Partial Differential Equations

Linear algebra is the mathematical guide of choice for implementing the principle of unit-economy⁶¹ applied to partial differential equations. The present chapter considers two kinds:

1. Single linear partial differential equations corresponding to the linear system
   $$A\vec u =\vec 0\, .$$
   However, instead of merely exhibiting general solutions to such a system, we shall take seriously the dictum which says that a differential equation is never solved until ``boundary'' conditions have been imposed on its solution. As identified in the ensuing section, these conditions are not arbitrary. Instead, they fall into three archetypical classes determined by the nature of the physical system which the partial differential equation conceptualizes.
2. Systems of pde's corresponding to an over-determined system

   $$A\vec u =\vec b\, .$$ (61)

   
   
   The idea for solving it takes advantage of the fundamental subspaces⁶²of $A$ [#!StrangLinearAlgebra!#]. Let $A$ be a $4\times 4$ matrix having rank $3$ . Such a matrix, we recall, has a vector $\vec u_r$ which satisfies $A\vec u_r =\vec 0$ , or, to be more precise

   $$A\vec u_rc_0 =\vec 0\, ,$$ (62)

   
   
   where $c_0$ is any non-zero scalar. Thus $\vec u_r$ spans $A$ 's one-dimensional nullspace
   $$\mathcal{N}(A)=span\{ \vec u_r \}~.$$
   This expresses the fact that the columnes of $A$ are linearly dependent.

   In addition we recall that the four rows of $A$ are linearly dependent also, a fact which is expressed by the existence of a vector $\vec u_\ell$ which satisfies

   $$\vec u_\ell^T A =\vec 0\, ,$$ (63)

   
   
   and which therefore spans $A$ 's one-dimensional left nullspace
   $$\mathcal{N}(A^T)=span\{ \vec u_\ell^T \}~.$$

In general there does not exist a solution to the over-determined system Eq.(6.1). However, a solution obviously does exist if and only if $\vec b$ satisfies

$$\vec u_\ell^T \vec b = 0 ~ .$$

Under such a circumstance there are infinitely many solutions, each one differing from any other merely by a multiple of the null vector $\vec u_r$ . The most direct path towards these solutions is via eigenvectors.

One of them is, of course, the vector $\vec u_r$ in Eq.(6.2). The other three, which (for the $A$ under consideration) are linearly independent, satisfy $A\vec v_i=\lambda_i \vec v_i$ with $\lambda_i \ne 0$ , or, in the interest of greater precision (which is needed in Section 6.2.3),

$$A\vec v_1 c_1 =\lambda_1 \vec v_1 c_1$$ (64)

$$A\vec v_2 c_2 =\lambda_2 \vec v_2 c_2$$ (65)

$$A\vec v_3 c_3 =\lambda_3 \vec v_3 c_3$$ (66)

where, like $c_0$ , the $c_i$ 's are any non-zero scalars. Because of the simplicity of $\vec u_\ell^T$ for the $A$ under consideration one can find the eigenvectors $\{ \vec v_1,\vec v_2,\vec v_3 \}$ , and hence their eigenvalues, by a process of inspection. These vectors span the range of $A$ ,

$${\mathcal{R}}(A)=span\{ \vec v_1,\vec v_2,\vec v_3\}~,$$

and therefore determine those vectors $\vec b$ for which there exists a solution to Eq.(6.1). Such vectors belong to $\mathcal{R}$ and thus have the form

$$\vec b=\vec v_1 b_1 +\vec v_2 b_2 +\vec v_3 b_3 ~.$$

These eigenvectors also serve to represent the corresponding solution,

$$\vec u=\vec u_r c_0 +\vec v_1 c_1 +\vec v_2 c_2 +\vec v_3 c_3 ~.$$

This, the fact that $A\vec u = \vec b$ , and the linear independence of the $\vec v_i's$ imply that the scalars $c_i$ satisfy the three equations

$$\lambda_1 c_1 =b_1$$ (67)

$$\lambda_2 c_2 =b_2$$ (68)

$$\lambda_3 c_3 =b_3~.$$ (69)

As expected, the contribution $c_4$ along the direction of the nullspace element is left indeterminate. These ideas from linear algebra and their application to solving a system, such as Eq.(6.1), can be extended to corresponding systems of partial differential equations. The Maxwell field equations, which we shall analalyze using linear algebra, is a premier example. In this extension the scalar entries of $A$ and $\vec u_\ell^T$ get replaced by differential operators, the vectors $\vec u$ and $\vec b$ by vector fields, the scalars $b_i$ and $c_i$ by scalar fields, the eigenvalues $\lambda_i$ by a second order wave operator, and the three Eqs.(6.7)-() by three inhomogeneous scalar wave equations corresponding to what in physics and engineering are called

• transverse electric ($TE$ ),
• transverse magnetic ($TM$ ), and
• transverse electric magnetic ($TEM$ ),

modes respectively.

Lecture 50

Footnotes

... unit-economy⁶¹

As identified in the footnote on Page .

... subspaces⁶²

Besides the domain and the target space, there are four of them:

(i)

the column space of $A$ , denoted by $\mathcal{R}(A)$ ,

(ii)

the nullspace (or kernel) of $A$ , denoted by $\mathcal{N}(A)$ , or also by $Ker(A)$ ,

(iii)

the row space of $A$ , denoted by $\mathcal{R}(A^T)$ , and

(iv)

the left nullspace of $A$ , denoted by $\mathcal{N}(A^T)$ .