Create Presentation
Download Presentation

Download Presentation
## How to solve ODEs using MATHEMATICA

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Plasma Application**Modeling POSTECH EECE490O How to solve ODEs using MATHEMATICA 2006. 03. 22 Gan-Young Park and Jae-Koo Lee Department of Electronic and Electrical Engineering, POSTECH**Plasma Application**Modeling POSTECH Contents • Basic usages of MATHEMATICA • Solving ODEs • - Euler’s method • - Predictor-Corrector method ( second-order ) • - Forth-order Runge-Kutta method • - Tridiagonal matrix method**Plasma Application**Modeling POSTECH References • Textbook • - ‘Numerical and Analytical Methods for Scientists and Engineers Using Mathematica’, Daniel Dubin, Wiley, 2003 • Web Lecture • : http://www.mathought.com/main.htm**Plasma Application**Modeling POSTECH Notebook : working window ( *.nb ) Standard Screen Palettes ( File – Palettes - ) : a collection of numerical expressions or characters**Plasma Application**Modeling POSTECH Basic Usages (1) • Shift + enter : Execution (shift + enter) (shift + enter) • In[1] indicates input contents in the line. • Out[1] indicates the result of In[1] • % : a symbol indicating the result obtained right before.**Plasma Application**Modeling POSTECH • Font size • : Format - Size - Basic Usages (2) • application of Palettes**Plasma Application**Modeling POSTECH Basic Usages (3) • The names of most of functions included start with a capital letter. • included constants • - π→ Pi • - ∞ → Infinite • - e → E • - i = → I • ? (function) or ?? (function) • : shows the usage of function or options**Plasma Application**Modeling POSTECH Basic Usages (4) • symbol calculation • The symbol “ * ” for a product can be replaced by “ blank”.**Plasma Application**Modeling POSTECH Common feature Different feature Basic Usages (5) • == ( equality ), = ( substitution ), := ( definition ) < g[x] = eq. & g[x] := eq. >**Plasma Application**Modeling POSTECH Common feature Different feature Basic Usages (6) < g[x_] = eq. & f[x_] := eq. >**Plasma Application**Modeling POSTECH Basic Usages (7) • Plot[ function, {variable, a, b}, options] : drawing 2-D graph between a and b**Plasma Application**Modeling POSTECH Basic Usages (8) • PlotStyle : a option for coloring • Input - Color Selector**Basic Usages (9)**• PlotLabel : a option for labeling at top of graph • AxesLabel : a option for labeling at axes • AspectRatio : a option for adjusting the ratio of vertical to horizontal • PlotRange : a option for restricting range to plot**Plasma Application**Modeling POSTECH Basic Usages (10) • DisplayFunction → Identity : a option for not showing a graph, just memorizing it. • Show : a function to display graphs which have been shown or memorized • DisplayFunction -> $DisplayFunction : a option for showing a graph memorized.**Plasma Application**Modeling POSTECH • Since path is assigned up to StandardPackages, just sub-paths • should be written. • The symbol ` is used in Mathematica instead of \. Basic Usages (11) • Package**Plasma Application**Modeling POSTECH Basic Usages (12) • Table [contents, {range of loop}]**Plasma Application**Modeling POSTECH • Since Do, For, While don’t show results and save results, functions such as Print[] should be used for checking results. • That’s a difference among Table and above functions. Basic Usages (13) • Do [exp, {i,min,max,d}] • For [start, test, i++, body] • While [test, body]**Plasma Application**Modeling POSTECH Basic Usages (14) • Module [ {local variables}, contents ] • : The variables written at {} in Module[ ] are used as local variables which doesn’t affect global variables with same characters and can’t be used out of Module[ ].**Plasma Application**Modeling POSTECH Solving ODEs - Euler’s method - Predictor-Corrector method ( second-order ) - Forth-order Runge-Kutta method - Solving tridiagonal matrix**Plasma Application**Modeling POSTECH Euler’s method (1)**Plasma Application**Modeling POSTECH Euler’s method (2)**Plasma Application**Modeling POSTECH Forth-order Runge-Kutta method ( based on the Simpson’s 1/3 rule )**Plasma Application**Modeling POSTECH , y(0)=1, y’(0)=0 Changing 2nd order ODE to 1st order ODE, y(0)=1 (1) z(0)=0 (2) Program 9-1 ( Nakamura )**Plasma Application**Modeling POSTECH Program 9-1 ( Nakamura ) To solve 2nd-order ODE using second-order Runge-Kutta method**Plasma Application**Modeling POSTECH y(0)=1 y(0)=0 Program 9-2 ( Nakamura ) To solve ODE using fourth-order Runge-Kutta method**Program 10-1 ( Nakamura )**Plasma Application Modeling POSTECH Solve difference equation, With the boundary conditions, x = 0 1 2 9 10 i = 0 1 2 9 10 known y(0)=1 Especially for i = 1,**Program 10-1 ( Nakamura )**For i = 10, Summarizing the difference equations obtained, we write Tridiagonal matrix**Plasma Application**Modeling POSTECH Solution Algorithm for Tridiagonal Equations (1) R2 Based on Gauss elimination R3**Plasma Application**Modeling POSTECH Solution Algorithm for Tridiagonal Equations (2)**Plasma Application**Modeling POSTECH Program 10-1 ( Nakamura ) To solve boundary-value problem using tridiagonal method