PDEs: L, H, and O

One of the math classes I’m taking this semester is on partial differential equations (or PDEs). In lieu of actually studying this troublesome topic, I think I’m going to write a few posts about it. Which will likely take more time, and which will count as studying anyway. Why am I so masochistic? Meh.

Anyways. So partial differential equations are just equations of partial derivatives of some function. As with ordinary differential equations (ODEs), the goal is to find that function. For example, you may have this equation: \frac{{\partial u}}{{\partial x_1 }} + \frac{{\partial u}}{{\partial x_2 }} = 0, written more concisely as u_{x_1 }  + u_{x_2 }  = 0. The goal, then, is to find the function u(x_1,x_2)

Just like ODEs, PDEs are difficult to solve in general. There are many different kinds, some which are easy to solve, some which are impossibly difficult. So this is usually the point where you need to distinguish PDEs into two very general categories: linear and nonlinear. A linear PDE is one which is linear in the solution and its derivatives. Essentially, all it means is that you shouldn’t multiply u with any of its derivatives, or any derivative of u with another derivative of u.

Linear: u_x  + x \cdot u_y  = 0
Nonlinear: u_x  + u \cdot u_y  = 0

The technical definition of linearity says that for any constants a and b, and for any functions f and g, the mapping \mathcal{L} satisfies the following property: \mathcal{L}[a\cdot f+b\cdot g] = a\cdot\mathcal{L}[f]+b\cdot\mathcal{L}[g]. In such a case, \mathcal{L} is called a linear mapping or a linear operator. (You’ve probably seen this before if you’ve studied calculus or vector algebra.) Since partial derivatives are themselves functions, we can see if a particular PDE is linear or not by just applying the definition.

If we denote \mathcal{L}[u] = u_x  + x \cdot u_y, which was our example above, we can check via the definition that it is indeed linear:

\mathcal{L}[a\cdot f+b\cdot g] = \frac{\partial }{{\partial x}}\left( {a\cdot f+b\cdot g} \right) + x\cdot \frac{\partial }{{\partial y}}\left( {a\cdot f+b\cdot g} \right) =
a \cdot f_x + b \cdot g_x  + ax\cdot f_y  + bx\cdot g_y  =
a \left(f_x + x\cdot f_y \right) + b \left(g_x + x\cdot g_y \right) = a\cdot\mathcal{L}[f] + b\cdot\mathcal{L}[g]. A similar exercise shows that the second example is not linear. I’m only going to talk about linear PDEs for the next few posts.

In that vein, you can further divide linear PDEs based upon their homogeneity. Homogeneity is another one of those general mathematical principles that pops up everywhere in a variety of different flavors. In our case, a homogeneous equation refers to when \mathcal{L}[u] is linear and is equal to zero, and an inhomogeneous equation is one in which \mathcal{L}[u] is linear, but is equal to some function f.

Another way to categorize PDEs is based upon their order. Order just refers to the highest derivative in the equation. Both of the examples above are first order, since they contain first derivatives only.

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