If heat is generated at a constant rate throughout a bar of...

If heat is generated at a constant rate throughout a bar of length $L=\pi$ with initial temperature $f(x)$ and the ends at $x=0$ and mare kept at temperature $0,$ the heat equation is $u_{1}=c^{2} u_{x x}+H$ with constant $H>0 .$ Solve this problem. Hint. Set $u=v-H x(x-\pi) /\left(2 c^{2}\right)$.

Key Concepts

Separation of Variables

This method involves assuming that the solution can be written as a product of functions, each of which depends on a single independent variable. For the homogeneous heat equation, this leads to splitting the PDE into ordinary differential equations in time and space, which can then be solved separately while satisfying the boundary conditions.

Eigenfunction Expansion

After separation of variables, the resulting spatial equation typically forms a Sturm?Liouville problem. Its eigenfunctions, which satisfy the prescribed boundary conditions, form an orthogonal basis. The solution to the homogeneous problem can then be expressed as a series expansion in these eigenfunctions, with coefficients determined by projecting the initial condition onto this basis.

Inhomogeneous Heat Equation

This concept refers to a heat equation that includes an additional term representing a source or sink of heat. In such equations, the temperature is influenced not only by spatial diffusion but also by internal generation (or consumption) of heat, making the problem more complex than the standard homogeneous case.

Change of Variables to Eliminate the Forcing Term

A common technique used to solve an inhomogeneous partial differential equation is to perform a change of variables that 'absorbs' the non-homogeneous term. By finding a particular solution that exactly satisfies the source term, one can subtract it from the original equation, thereby reducing the problem to a homogeneous one which is easier to solve.

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