Boundary Value Problems
A few topics related to Boundary Value Problems. Starting with Fourier Series, Sine Series and Cosine Series.
Table of Contents
- Functions
- Introduction to Functions
- Function Fun, Shifting
- Euler Equation Solutions
- Bessel Function of First Kind
- Fourier Series
- Even and Odd Periodic Extensions
- Fourier Sine Series
- Fourier Sine and Cosine Series
- Fourier Series Function Guess
- Fourier Cosine Series to Fourier Cosine Transform
- Making Complex Fourier Transform Real
- Fourier Transform of e^(-a |x| )
- Diffusion Equations
- Green's Function
- Hyperbolic Equations
- Convection of a Line
- Convection of a Sine Wave
- Semi-Infinite Wave, D'Alembert Solution
Introduction to Functions
This applet illustrates the relation between the independent variable [math](x)[/math] and the dependent variable [math](y)[/math] for functions.[br][br]The drop down list allows you to chose a function to explore. The last function can be edited in the box that appears.[br][br]You can move the dot on the [math]x[/math] axis to change the [math]x[/math] value. The output [math]y[/math] value is shown along with arrows indicating the flow of information.[br][br]The middle mouse or shift mouse allows you to move the graph. The mouse wheel scales the graph. You scale the axes independently with the middle mouse over the axis.
Choose a function from the dropdown box. Choosing the last function in the list will allow you to enter your own function in the entry box. You can use some of the suggestions.[br][br]Are there [math]x[/math] values where the [math]y[/math] value is undefined or does not exist?[br][br]Are there [math]y[/math] values you cannot get with any [math]x[/math] value?[br][br]Are there more than one[math]y[/math] values for any [math]x[/math] values?[br][br]Why is the [math]x[/math] value called the independent variable?[br][br]If the [math]y[/math] value is undefined, then the corresponding [math]x[/math] value cannot be in the [u]domain[/u] of the function.[br]If a [math]y[/math] value cannot be obtained with any [math]x[/math] value ( in the domain ) it is not in the [u]range[/u] of the function.
Even and Odd Periodic Extensions
Periodic Extensions
Description
Enter a function of x for f(x) and a limit L. Selecting one of the check boxes will show periodic extensions of this function. [br][list][*]-L<x<L Periodic will show the 2L periodic extension[/*][*]0<x<L Periodic will show the periodic extension with period L[/*][*]Even Extension will show the even periodic extension with period 2L[/*][*]Odd Extension will show the odd extension with period 2L[/*][/list]
Green's Function
Description
Here Green's Function is the response to a unit impulse function at [math]x=\xi[/math] of the one dimensional heat equation [math]u_t=\alpha^2u_{xx}[/math]. It can be used to solve the heat equation with an arbitrary initial condition as a convolution integral [math]u\left(x,t\right)=\frac{1}{2\alpha\sqrt{\pi t}}\int_{-\infty}^{\infty}f\left(x\right)e^{-\frac{\left(x-\xi\right)^2}{4\alpha^2t}}d\xi[/math].
Two Impulses
Because the boundary conditions are zero at [math]\pm\infty[/math] and the heat equation solution results in zero for [math]u_t-\alpha^2u_{xx}=0[/math] , solutions can be added to obtain a new solution. By using Greens Function an initial condition plus a forcing function solution can be obtained through multiple integrations. To illustrate, the applet below shows an impulse response at [math]t=0[/math] and [math]x=\xi_1[/math] followed by another impulse at [math]t=t_2[/math] and [math]x=\xi_2[/math]. The final solution is the sum of the other two solutions. In this manor a general solution to the heat equation with forcing is[br][br]
Convection of a Line
Description
The Convection Equation is [math]u_t=-Vu_x[/math] where [math]V[/math] is a constant.[br]For an initial condition of a line [math]u\left(x,0\right)=mx+b[/math] substitution of [math]u_x=m[/math] gives[br][math]u_t=-Vm[/math] which, since [math]V[/math] and [math]m[/math] are both constants, can be integrated from the initial time to give[br][br][size=150][math]u\left(x,t\right)=u\left(x,0\right)-Vmt[/math][/size][br][br]This applet graphs the initial condition and the value at a later time, [math]t[/math]. Two arrows indicate the change. One arrow indicates the change in [math]u[/math] at [math]x=1[/math] and the other indicates the line movement in the [math]x[/math] direction.[br][br]Several sliders are available to change all of the parameters.
Activities
Observe what happens as you adjust each slider.[br][br]Particularly interesting is what happens to the [math]Vt[/math] displacement as the slope [math]m[/math] is changed.[br][br]A curve can be considered as a set of short connected line segments so how would a curve 'move'?[br][br]