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Answer :
The long-term behavior of the solution can be observed by analyzing the solution at different time points. It can be seen if the temperature distribution reaches a stable pattern or approaches a steady-state where there is no further change over time.
The given problem represents the 2D heat equation with the diffusion coefficient k. The equation is k * (uₓₓ + uᵧᵧ) = uₜ, where u represents the temperature distribution in the x-y plane at time t. The boundary conditions for this problem are:
u(0, y, t) = u(a, y, t) = 0: This means that the temperature at the boundaries of the x-axis (x = 0 and x = a) remains fixed at zero. It implies that there is no heat flow across these boundaries.
uₕᵧ(1, 0, t) = uₕᵧ(r, b, t) = 0: This condition states that the temperature gradient with respect to y (uₕᵧ) at the boundaries of the y-axis (y = 0 and y = b) is zero. It indicates that there is no heat flow perpendicular to the y-axis at these boundaries.
u(x, y, 0) = f(x, y): This specifies the initial temperature distribution f(x, y) at time t = 0. It serves as the starting point for the time evolution of the temperature distribution.
The physical interpretation of these boundary conditions is as follows:
The zero temperature boundary conditions at x = 0 and x = a imply that the two ends of the material are kept at a constant temperature of zero. This could represent a situation where the material is in contact with a heat sink or an environment maintained at a fixed low temperature.
The zero temperature gradient boundary conditions at y = 0 and y = b indicate that there is no heat flow perpendicular to the y-axis at the boundaries. This can represent situations where the material is insulated or has a symmetrical heat distribution with respect to the y-axis.
For the extra credit part, let's consider the specific values provided: k = 0.5, a = b = 10, and 1: 4 < x < 6, 4 < y < 6, f(x, y) = 0 otherwise.
To solve the problem and visualize the dynamics, we can use a series solution approach such as the Fourier series method. The series solution will involve an infinite sum of terms, but for simplicity, let's consider the first 10 terms of the series.
By solving the 2D heat equation with the given boundary conditions and initial condition, we can obtain the solution u(x, y, t) at different time points: t = 0, 0.1, 0.2, 1, 10, 25, 50, 100, and 500.
The long-term behavior of the solution can be observed by analyzing the solution at different time points. It can be seen if the temperature distribution reaches a stable pattern or approaches a steady-state where there is no further change over time.
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