Made progress on both PDE examples and system of non-linear equations examples

3 files changed, 63 insertions, 9 deletions

diff --git a/tutorials/module_3/3_system_of_equations.md b/tutorials/module_3/1_linear_equations.md
index 9f50b97..9f50b97 100644
--- a/tutorials/module_3/3_system_of_equations.md
+++ b/tutorials/module_3/1_linear_equations.md
diff --git a/tutorials/module_3/2_roots_optimization.md b/tutorials/module_3/2_roots_optimization.md
index ee8f8c6..a91a464 100644
--- a/tutorials/module_3/2_roots_optimization.md
+++ b/tutorials/module_3/2_roots_optimization.md
@@ -236,4 +236,3 @@ Numerical methods for Engineers 7th Edition Case study 8.4.
 
 # Systems of Non-linear equations
 
-
diff --git a/tutorials/module_3/6_pde.md b/tutorials/module_3/6_pde.md
index 95d63c4..2980fa7 100644
--- a/tutorials/module_3/6_pde.md
+++ b/tutorials/module_3/6_pde.md
@@ -33,7 +33,7 @@ $$
 Finite-different solutions
 - Laplacian Difference equations in dimension $x$ and $y$:
 $$
-\frac{\partial^2T}{\partial x^2}= \frac{T_{i+1,j}-2T_{i,j}+T_{i-1,j}}{\Delta x^2}
+\boxed{\frac{\partial^2T}{\partial x^2}= \frac{T_{i+1,j}-2T_{i,j}+T_{i-1,j}}{\Delta x^2}}
 $$
 $$
 \frac{\partial^2T}{\partial y^2}= \frac{T_{i+1,j}-2T_{i,j}+T_{i-1,j}}{\Delta y^2}
@@ -64,9 +64,13 @@ $$
 
 Explicit methods
 
-Forward-time central-space
+Forward-time central-space (FTCS Scheme)
 	based on forward euler method and central difference in space. 
-	
+
+One dimensional Heat conduction example
+	We need to measure the insulation of a wall and measure the change of temperature through the 
+
+
 Simple Implicit methods
 Crank-Nicolson
 ADI
@@ -107,9 +111,10 @@ General Approach
 3. Assembly
 4. Boundary Conditions
 5. Solutions
-6. Postprocessing
+6. Post-processing
 ### One-dimensional analysis
 
+
 ### Two-dimensional Analysis
 
 
@@ -128,18 +133,68 @@ General Approach
 
 
 
-# Problem 1: Finite-Element Solution of a Series of Springs
+## 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, determine the displacement of the springs.
 
+Solution:
+Setup: Let's partition the system to treat each spring as an element. Thus, the system consists for 4 elements and 5 nodes.
+<img 
+    style="display: block; 
+           margin-left: auto;
+           margin-right: auto;
+           width: 50%;"
+    src="example32-4.png" 
+    alt="Diagram of system of springs example">
+Element Equations: Analyzing element 1 we get the following free body diagram: 
+<img 
+    style="display: block; 
+           margin-left: auto;
+           margin-right: auto;
+           width: 50%;"
+    src="example32-4element1.png" 
+    alt="Diagram of element 1">
 
-
-
-
+Applying Hook's law to the element we get:
+$$
+F=kx
+$$
+$$
+F=k(x_1-x_2)
+$$
+where $(x_1-x_2)$ is how much the first spring is stretched out.
+Re-writing this equation:
+$$
+F_1 = kx_1 - kx_2
+$$
+$$
+F_2=-kx_1+kx_2
+$$
+$$
+\begin{bmatrix} k & -k \\ -k & k \end{bmatrix} 
+\begin{Bmatrix} x_{1} \\
+x_{2} \end{Bmatrix}
+=
+\begin{Bmatrix} F_{1} \\ F_{2} \end{Bmatrix}
+$$
+$$
+[k]\{x\}=\{F\}
+$$
+where $[k]$ is the element property matrix or in this case element stiffness matrix. $x$ is a column vector of unknowns (in this case position of each) and $F$ is a vector column with external influence applied at the nodes. 
+Assembly: Once individual element equations are derived we will link them together using assembly.
+$$
+[k]\{x'\}=\{F'\}
+$$
+where $[k]$ is the assemblage property matrix and $\{u'\}$ and $\{F'\}$ column vectors are unknowns and external forces that are marked with primes to denote that they are an assemblage of the vectors $\{u\}$ and $\{F\}$.
+$$
+\begin{bmatrix} k & -k \\ -k & k \end{bmatrix} 
+\begin{Bmatrix} x_{1} \\ x_2 \end{Bmatrix}
+$$
 
 ## Problem 2: Finite 
 
 Solve the non-dimensional transient heat conduction equation in two dimensions, which represents the transient temperature distribution in an insulated plate
 
+---
\ No newline at end of file