Final answer:
The student is looking for a finite-difference solution to the heat equation, ∂u/∂t = ∂²u/∂x², subject to certain initial and boundary conditions. The problem is discretized and solved numerically using difference equations that approximate the heat equation's derivatives over a grid.
Step-by-step explanation:
Finite-Difference Solution for a Heat Equation
The student is asking for a finite-difference solution to a partial differential equation, specifically the heat equation, which models the distribution of heat (or variations in temperature) in a given region over time. The equation given is the one-dimensional, time-dependent heat equation without sources, ∂u/∂t = ∂²u/∂x², where u(x, t) is the temperature distribution function of position x and time t.
To calculate the finite-difference solution, we typically discretize both the spatial domain (x) and the temporal domain (t) using a grid. The boundaries at x=0 and x=1 are held at constant temperature, namely u(0, t) = u(1, t) = 0 for all t > 0, which are the Dirichlet boundary conditions. The initial condition is provided by u(x, 0) = sin(πx), setting the initial distribution of temperature along the rod.
The finite-difference method involves approximating the derivatives in the heat equation with difference equations. For instance, the temporal derivative ∂u/∂t could be approximated by [u(x, t+Δt) - u(x, t)]/Δt, and the spatial derivative ∂²u/∂x² could be approximated by [u(x+Δx, t) - 2u(x, t) + u(x-Δx, t)]/(Δx)². By solving these difference equations iteratively, one can obtain a numerical approximation of the equation's solution.