_1 = 0$ and $\lambda$ to be quantized as $\frac{n\pi}{L}$ , leading to eigenfunctions $X_n(x) = B_n \sin(\frac{n\pi x}{L})$ .

Step 4: Solve the Temporal Equation

The temporal equation $T'(t) + \alpha^2 \lambda^2 T(t) = 0$ is a first-order linear differential equation. Its solution is an exponential decay function, where $\lambda$ takes the values determined from the spatial problem.

Step 5: Form the General Solution

Combining the spatial and temporal solutions, and using the principle of superposition, we form the general solution as an infinite series. Each term in the series corresponds to a particular value of $n$ .

Step 6: Apply Initial Condition

To find the coefficients $B_n$ , we apply the initial condition $u(x,0) = \sin(\frac{\pi x}{L})$ . This involves using the orthogonality of sine functions. For the given initial condition, only the $n=1$ term will be non-zero.

Step 7: Determine Coefficients and Final Solution

Evaluating the integral for $B_n$ , we find that $B_1 = 1$ and all other $B_n = 0$ . Substituting these coefficients back into the general solution gives the particular solution that satisfies all given conditions.

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