A Narrow, Technical Problem in Partial Differential Equations

While I was in school, one of my professors set this problem to me and my classmates and challenged us to solve it over the next few days. I found the challenge intriguing and it fascinated me, so I thought it was worth sharing. The problem was this:


Show that

\displaystyle v(x,t) = \int_{-\infty}^{\infty} f(x-y,t)g(y)dy,    (1.1)

where \displaystyle g(y) has finite support and also satisfies the PDE

\displaystyle \frac{\partial v}{\partial t} = -\kappa \frac{\partial^{2}v}{\partial x^{2}}.   (1.2)



First off, what does finite support mean? Mathematically speaking, a function has support which is characterized by a subset of its domain whose members do not map to zero, and yet are finite. (Just as a quick note: much of the proper definitions require an understanding in mathematical analysis and measure theory, something which I have not studied in detail, so take that explanation with a grain of salt.)

As for the solution, we can rewrite the given PDE as

\displaystyle \frac{\partial v}{\partial t} - \kappa \frac{\partial^{2}v}{\partial x^{2}} = 0.    (2)

The PDE requires a first-order time derivative and a second-order spatial derivative.

\displaystyle \therefore \frac{\partial v}{\partial t} = \frac{\partial}{\partial t}\int_{-\infty}^{\infty} f(x-y,t)g(y)dy,   (3.1)

and

\displaystyle \frac{\partial^{2} v}{\partial x^{2}} = \frac{\partial^{2}}{\partial x^{2}}\int_{-\infty}^{\infty} f(x-y,t)g(y)dy.    (3.2)

Next, we substitute Eqs. (3.1) and (3.2) into Eq.(2), yielding

\displaystyle \frac{\partial}{\partial t}\int_{-\infty}^{\infty} f(x-y,t)g(y)dy -\kappa \frac{\partial^{2}}{\partial x^{2}}\int_{-\infty}^{\infty} f(x-y,t)g(y)dy = 0.    (4)

Note that taking the derivative of a function and then integrating that function is equivalent to integrating the function and differentiating the same function, in conjunction with the fact that the sum or difference of the integrals is the integral of the sum or difference (proofs of these facts are typically covered in a course in real analysis). Taking advantage of these gives

\displaystyle \int_{-\infty}^{\infty} \bigg\{\frac{\partial}{\partial t}f(x-y,t)-\kappa\frac{\partial^{2}}{\partial x^{2}}f(x-y,t)\bigg\}g(y)dy = 0.   (5)

Notice that the terms contained in the brackets equate to \displaystyle 0. This means that

\displaystyle \int_{-\infty}^{\infty} 0 \cdot g(y)dy = 0.   (6)

This implies that the function \displaystyle v(x,t) does satisfy the given PDE (Eq.(2)).




References:

Definition of Support in Mathematics: https://en.wikipedia.org/wiki/Support_(mathematics)

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