1.2: PDE’s
- Page ID
- 8368
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Rather than giving a strict mathematical definition, let us look at an example of a PDE, the heat equation in 1 space dimension
\[\dfrac{\partial^2 u(x,t)}{\partial x^2} = \frac{1}{k} \dfrac{\partial u(x,t)}{\partial t}. \label{eq:I:heat} \]
It is a PDE since partial derivatives are involved.
To remind you of what that means: \(\dfrac{\partial}{\partial x}{u(x,t)}\) denotes the differentiation of \(u(x,t)\) w.r.t. \(x\) keeping \(t\) fixed,
\[\dfrac{\partial}{\partial x} (x^2t+xt^2) = 2xt+t^2. \nonumber \]
- Equation \ref{eq:I:heat} called linear since \(u\) and its derivatives appear linearly, i.e., once per term. No functions of \(u\) are allowed. Terms like \(u^2\), \(\sin(u)\), \(u\dfrac{\partial}{\partial x}{u}\), etc., break this rule, and lead to non-linear equations. These are interesting and important in their own right, but outside the scope of this course.
- Equation \ref{eq:I:heat} is also homogeneous (which just means that every term involves either \(u\) or one of its derivatives, there is no term that does not contain \(u\)). The equation \[\dfrac{\partial^2}{\partial x^2}{u(x,t)} = \frac{1}{k} \dfrac{\partial}{\partial t}{u(x,t)}+\sin(x) \nonumber \] is called inhomogeneous, due to the \(\sin(x)\) term on the right, that is independent of \(u\).
Why is all that so important? A linear homogeneous equation allows superposition of solutions. If \(u_1\) and \(u_2\) are both solutions to the heat equation,
\[\dfrac{\partial^2}{\partial x^2}{u_1(x,t)} - \frac{1}{k} \dfrac{\partial}{\partial t}{u_1(x,t)}= \dfrac{\partial}{\partial t}{u_2(x,t)} - \frac{1}{k} \dfrac{\partial^2}{\partial x^2}{u_2(x,t)}=0, \label{eq:I:heatsol} \]
any combination is also a solution,
\[\dfrac{\partial^2}{\partial x^2}{[a u_1(x,t)+bu_2(x,t)]} - \frac{1}{k} \dfrac{\partial}{\partial t}{[au_1(x,t)+b u_2(x,t)]}=0. \nonumber \]
For a linear inhomogeneous equation this gets somewhat modified. Let \(v\) be any solution to the heat equation with a \(\sin(x)\) inhomogeneity,
\[\dfrac{\partial^2}{\partial x^2}{v(x,t)} - \frac{1}{k} \dfrac{\partial}{\partial t}{v(x,t)}=\sin(x). \nonumber \]
In that case \(v+au_1\), with \(u_1\) a solution to the homogeneous equation, see Equation \ref{eq:I:heatsol}, is also a solution,
\[\begin{aligned} \dfrac{\partial^2}{\partial x^2}{[v(x,t)+a u_1(x,t)]} - \frac{1}{k} \dfrac{\partial}{\partial t}{[v(x,t)+a u_1(x,t)]}&=&\nonumber\\ \dfrac{\partial^2}{\partial x^2}{v(x,t)} - \frac{1}{k} \dfrac{\partial}{\partial x}{v(x,t)} +a \left(\dfrac{\partial}{\partial x}{u_1(x,t)} - \frac{1}{k} \dfrac{\partial}{\partial t}{u_1(x,t)}\right)&=& \sin(x).\end{aligned} \nonumber \]
Finally we would like to define the order of a PDE as the power in the highest derivative, even it is a mixed derivative (w.r.t. more than one variable).
Which of these equations is linear? and which is homogeneous?
- \(\dfrac{\partial^2 u}{\partial x^2} + x^2 \dfrac{\partial u}{\partial y} = x^2 + y^2\)
- \(y^2\dfrac{\partial^2 u}{\partial x^2}+u\dfrac{\partial u}{\partial x} + x^2\dfrac{\partial^2 u}{\partial y^2} = 0\)
- \(\dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial x^2}=0\)
- Answer
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TBA
What is the order of the following equations?
- \(\dfrac {\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} = 0\)
- \(\dfrac{\partial^2 u}{\partial x^2} - 2\dfrac{\partial^4 u}{\partial^3 x \partial y} + \dfrac{\partial^2 u}{\partial y^2}= 0\)
- Answer
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TBA