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title = "SU-MATH53 FEB212024" | ||
author = ["Houjun Liu"] | ||
draft = false | ||
+++ | ||
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A [Partial Differential Equation]({{< relref "KBhpartial_differential_equations.md" >}}) is a [Differential Equation]({{< relref "KBhdiffeq_intro.md" >}}) which has more than one **independent variable**. | ||
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For instance: | ||
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\begin{equation} | ||
\pdv{U}{t} = \alpha \pdv[2]{U}{x} | ||
\end{equation} | ||
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## Linear Partial Differential Equation {#linear-partial-differential-equation} | ||
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A PDE is a [Linear PDE](#linear-partial-differential-equation) when it takes on the form of: | ||
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\begin{equation} | ||
a\_{k,j}(t,x) \frac{\dd[k+j]{u}}{\dd[k]{x}\dd[j]{t}} | ||
\end{equation} | ||
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meaning, it overall takes on the form of: | ||
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\begin{equation} | ||
a\_{m,n}(t,x) \frac{\partial^{m+n}{u}}{\partial^{m}{x}\partial^{n}{t}} + \dots + a\_{1,0}(t,x) \pdv{u}{x} + a\_{0,1}(t,x) \pdv{u}{t} + a\_{0,0}(t,x) u = f(t,x) | ||
\end{equation} | ||
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### [superposition principle]({{< relref "KBhordinary_differential_equations.md#superposition-principle" >}}) {#superposition-principle--kbhordinary-differential-equations-dot-md} | ||
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like homogeneous linear [ODE]({{< relref "KBhordinary_differential_equations.md" >}})s, homogeneous [Linear PDE](#linear-partial-differential-equation)s also satisfy the superposition principle: any linear combinations of solutions are a solution. | ||
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## Traveling Wave {#traveling-wave} | ||
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For two-variable [PDE]({{< relref "KBhpartial_differential_equations.md" >}})s, it is called a [Traveling Wave](#traveling-wave) if solutions to \\(u\\) takes on the form: | ||
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\begin{equation} | ||
u(t,x) = w(x-ct) | ||
\end{equation} | ||
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for some constant \\(c\\), and where \\(w(x)\\) is a function which depends on only one of the two variables. |
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