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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