# Elliptic and Hyperbolic Equations

Partial differential equations are differential equations that involve more that one independent variable.

## Contents |

## Introduction

All linear second order partial differential equations fall into one of three categories.

The discriminant (∆ = b^{2}-4*a*c) determines which category this is.

- When ∆ < 0, it's
**Elliptic**, therefore use a**Laplace Equation**.

- When ∆ = 0, it's
**Parabolic**, therefore use a**Heat Conduction Equation**.

- When ∆ > 0, it's
**Hyperbolic**, therefore use a**Wave Equation**.

However, only very select PDE’s have analytical solutions and therefore we use numerical methods to find solutions.

The following are the steps to solution:

- Equation: For any question you will be given an equation in terms of partial differentials.
- Boundary Conditions: Provide the starting point values which will be used to build the set of equations used to solve numerically.
- Difference Formulas: Used depending on what the order the differential terms are, first or second, and what the differentials are in terms of. (Eg. time, x, y, etc.)
- Central Difference: Second order in terms of x or y.
- FTCS (Forward Time, Central Space): First order in terms of time and second order in terms of x or y
- Build a series of equations based on this information
- Create a matrix and solve using Gaussian elimination

## Example

The following is an Elliptic example.

### Step 1

Equation:

### Step 2

Boundary Conditions with ‘a’ being the boundary limit line on the x axis and ‘b’ being the boundary limit line on the y axis.

### Step 3

Change the boundary conditions using backwards difference formula.

Where m and n are the defined number of steps.

Or, we can use a FTCS method using a step size half as long.

### Step 4 and 5

Using the difference formula, for step size m and n, values for the equation can be found for all i and j values.

From these series of equations, for i = 1, 2, … , a matrix can be formed in the usual way and solved.

## End

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