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# Project: Numerical Solutions to Ordinary Differential Equations in Hardware PowerPoint PPT Presentation

Project: Numerical Solutions to Ordinary Differential Equations in Hardware. Joseph Schneider EE 800 March 30, 2010. Ordinary Differential Equations. Function with one independent variable and derivatives of the dependent variable Eg . y’ = 1 + y/x

Project: Numerical Solutions to Ordinary Differential Equations in Hardware

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## Project: Numerical Solutions to Ordinary Differential Equations in Hardware

Joseph Schneider

EE 800

March 30, 2010

### Ordinary Differential Equations

• Function with one independent variable and derivatives of the dependent variable

• Eg. y’ = 1 + y/x

• Requires some initial condition in order to be solved

• Eg. y(1) = 2

### Ordinary Differential Equations

• Found in many areas of engineering

• Radioactive decay, heat equation, motion…

• In electrical engineering, charge, flux, voltage, current, all intertwined by differential equations

### Ordinary Differential Equations

• In some cases, original equation can be derived with relative ease; Exact solutions are then available

• In other cases, we will only have the ODE to work with

• Numerical solutions have been developed to deal with these cases

### Ordinary Differential Equations

• Most basic case: Euler’s Method

• Selecting a step size h, iterate from initial value to desired value using the derivative function

### Ordinary Differential Equation

• Euler’s Method most basic case – Simple, but inaccurate

• Variety of other methods that have been developed

### Ordinary Differential Equations

• Error directly linked with step size

• As step size decreases, error decreases; However, takes longer for process to complete

• Implemented in software (eg. Matlab), more accurate methods take several seconds to complete for smaller scale cases; Several minutes for larger cases

### Ordinary Differential Equations

• Original project goal: Implement the Runge-Kutta 4th order method in hardware for improved speed

### Project

• Euler’s method also implemented for comparisons on area, timing, and accuracy.

• Current implementation: Uses fixed-point number representation, state machine

• Further steps on this implementation

• Variable-point number representation to improve accuracy

• Modify for parallelism to examine impacts on area, timing

### Project

• Second implementation: Error-controlled design of Runge-Kutta method

• Error is specified at beginning of process, step size is varied to ensure final result meets error specifications

### Project

• For second implementation, desired to use a hardware design to improve on bottleneck of software design

• Comparisons to software on time vs. error threshold