Differential Equations Demos Table of Contents (ToC):

Transcription

1 2 DiffEqnsDemos.nb Differential Equations Demos Table of Contents (ToC): * Mathematica Demo Central (Welcome) * Mathematica Demo Central (Quick Facts) * Gravitational Field (DEs Motivation) * First Order DE Solution Finder and Plotter * Exact Equations * First Order Linear Equations * Bifurcation Diagram * Euler s Method * Mass Spring Damper Graphical Explorer * Mass Spring Damper Complexification Method * Plot of Vector Field * Higher Order Numerical Methods * Euler s Method System * Student Solution RK4 System * Mass Spring Damper Analytical Explorer * First Order Linear System * Laplace Transform (First Order) * Laplace Transform (Second Order) * Impulse Response Explorer * Impulse Response Approximation Method Welcome to the Mathematica Demo Central (Differential Equations) Demos Directions: You can view this Slide Show in full screen mode by following Palettes Slide Show Start Presentation (see below screen shot) (1) You can tab through the Slides using the navigation bar in the slide show navigator. (2) After selecting your demo, select the cell right below the comment (*main code hiding below*) which looks and evaluate it using "# (3) You will then be free to explore any interactive graphics that are available to you.

2 DiffEqnsDemos.nb 3 4 DiffEqnsDemos.nb (Quick Facts and Examples) Description: In this demo, some quick facts and examples are given to introduce you to Mathematica. Quick Facts: (Motivation) Motion in a Gravitational Force Field Directions: For a defined force field and initial conditions, explore the trajectories over time. * Site license for student and faculty personal machines: ( * Hands-On Start to Mathematica video tutorials: ( * Sample Code (screen shot):

3 DiffEqnsDemos.nb 5 6 DiffEqnsDemos.nb (Differential Equations) Solution Finder and Plotter Directions: For a defined first order Differential Equations (DE) with initial conditions (IC) find the exact (if it exists) and numerical solutions and show time plots. (Differential Equations) Exact Equations Directions: For a given exact equation, explore the relationship between the potential function surface, the specific contour, and vector field Note: If no exact solution exists, go to Higher Order Numerical Methods demo below.!

4 DiffEqnsDemos.nb 7 8 DiffEqnsDemos.nb (Differential Equations) First Order Linear Directions: For a defined first order linear DE, explores the relationship of the particular and homogenous solutions. (Differential Equations) Bifurcation Diagram Explorer Description: Input the right-hand-side of an autonomous DE dy = f (y, μ) and dt explore the behavior of the slope field for various values of μ

5 DiffEqnsDemos.nb 9 10 DiffEqnsDemos.nb (Differential Equations) Euler s Method Description: For a defined vector field, explore graphically and numerically Euler s Method for a first order DE. This example is the first in a series of steps that culminates with Runge-Kutta Order 4 for not only a first order DE but also a system of first order DEs. Note: This Euler s numerical method demo compares the numerical and exact solutions of a DE. It therefore works only if there is an exact solution to the DE. Use Higher Order Methods if there is no exact solution to see only the numerical solution. (Differential Equations) MassSpringDamper Graphical Explorer Description: Explores the second order, MassSpringDamper differential equation with a sinusoidal forcing function to build intution about the meaning of the parameters A (amplitude of forcing), Ω (forcing frequency), ω (natural frequency), and δ (damping).

6 DiffEqnsDemos.nb DiffEqnsDemos.nb (Differential Equations) Forced Mass Spring Damper (Complexification Method) Description: For a defined mass (m), spring (k), damper (b) system with sinusoidal forcing solve the second order MSD by complexifying the forcing function. (Differential Equations) Plot of Vector Field Directions: For a chosen vector field F(x,y)={F1(x1,x2),F2(x1,x2)} plot its vector field on a square centered about a chosen point.

7 DiffEqnsDemos.nb DiffEqnsDemos.nb (Differential Equations) Higher Order Numerical Methods Directions: For a defined DE, explore the higher order methods of (a) Three-Term-Taylor (3TT) (b) Runge-Kutta 2 (RK2) and (c) Runge-Kutta 4 (RK4) (Differential Equations) Euler s Method (System) Description: For a defined vector field, explore graphically and numerically Euler s Method for a system of two first order DEs. This example is the second in a series of steps that culminates with Runge-Kutta Order 4 for a system of first order DEs.

8 DiffEqnsDemos.nb DiffEqnsDemos.nb (Differential Equations) Student Solution for RK4 System Description: Culmination of numerical methods approximation section of DE course. Students will use their code code to tackle some problem related to a nonlinear system of first order DEs (either SIR or some game-theoretic or some model of their interest) (Differential Equations) MassSpringDamper Explorer Description: For a linear system of MSD form, graphically explores the solutions for various initial conditions and exhibit exact solutions using the eigensystem of the matrix of the linear system.

9 DiffEqnsDemos.nb DiffEqnsDemos.nb (Differential Equations) First Order Linear System Description: For a defined 2x2 matrix A of the first order linear system of DEs, use the eigensystem of A to compute analytic solutions. (Differential Equations) Laplace Transform of 1st Order DE Description: Solve a first order initial value problem αy +βy=f(t); y(0)=y0 using the method of Laplace Transform.

10 DiffEqnsDemos.nb DiffEqnsDemos.nb (Differential Equations) Laplace Transform of 2nd Order DE Description: Solve a second order initial value problem my +by +cy=f(t); y(0)=y0, y (0)=v0 using the method of Laplace Transform. (Differential Equations) Impulse Response Explorer Description: Explores the second order, MassSpringDamper differential equation with a delta function as forcing function to build intuition and lay the groundwork for the impulse response approximation method for any forcing function. (ToUpdate)

11 DiffEqnsDemos.nb DiffEqnsDemos.nb (Differential Equations) Impulse Response Method Description: Explores the approximate solutions to a MSD system with defined forcing function to build intuition about the convolution integral. (ToUpdate) (Differential Equations) Series Approximations Description: For a defined second order, initial value problem and defined number of terms, explore the graphical meaning of series solutions and interval of convergence.

12 DiffEqnsDemos.nb DiffEqnsDemos.nb 1D Wave Equation, u tt - c 2 u xx = 0 Directions: Let c=1 with clamped boundary conditions u(0,t)=u(l,t)=0 t 2D Wave Equation, u tt - c 2 u xx + u yy = 0 Directions: Let c=1, On a rectangle with clamped boundary conditions