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content/posts/KBhodes_index.md

@@ -28,9 +28,10 @@ Prof. Rafe Mazzeo
 ## Review {#review}
 
 1.  it suffices to study [First Order ODEs]({{< relref "KBhfirst_order_odes.md" >}}) because we can convert all [higher order functions]({{< relref "KBhgeneric.md#higher-order-functions" >}}) into a [First Order ODEs]({{< relref "KBhfirst_order_odes.md" >}})
-2.  linear systems \\(y'=Ay\\) can be solved using [eigenvalue]({{< relref "KBheigenvalue.md" >}}), [matrix exponentiation]({{< relref "KBhmatrix_exponentiation.md" >}}), etc. (recall that **special cases exists** where repeated eigenvalues, etc.)
+2.  homogeneous linear systems \\(y'=Ay\\) can be solved using [eigenvalue]({{< relref "KBheigenvalue.md" >}}), [matrix exponentiation]({{< relref "KBhmatrix_exponentiation.md" >}}), etc. (recall that **special cases exists** where repeated eigenvalues, etc.)
 3.  inhomogeneous systems \\(y' = Ay +f(t)\\) can be solved using [intergrating factor]({{< relref "KBhlinear_non_seperable_equation.md#solving-differential-equations" >}}) or [variation of parameters method]({{< relref "KBhnon_homogeneous_linear_differential_equation.md#variation-of-parameters-method" >}})
-4.  general analysis of \\(y'=f(y)\\): we can talk about stationary solutions (1. linearize each \\(y\_0\\) stationary solutions to figure local behavior 2. away from stationary solutions, use [Lyapunov Function]({{< relref "KBhnon_linear_ode.md#monotone-function" >}})s to discuss)
+4.  general analysis of non-linear \\(y'=f(y)\\): we can talk about stationary solutions (1. linearize each \\(y\_0\\) stationary solutions to figure local behavior 2. away from stationary solutions, use [Lyapunov Function]({{< relref "KBhnon_linear_ode.md#monotone-function" >}})s to discuss), or liapenov functions
+5.  for variable-coefficient ODEs, we decry sadness and [Solving ODEs via power series]({{< relref "KBhsu_math53_feb122024.md#solving-odes-via" >}})
 
 
 ## Content {#content}
@@ -75,6 +76,11 @@ What we want to understand:
 -   [SU-MATH53 FEB162024]({{< relref "KBhsu_math53_feb162024.md" >}})
 
 
+### PDEs {#pdes}
+
+-   [SU-MATH53 FEB212024]({{< relref "KBhsu_math53_feb212024.md" >}})
+
+
 ## Midterm Sheet {#midterm-sheet}
 
 -   [SU-MATH53 Midterm Sheet]({{< relref "KBhsu_math53_midterm_sheet.md" >}})

content/posts/KBhsu_math53_feb212024.md

@@ -0,0 +1,44 @@
++++
+title = "SU-MATH53 FEB212024"
+author = ["Houjun Liu"]
+draft = false
++++
+
+A [Partial Differential Equation]({{< relref "KBhpartial_differential_equations.md" >}}) is a [Differential Equation]({{< relref "KBhdiffeq_intro.md" >}}) which has more than one **independent variable**.
+
+For instance:
+
+\begin{equation}
+\pdv{U}{t} = \alpha \pdv[2]{U}{x}
+\end{equation}
+
+
+## Linear Partial Differential Equation {#linear-partial-differential-equation}
+
+A PDE is a [Linear PDE](#linear-partial-differential-equation) when it takes on the form of:
+
+\begin{equation}
+a\_{k,j}(t,x) \frac{\dd[k+j]{u}}{\dd[k]{x}\dd[j]{t}}
+\end{equation}
+
+meaning, it overall takes on the form of:
+
+\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}
+
+
+### [superposition principle]({{< relref "KBhordinary_differential_equations.md#superposition-principle" >}}) {#superposition-principle--kbhordinary-differential-equations-dot-md}
+
+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.
+
+
+## Traveling Wave {#traveling-wave}
+
+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:
+
+\begin{equation}
+u(t,x) = w(x-ct)
+\end{equation}
+
+for some constant \\(c\\), and where \\(w(x)\\) is a function which depends on only one of the two variables.

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