Elliptic and Hyperbolic Equations

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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 (∆ = b2-4*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:

  1. Equation: For any question you will be given an equation in terms of partial differentials.
  2. Boundary Conditions: Provide the starting point values which will be used to build the set of equations used to solve numerically.
  3. 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.)
  4. Central Difference: Second order in terms of x or y.
  5. FTCS (Forward Time, Central Space): First order in terms of time and second order in terms of x or y
  6. Build a series of equations based on this information
  7. Create a matrix and solve using Gaussian elimination

Example

The following is an Elliptic example.

Step 1

Equation:

Elliptic Equation.PNG

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.

Boundary Conditions.PNG

Step 3

Change the boundary conditions using backwards difference formula.

Difference Formula.PNG

Where m and n are the defined number of steps.

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

FTCS Elliptic Equation.PNG

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

This is the end of this topic. Click here to go back to the main subject page for Numerical Methods & Statistics.

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