# Partial Differential Equation

## Finite Difference 
### Elliptic Equations
- Used for steady-state, boundary value problems


Description of how the Laplace equations works
$$
\frac{\partial^2T}{\partial x^2}+\frac{\partial^2T}{\partial y^2}=0
$$

Finite-different solutions
- Laplacian Difference equation
$$
\frac{\partial^2T}{\partial x^2}= \frac{T_{i+1,j}-2T_{i,j}+T_{i-1,j}}{\Delta x^2}
$$
and
$$
\frac{\partial^2T}{\partial y^2}= \frac{T_{i+1,j}-2T_{i,j}+T_{i-1,j}}{\Delta y^2}
$$

Boundary Conditions


Control-Volume approach
<img 
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    src="control_volume_approach.png" 
    alt="Two Different perspsectives for developing approximate solutions of PDE: (a) finite-difference or node and (b) control-volume based">
Computer Algorithms

### Parabolic Equations
- Used for unstead-state, initial + boundary conditions problems
For parabolic PDE equations we also consider the change in time as well as space.

Heat-conduction equation
Explanation of heat equation
$$
k\frac{\partial^2T}{\partial x^2}=\frac{\partial T}{\partial t}
$$

Explicit methods

Simple Implicit methods
Crank-Nicolson
ADI


### Hyperbolic Equations
MacCormack Method


## Finite-Element Method
General Approach
### One-dimensional analysis

### Two-dimensional Analysis

















# Problem 1: Finite-Element Solution of a Series of Springs

Problem 32.4 from Numerical Methods for Engineers 7th Edition Steven C. Chapra and Raymond P. Canale

A series of interconnected strings are connected to a fixed wall where the other is subject to a constant force F. Using the step-by-step procedure from above, dertermine the displacement of the springs.






## Problem 2: Finite 

Solve the non-dimensional transient heat conduction equation in two dimensions, which represents the transient temperature distribution in an insulated plate