Resolved the link issue

Author: pritkc

doc/sphinx/source/examples-doc/Elliptic/1D/Elliptic1D.md

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+### Elliptic1D
+
+Solves the 1D Poisson equation with Robin boundary conditions.
+
+$$
+\nabla^2 u(x) = f(x)
+$$
+
+with $x\in[0,1]$, and $f(x) =e^x$. The boundary conditions are given by
+
+$$
+au + b\frac{du}{dx} = g
+$$
+
+with $a=1$, $b=1$, and $g=0$, and
+
+$$
+au(0) + b\frac{du(0)}{dx} = 0
+$$
+
+$$
+au(1) + b\frac{du(1)}{dx} = 2e
+$$
+
+This corresponds to the call to robinBC2D of `robinBC2D(k, m, dx, a, b)`.
+
+---
+
+This example is implemented in:
+- [MATLAB](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab/Elliptic/1D/elliptic1D.m)
+- [C++](https://github.com/csrc-sdsu/mole/blob/main/examples/cpp/Elliptic/1D/elliptic1D.cpp)
+
+Additional MATLAB variants of this example with different boundary conditions:
+- [Homogeneous Dirichlet](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab/Elliptic/1D/elliptic1DHomogeneousDirichlet.m)
+- [Non-Homogeneous Dirichlet](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab/Elliptic/1D/elliptic1DNonHomogeneousDirichlet.m)
+- [Left Dirichlet, Right Neumann](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab/Elliptic/1D/elliptic1DLeftDirichletRightNeumann.m)
+- [Left Dirichlet, Right Robin](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab/Elliptic/1D/elliptic1DLeftDirichletRightRobin.m)
+- [Left Neumann, Right Neumann](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab/Elliptic/1D/elliptic1DLeftNeumannRightNeumann.m)
+- [Left Neumann, Right Robin](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab/Elliptic/1D/elliptic1DLeftNeumannRightRobin.m)
+- [Left Robin, Right Robin](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab/Elliptic/1D/elliptic1DLeftRobinRightRobin.m)
+- [Periodic Boundary Conditions](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab/Elliptic/1D/elliptic1DPeriodicBC.m)
+- [Non-Periodic Boundary Conditions](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab/Elliptic/1D/elliptic1DNonPeriodicBC.m)

doc/sphinx/source/examples-doc/Elliptic/1D/Poisson1D.md

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+### Poisson1D
+
+Solves the 1D Poisson equation with Robin boundary conditions.
+
+$$
+-\frac{d^2 C}{d x^2} = f(x)
+$$
+
+The boundary conditions are given by
+
+$$
+au + b\frac{du}{dx} = g
+$$
+
+The equation is discretized using mimetic operators.

doc/sphinx/source/examples-doc/Elliptic/2D/Elliptic2D-Case-2.md

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+### Elliptic2D Case 2
+
+Solves the 2D Poisson equation with Robin boundary conditions on a nonuniform sinusoidal grid.
+
+$$
+\nabla^2 u(x,y) = f(x,y)
+$$
+
+with $x\in[-\pi, 2\pi], y\in[-\pi, \pi]$, and
+
+$$
+f(x,y) = \begin{cases}
+    \sin(x)\sin(y) & \text{ along boundaries } \\
+    -2\sin(x)\sin(y) & \text{ otherwise }
+\end{cases}
+$$
+
+The boundary conditions are given by
+
+$$
+au + b\nabla u = g
+$$
+
+with $a=1$, $b=0$, and $g=0$, which is equivalent to Dirichlet conditions along each boundary.
+This corresponds to the call to robinBC2D of `robinBC2D(k, m, 1, n, 1, 1, 0)`.

doc/sphinx/source/examples-doc/Elliptic/2D/Elliptic2D-Nodal-Curv-Sinusoidal.md

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+### Elliptic2D Nodal Curv Sinusoidal
+
+Solves the 2D Poisson equation with Robin boundary conditions on a curvilinear sinusoidal grid using the nodal mimetic operator. This requires manually setting the boundary condition in the Laplacian, as there is no boundary condition operator for the nodal curvilinear operators.
+
+$$
+\nabla^2 u(x,y) = f(x,y)
+$$
+
+with $x\in[-\pi, 2\pi], y\in[-\pi, \pi]$, and
+
+$$
+f(x,y) = \begin{cases}
+    \sin(x)+\cos(y) & \text{ along boundaries } \\
+    -\sin(x) - \cos(y) & \text{ otherwise }
+\end{cases}
+$$
+
+The boundary conditions are given by
+
+$$
+au + b\nabla u = g
+$$
+
+with $a=1$, $b=0$, and $g=0$, which is equivalent to Dirichlet conditions along each boundary.
+The MATLAB code uses the function `boundaryIdx2D` to find the correct locations for the weights of the boundary condition in the Laplacian node. The code then sets the appropriate values to $0$ or $1$.

doc/sphinx/source/examples-doc/Elliptic/2D/Elliptic2D-Nodal-Curv.md

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+### Elliptic2D Nodal Curv
+
+Solves the 2D Poisson equation with Robin boundary conditions on a curvilinear grid using the nodal mimetic operator. This requires manually setting the boundary condition in the Laplacian, as there is no boundary condition operator for the nodal curvilinear operators.
+
+$$
+\nabla^2 u(x,y) = f(x,y)
+$$
+
+with $x\in[0,50], y\in[0,50]$, and
+
+$$
+f(x,y) = \begin{cases}
+    (x-0.5)^2+(y-0.5)^2 & \text{ along boundaries } \\
+    4 & \text{ otherwise }
+\end{cases}
+$$
+
+The boundary conditions are given by
+
+$$
+au + b\nabla u = g
+$$
+
+The MATLAB code uses the function `boundaryIdx2D` to find the correct locations for boundary condition weights in the nodal Laplacian. The code then sets the appropriate values to $0$ or $1$.

doc/sphinx/source/examples-doc/Elliptic/2D/Elliptic2D.md

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+### Elliptic2D
+
+Solves the 2D Poisson equation with Robin boundary conditions on a nonuniform grid.
+
+$$
+\nabla^2 u(x,y) = f(x,y)
+$$
+
+with $x\in[0,10], y\in[0,10]$, and
+
+$$
+f(x,y) = \begin{cases}
+    (x-0.5)^2+(y-0.5)^2 & \text{ along boundaries } \\
+    4 & \text{ otherwise }
+\end{cases}
+$$
+
+The boundary conditions are given by
+
+$$
+au + b\nabla u = g
+$$
+
+with $a=1$, $b=0$, and $g=0$, which is equivalent to Dirichlet conditions along each boundary.
+This corresponds to the call to robinBC2D of `robinBC2D(k, m, 1, n, 1, 1, 0)`.

doc/sphinx/source/examples-doc/Elliptic/2D/Minimal-Poisson-2D.md

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+### Minimal Poisson 2D
+
+Solves the 2D Poisson equation with Robin boundary conditions on a uniform grid where $\Delta x=\Delta y = 1$.
+
+$$
+\nabla^2 u(x,y) = f(x,y)
+$$
+
+with $x\in[0,5], y\in[0,5]$, and
+
+$$
+f(x,y) = \begin{cases}
+    100 & \text{ if y = 0} \\
+    0 & \text{ otherwise }
+\end{cases}
+$$
+
+The boundary conditions are given by
+
+$$
+au + b\nabla u = g
+$$
+
+with $a=1$, $b=0$, and $g=0$.
+This corresponds to the call to robinBC2D of `robinBC2D(k, m, 1, n, 1, 1, 0)`.

doc/sphinx/source/examples-doc/Elliptic/3D/Elliptic3D.md

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+### Elliptic3D
+
+Similar to `minimal_poisson2D`, this solves a 3D problem using mimetic Laplacian, where one side of the domain is set to 100, and allowed to diffuse.
+
+$$
+\nabla^2 u(x,y,z) = f(x,y,z)
+$$
+
+with $x\in[0,5], y\in[0,6], z\in[0,7]$, and
+
+$$
+f(x,y) = \begin{cases}
+    100 & \text{ if z = 0} \\
+    0 & \text{ otherwise }
+\end{cases}
+$$
+
+The boundary conditions are given by
+
+$$
+au + b\nabla u = g
+$$
+
+with $a=1$, $b=0$, and $g=0$.
+This corresponds to the call to robinBC3D of `robinBC2D(k, m, 1, n, 1, o, 1, 1, 0)`.

doc/sphinx/source/examples-doc/Hyperbolic/1D/Burgers1D.md

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+### Burgers1D
+
+This example deals with the conservative form of the inviscid Burgers equation in 1D.
+
+$$
+\frac{\partial U}{\partial t} + \frac{\partial}{\partial x}\Big(\frac{U^2}{2}\Big) = 0
+$$
+
+with $U = u(x,t)$ defined on the domain $x\in[-15,15]$, from time $t\in[0,10]$ and initial conditions
+
+$$
+u(x,0) = e^{\frac{-x^2}{50}}
+$$
+
+The wave is allowed to propagate across the domain while the area under the curve is calculated. 
+
+---
+
+This example is implemented in:
+- [MATLAB](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab/Hyperbolic/1D/burgers1D.m)
+- [C++](https://github.com/csrc-sdsu/mole/blob/main/examples/cpp/Hyperbolic/1D/Burgers1D.cpp) 

doc/sphinx/source/examples-doc/Hyperbolic/1D/Hyperbolic1D-Lax-Friedrichs.md

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+### Hyperbolic1D Lax Friedrichs
+
+Solves the 1D advection equation with periodic boundary conditions using a sided nodal mimetic operator and the Lax-Friedrichs scheme for time integration. The main feature of this code is the lack of an interpolator. This is first order in both time and space, even though a second order mimetic operator is used.
+
+$$
+\frac{\partial U}{\partial t} + a\frac{\partial U}{\partial x} = 0
+$$
+
+where $U=u(x,t)$ and $a=1$ is the advection velocity. The domain $x\in[0,1]$ and $t\in[0,1]$ with initial condition
+
+$$
+u(x,0) = \sin(2\pi x)
+$$
+
+Periodic boundary conditions are used
+
+$$
+u(0,t) = u(1,t)
+$$
+
+Using finite differences for the time derivative
+
+$$
+\frac{\partial U}{\partial t} = \frac{U^{n+1}_i - U^{n}_i}{\Delta t}
+$$
+
+where $U_i^n$ is $u(x_i, t_n)$, and the mimetic operator $\mathbf{D}$ for the space derivative.
+
+$$
+\frac{U^{n+1}_i - U^{n}_i}{\Delta t} + a \mathbf{D} U^n_i = 0
+$$
+
+$$
+\frac{U^{n+1}_i - U^{n}_i}{\Delta t} = -a \mathbf{D} U^n_i
+$$
+
+$$
+U^{n+1}_i = \mathbf{U}^n_i - a \Delta t \mathbf{D} U^n_i
+$$
+
+The last line $\mathbf{U}$ is the derived (average) value of the U from the $U^n$ timestep.