A laterally insulated bar of length 10 cm and constant...

A laterally insulated bar of length $10 \mathrm{cm}$ and constant cross-sectional area $1 \mathrm{cm}^{2}$, of density $10.6 \mathrm{gm} / \mathrm{cm}^{3}$, thermal conductivity $1.04 \mathrm{cal} / \mathrm{cm}$ sec $^{\circ} \mathrm{C}$, and specific heat conductor $)$ has initial temperature $f(x)$ and is kept at $0^{\circ} \mathrm{C}$ at the ends $x=0$ and $x=10$, Find the temperature $u(x, t)$ at later times, Here, $f(x)$ equals: $$f(x)=1-0.2|x-5|$$

Key Concepts

Fourier Series

Fourier series are used to express a function as a sum of sine and cosine terms. In the context of the heat equation, the initial temperature distribution is often expanded into a Fourier series in order to match the spatial eigenfunctions obtained from the separation of variables method. This expansion allows for the construction of the general solution as a series whose coefficients are determined by the initial condition.

Heat Equation

The heat equation is a partial differential equation that describes how the distribution of heat evolves over time within a given region. It is a parabolic PDE and serves as a mathematical model for the diffusion of heat (or variation in temperature) in a medium. This equation typically takes the form u_t = k*u_xx, where u(x,t) is the temperature, k is the thermal diffusivity, and the subscripts denote partial derivatives with respect to time and space.

Initial and Boundary Conditions

In problems involving the heat equation, the initial condition specifies the temperature distribution at the start of the observation, while the boundary conditions define the behavior of the temperature at the boundaries of the domain at all later times. These conditions are crucial to determine a unique solution to the heat equation. For instance, one may have fixed (Dirichlet) boundary conditions that maintain the temperature at a constant value along the boundaries.

Separation of Variables

Separation of variables is a powerful method for solving partial differential equations like the heat equation. The technique assumes that the solution can be written as a product of functions, each depending on a single independent variable. By substituting this product into the heat equation, the PDE can be separated into a set of ordinary differential equations, each of which is easier to solve individually.

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