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# 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
style="display: block;
margin-left: auto;
margin-right: auto;
width: 50%;"
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
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