I am preparing for a big PDE exam, but I find myself woefully lost in doing even the most simple computational problems. I'm using Evans, (2nd edition), and—while it is great for theory—it is absolutely worthless, as far as I am concerned, when it comes to giving useful practical formalisms.
I have a deep fondness for algorithims and purely mechanical procedures. I am also somewhat impatient. That being said, I would very much like to know how to solve certain nice types of simple second order PDEs formally.
Here is an example of what I would consider to be an formal method—in this case, for solving an ODE with an integrating factor:
Step 1: $y^{\prime}+p\left(t\right)y=f\left(t\right)$
Step 2: $\mu\left(t\right)=e^{\int p\left(t\right)dt}$
Step 3: $\left(y\left(t\right)\mu\left(t\right)\right)^{\prime}=\mu\left(t\right)f\left(t\right)$
Step 4: $y\left(t\right)\mu\left(t\right)=C+\int\mu\left(t\right)f\left(t\right)dt$
Step 5: $y\left(t\right)=\frac{C}{\mu\left(t\right)}+\frac{1}{\mu\left(t\right)}\int\mu\left(t\right)f\left(t\right)dt$
Step 6: Use initial condition $y_{0}=y\left(t_{0}\right)$ to solve for $C$ .
Caveats: $p\left(t\right)$ and $f\left(t\right)$ have to be integrable. Need initial condition $y_{0}=y\left(t_{0}\right)$ to find a specific solution.
In other words, I'm looking for solution methods that utilize patterns of symbols, and nothing more.
• Example 1: Scanning through questions on Stack Exchange, I saw someone use the following factorization for the wave equation. Taking $u_{tt}=c^{2}u_{xx}$ , they wrote $$0=u_{tt}-c^{2}u_{xx}=\left(\frac{\partial}{\partial t}-c\frac{\partial}{\partial x}\right)\left(\frac{\partial}{\partial t}+c\frac{\partial}{\partial x}\right)$$ They then conclude to make the change of variables $\xi=x-ct,\eta=x+ct$. I, however, note that:$$\frac{\partial}{\partial t}-c\frac{\partial}{\partial x}=0\Rightarrow\frac{\partial}{\partial t}=c\frac{\partial}{\partial x}\Rightarrow\frac{\partial x}{\partial t}=c\Rightarrow x=?+ct\Rightarrow?=x-ct$$ and:$$\frac{\partial}{\partial t}+c\frac{\partial}{\partial x}=0\Rightarrow\frac{\partial}{\partial t}=-c\frac{\partial}{\partial x}\Rightarrow\frac{\partial x}{\partial t}=-c\Rightarrow x=??-ct\Rightarrow??=x+ct$$ which are of course the correct characteristics.
Does this work in more general circumstances? For instance, suppose:
$$\left(\frac{\partial}{\partial x}-a\frac{\partial}{\partial y}\right)\left(\frac{\partial}{\partial x}-b\frac{\partial}{\partial y}\right)u=F$$ Then, are the characteristics the expressions:$$\frac{\partial}{\partial x}-a\frac{\partial}{\partial y}=0\Rightarrow\frac{\partial}{\partial x}=a\frac{\partial}{\partial y}\Rightarrow\frac{\partial y}{\partial x}=a\Rightarrow y=\xi+\int adx\Rightarrow\xi=y-\int adx$$ and $$\frac{\partial}{\partial x}-b\frac{\partial}{\partial y}=0\Rightarrow\frac{\partial}{\partial x}=b\frac{\partial}{\partial y}\Rightarrow\frac{\partial y}{\partial x}=b\Rightarrow y=\eta+\int bdx\Rightarrow\eta=y-\int bdx$$ ?
(Because if they are, then this is yet more beautiful mathematics that my professors kept secret from me.)
Moreover, supposing you have a PDE in higher dimensions (say, $\mathbb{R}^{d}$ ) (or of higher order) where the differential operator “factors” into something like:$$\left(\prod_{k=1}^{K}\left(\frac{\partial}{\partial x_{m_{k}}}-a_{k}\frac{\partial}{\partial x_{n_{k}}}\right)\right)u=F$$ where $m_{1},...,m_{K}$ and $n_{1},...,n_{k}$ are sequences in $\left\{ 1,2,...,d\right\}$ , can we then obtain characteristics: $$\psi_{k}\left(x_{1},...,x_{d}\right)=x_{n_{k}}-\int a_{k}\left(x_{1},...,x_{d}\right)dx_{m_{k}}?$$ or by some variation of this method?
• Example 2: Although Evans explains how to solve $$\begin{cases} u_{t}-\nabla^{2}u=f & \textrm{in }\mathbb{R}^{n}\times\left(0,\infty\right)\\ u=g & \textrm{on }\mathbb{R}^{n}\times\left\{ t=0\right\} \end{cases}$$ on $\mathbb{R}^{n}\times\left(0,\infty\right)$ with $u=g$ on $\mathbb{R}^{n}\times\left\{ t=0\right\}$ and $c\in\mathbb{R}$ by way of the almighty heat kernel, $$K\left(\mathbf{x},t\right) :u\left(\mathbf{x},t\right)=\left(K*g\right)\left(\mathbf{x},t\right)+\int_{0}^{t}\left(K*f\right)\left(\mathbf{x},t-s\right)ds$$ he actually has the gall to leave the solution method behind: $$\begin{cases} u_{t}-\nabla^{2}u+cu=f & \textrm{in }\mathbb{R}^{n}\times\left(0,\infty\right)\\ u=g & \textrm{on }\mathbb{R}^{n}\times\left\{ t=0\right\} \end{cases}$$ as an (unanswered!) exercise!
From my perspective, if a given formalism works for a certain situation, then there should be a family of “perturbations” (i.e., changes/variations) one can make to the formulas that cause corresponding perturbations in the formula of the solution. As such, telling someone the base formula (ex: the exact solution to the non-homogeneous heat equation) without also explaining how the formula behaves under “perturbations”—and, for that matter, without explaining the kinds of perturbations that break the formula—is like giving them only the first page of a twenty page instruction manual.
I have a feeling that this particular problem be solved by making a substitution $v=\textrm{stuff involving }u$ , but the lack of any clear formalistic guidelines to how to approach such a problem are utterly maddening. Having an answer for how to do this would be quite nice, as would be knowing the limits of that method's adaptability to “perturbations” Moreover, to address more possible perturbations, what about expressions such as:$$u_{t}-\nabla^{2}u+u_{x}+u=f$$ $$u_{t}-\nabla^{2}u+t^{2}u=f$$
$$u_{t}-\nabla^{2}u+\left(ax+b+x_{1}^{2}\right)u=f$$ $$u_{t}-\nabla^{2}u+tu_{x}+3\sin\left(tx_{1}\right)u=f$$ and so on?
Obviously, we'll break the formulas if we add non-linear terms like $u^{2}$ or $uu_{x_{2}}$ , or if we add derivatives of order $3$ or higher. (But, can we add derivatives of order $2$ without breaking the method? If so, what would that entail?)
Admittedly, a lot of this is coming from my own frustration and worries. I know that the staggering diversity and variation among PDEs precludes any hopes of universally-applicable theories or solution methods, and so, especially for those doing work in the field, the kinds of questions and concerns that I'm raising here are mostly non-sequiturs.
That being said, what I'm hoping for in making this post are:
(1) The solution technique for: $$\begin{cases} u_{t}-\nabla^{2}u+cu=f & \textrm{in }\mathbb{R}^{n}\times\left(0,\infty\right)\\ u=g & \textrm{on }\mathbb{R}^{n}\times\left\{ t=0\right\} \end{cases}$$
(2) Are: $$\frac{\partial}{\partial x}-a\frac{\partial}{\partial y}=0\Rightarrow\frac{\partial}{\partial x}=a\frac{\partial}{\partial y}\Rightarrow\frac{\partial y}{\partial x}=a\Rightarrow y=\xi+\int adx\Rightarrow\xi=y-\int adx$$ and $$\frac{\partial}{\partial x}-b\frac{\partial}{\partial y}=0\Rightarrow\frac{\partial}{\partial x}=b\frac{\partial}{\partial y}\Rightarrow\frac{\partial y}{\partial x}=b\Rightarrow y=\xi+\int bdx\Rightarrow\xi=y-\int bdx$$ valid methods of obtaining characteristics for one-dimensional PDEs?
(3) Any resources (especially free ones!) that addresses these kind of “formal” questions in a transparent, easy-to-read manner?
Thanks!