Abstract
We study the oscillation behavior of solutions to the one-dimensional heat equation and give some interesting examples. We also demonstrate a simple ODE method to find explicit solutions of the heat equation with certain particular initial conditions.
Highlights
Consider the one-dimensional heat equation with initial condition ut (x, t) = uxx (x, t), x ∈ R, t > 0 (1) u (x, 0) = φ (x), x ∈ R.It is known that if φ (x) is a bounded continuous function defined on R, the function given by convolution integral u (x, t) = ∞1 √ e−(x−y)2 4t φ (y) dy, x ∈ R, t>0 (2)−∞ 4πt is a smooth solution of the heat equation on R × (0, ∞) with lim(x,t)→(x0,0) u (x, t) = φ (x0) for any x0 ∈ R
On, when we say u (x, t) is “the solution” of the heat equation with u (x, 0) = φ (x), x ∈ R, we always mean that it is given by the convolution integral (2)
The solution u (x, t) of the heat equation with initial data φ (x) = α + λx sin x is given by u (x, t) = α + λ e−tx sin x + 2te−t cos x, x ∈ R, t > 0, (36)
Summary
We first note that if we add an unbounded odd function, say x cos x, to the initial data in the two examples, it will not affect the values of The solution u (x, t) of the heat equation with initial data φ (x) = α + λx sin x is given by u (x, t) = α + λ e−tx sin x + 2te−t cos x , x ∈ R, t > 0, (36)
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