A semi-infinite bar is initially at temperature 100^∘ for 0^' 1. Starting at

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A semi-infinite bar is initially at temperature 100^∘ for...

A semi-infinite bar is initially at temperature $100^{\circ}$ for $0^{\prime}<x<1$, and $0^{\circ}$ for $x>1$. Starting at $t=0$, the end $x=0$ is maintained at $0^{\circ}$ and the sides are insulated. Find the temperature in the bar at time $t$, as follows. Separate variables in the heat flow equation and get clementary solutions $e^{a^{2} \lambda^{2} t} \sin k x$ and $e^{-a^{2} k^{21}} \cos k x$. Discard the cosines since $u=0$ at $x=0$. Leok for a solution $$ u(x, t)=\int_{0}^{\infty} B(k) e^{-k^{2} a^{2 t}} \sin k x d k $$ and proceed as in Example 2. I eave your answer as an integral.

A semi-infinite bar is initially at temperature $100^{\circ}$ for $0^{\prime}<x<1$, and $0^{\circ}$ for $x>1$. Starting at $t=0$, the end $x=0$ is maintained at $0^{\circ}$ and the sides are insulated. Find the temperature in the bar at time $t$, as follows. Separate variables in the heat flow equation and get clementary solutions $e^{a^{2} \lambda^{2} t} \sin k x$ and $e^{-a^{2} k^{21}} \cos k x$. Discard the cosines since $u=0$ at $x=0$. Leok for a solution
$$
u(x, t)=\int_{0}^{\infty} B(k) e^{-k^{2} a^{2 t}} \sin k x d k
$$
and proceed as in Example 2. I eave your answer as an integral.

Key Concepts

Fourier Sine Transform

The Fourier sine transform is a tool used for solving problems defined on semi-infinite or finite intervals, particularly when the solution has to satisfy homogeneous Dirichlet boundary conditions (i.e., the solution vanishes at the boundary). It expresses functions as integrals of sine functions which naturally satisfy the zero-boundary condition, making it a powerful method for representing the solution in integral form.

Separation of Variables

Separation of variables is a technique used to reduce a partial differential equation to a set of simpler ordinary differential equations by assuming that the solution can be written as a product of functions, each depending on a single coordinate. This method is effective in problems with homogeneous boundary conditions and can lead to a series of eigenvalue problems each associated with their own eigenfunction.

Eigenfunction Expansion

Once the variables are separated and the resulting ordinary differential equations are solved, the solution is typically expressed in terms of an infinite series or a continuous integral of eigenfunctions. These eigenfunctions are determined by the boundary conditions and are orthogonal with respect to an inner product, allowing the initial conditions to be represented as a sum or integral of these functions.

Boundary and Initial Conditions

In solving partial differential equations like the heat equation, boundary and initial conditions play a crucial role in determining the unique solution. Boundary conditions can fix the temperature at the edges of the domain, while initial conditions specify the temperature distribution at the start of the observation. Together, they constrain the system and ensure that the mathematical solution corresponds to the physical problem.

Heat Equation

The heat equation is a fundamental partial differential equation that describes how heat diffuses through a material over time. Its study involves understanding how temperature changes with both space and time, and it is used to model a variety of physical processes besides heat flow, such as diffusion and the behavior of other scalar fields.

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