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

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### Elliptic1D Homogenous Dirichlet Boundary Conditions
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Solves the 1D Poisson equation with homogeneous Dirichlet boundary conditions.
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$$
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-\nabla^2 u(x) = 1
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$$
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with $x\in[0,1]$. The boundary conditions are given by
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$$
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au + b\frac{du}{dx} = g
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$$
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with
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$$
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1u(0) + b\frac{du(0)}{dx} = 0
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$$
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$$
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1u(1) + b\frac{du(1)}{dx} = 0
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$$
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This corresponds to the call to addScalarBC1D of `addScalarBC1D(A,b,k,m,dx,dc,nc,v)`, where `dc`, `nc`, and `vc` are vectors which hold the coefficients for $a$, $b$, and $g$ in the above system of equations. $a=[1,1]$, $b=[0,0]$ and $g=[0,0]$. Substituting these values in gives:
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$$
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u(0) = 0
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$$
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$$
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u(1) = 0
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$$
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The true solution is
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$$
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u(x) = \frac{x(1-x)}{2}
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$$
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---
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This example is implemented in:
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- [MATLAB/ OCTAVE](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DHomogeneousDirichlet.m)
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Additional MATLAB/ OCTAVE variants of this example with different boundary conditions:
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- [Non-Homogeneous Dirichlet](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DNonHomogeneousDirichlet.m)
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- [Left Dirichlet, Right Neumann](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftDirichletRightNeumann.m)
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- [Left Dirichlet, Right Robin](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftDirichletRightRobin.m)
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- [Left Neumann, Right Neumann](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftNeumannRightNeumann.m)
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- [Left Neumann, Right Robin](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftNeumannRightRobin.m)
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- [Left Robin, Right Robin](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftRobinRightRobin.m)
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- [Periodic Boundary Conditions](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DPeriodicBC.m)
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- [Non-Periodic Boundary Conditions](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DNonPeriodicBC.m)

‎doc/sphinx/source/examples/Elliptic/1D/Elliptic1D-Left-Dirichlet-Right-Neumann.md‎

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### Elliptic1D Left Dirichlet and Right Neumann Boundary Conditions
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Solves the 1D Poisson equation with left Dirichlet and right Neumann boundary conditions.
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$$
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-\nabla^2 u(x) = 1
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$$
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with $x\in[0,1]$. The boundary conditions are given by
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$$
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a_nu + b_n\frac{du}{dx} = g_n
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$$
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with the left hand side boundary condition (Neumann) satisfying
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$$
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0u(0) + 1\frac{du(0)}{dx} = 0
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$$
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and the right hand boundary condition (Dirichlet) satisfying
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$$
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1u(1) + 0\frac{du(1)}{dx} = 0
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$$
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This corresponds to the call to addScalarBC1D of `addScalarBC1D(A,b,k,m,dx,dc,nc,v)`, where `dc`, `nc`, and `vc` are vectors which hold the coefficients for $a$, $b$, and $g$ in the above system of equations. $a=[0,1]$, $b=[1,0]$ and $g=[0,0]$. Substituting these values in gives:
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$$
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\frac{du(0)}{dx} = 0
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$$
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$$
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u(1) = 0
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$$
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The true solution is:
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$$
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u(x) = \frac{1-x^2}{2}
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$$
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---
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This example is implemented in:
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- [MATLAB/ OCTAVE](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftDirichletRightNeumann.m)
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Additional MATLAB/ OCTAVE variants of this example with different boundary conditions:
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- [Homogeneous Dirichlet](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DHomogeneousDirichlet.m)
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- [Non-Homogeneous Dirichlet](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DNonHomogeneousDirichlet.m)
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- [Left Dirichlet, Right Robin](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftDirichletRightRobin.m)
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- [Left Neumann, Right Neumann](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftNeumannRightNeumann.m)
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- [Left Neumann, Right Robin](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftNeumannRightRobin.m)
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- [Left Robin, Right Robin](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftRobinRightRobin.m)
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- [Periodic Boundary Conditions](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DPeriodicBC.m)
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- [Non-Periodic Boundary Conditions](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DNonPeriodicBC.m)

‎doc/sphinx/source/examples/Elliptic/1D/Elliptic1D-Left-Dirichlet-Right-Robin.md‎

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### Elliptic1D Left Dirichlet and Right Robin Boundary Conditions
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Solves the 1D Poisson equation with left Dirichlet and right Robin boundary conditions.
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$$
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-\nabla^2 u(x) = \pi^2 \sin(\pi x)
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$$
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with $x\in[0,1]$.
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The boundary conditions are given by
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$$
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a_nu + b_n\frac{du}{dx} = g_n
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$$
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with the left hand side boundary condition (Dirichlet) satisfying
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$$
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1u(0) + 0\frac{du(0)}{dx} = 10
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$$
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and the right hand boundary condition (Robin) satisfying
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$$
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400u(1) + 1\frac{du(1)}{dx} = 15
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$$
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This corresponds to the call to addScalarBC1D of `addScalarBC1D(A,b,k,m,dx,dc,nc,v)`, where `dc`, `nc`, and `vc` are vectors which hold the coefficients for $a$, $b$, and $g$ in the above system of equations. $a=[1,400]$, $b=[0,1]$ and $g=[10,15]$.
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Substituting these values in gives:
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$$
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u(0) = 10
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$$
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$$
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400u(1) + \frac{du(1)}{dx} = 15
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$$
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The true solution is:
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$$
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u(x) = \sin(\pi x) + \frac{\pi - 3985}{401}x + 10
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$$
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---
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This example is implemented in:
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- [MATLAB/ OCTAVE](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftDirichletRightRobin.m)
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Additional MATLAB/ OCTAVE variants of this example with different boundary conditions:
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- [Homogeneous Dirichlet](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DHomogeneousDirichlet.m)
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- [Non-Homogeneous Dirichlet](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DNonHomogeneousDirichlet.m)
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- [Left Dirichlet, Right Neumann](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftDirichletRightNeumann.m)
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- [Left Neumann, Right Neumann](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftNeumannRightNeumann.m)
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- [Left Neumann, Right Robin](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftNeumannRightRobin.m)
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- [Left Robin, Right Robin](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftRobinRightRobin.m)
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- [Periodic Boundary Conditions](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DPeriodicBC.m)
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- [Non-Periodic Boundary Conditions](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DNonPeriodicBC.m)

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

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### Elliptic1D Pure Neumann Boundary Conditions
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Solves the 1D Poisson boundary value problem with Neumann boundary conditions.
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$$
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-\nabla^2 u(x) = x - \frac{1}{2}
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$$
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with $x\in[0,1]$.
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The boundary conditions are given by
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$$
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au + b\frac{du}{dx} = g
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$$
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with the left hand side boundary condition (Neumann) satisfying
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$$
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0u(0) + 1\frac{du(0)}{dx} = 0
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$$
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and the right hand boundary condition (Neumann) satisfying
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$$
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0u(1) + 1\frac{du(1)}{dx} = 0
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$$
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This corresponds to the call to addScalarBC1D of `addScalarBC1D(A,b,k,m,dx,dc,nc,v)`, where `dc`, `nc`, and `vc` are vectors which hold the coefficients for $a$, $b$, and $g$ in the above system of equations. $a=[0,0]$, $b=[1,1]$ and $g=[0,0]$.
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Substituting these values in gives:
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$$
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\frac{du(0)}{dx} = 0
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$$
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$$
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\frac{du(1)}{dx} = 0
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$$
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The exact solution ( with constant $C$ ) is:
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$$
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u(x) = C + \frac{x^2}{4} - \frac{x^3}{6}
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$$
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---
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This example is implemented in:
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- [MATLAB/ OCTAVE](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftNeumannRightNeumann.m)
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Additional MATLAB/ OCTAVE variants of this example with different boundary conditions:
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- [Homogeneous Dirichlet](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DHomogeneousDirichlet.m)
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- [Non-Homogeneous Dirichlet](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DNonHomogeneousDirichlet.m)
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- [Left Dirichlet, Right Neumann](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftDirichletRightNeumann.m)
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- [Left Dirichlet, Right Robin](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftDirichletRightRobin.m)
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- [Left Neumann, Right Robin](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftNeumannRightRobin.m)
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- [Left Robin, Right Robin](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftRobinRightRobin.m)
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- [Periodic Boundary Conditions](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DPeriodicBC.m)
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- [Non-Periodic Boundary Conditions](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DNonPeriodicBC.m)

‎doc/sphinx/source/examples/Elliptic/1D/Elliptic1D-Left-Neumann-Right-Robin.md‎

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### Elliptic1D Left Neumann Right Robin Boundary Conditions
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Solves the 1D Poisson boundary value problem with left Neumann and right Robin boundary conditions.
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$$
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-\nabla^2 u(x) = \pi^2 \sin(\pi x)
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$$
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with $x\in[0,1]$.
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The boundary conditions are given by
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$$
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au + b\frac{du}{dx} = g
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$$
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with the left hand side boundary condition (Neumann) satisfying
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$$
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0u(0) + 1\frac{du(0)}{dx} = 10
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$$
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and the right hand boundary condition (Robin) satisfying
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$$
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400u(1) + 1\frac{du(1)}{dx} = 15
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$$
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This corresponds to the call to addScalarBC1D of `addScalarBC1D(A,b,k,m,dx,dc,nc,v)`, where `dc`, `nc`, and `vc` are vectors which hold the coefficients for $a$, $b$, and $g$ in the above system of equations. $a=[0,400]$, $b=[1,1]$ and $g=[10,15]$.
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Substituting these values in gives:
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$$
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\frac{du(0)}{dx} = 10
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$$
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$$
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400u(1) + \frac{du(1)}{dx} = 15
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$$
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The exact solution is:
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$$
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u(x) = \sin(\pi x) + -(10 + \pi)x + \frac{402\pi + 4025}{400}
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$$
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The example is taken from [this paper](https://www.scirp.org/journal/paperinformation?paperid=50586
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)
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---
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This example is implemented in:
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- [MATLAB/ OCTAVE](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftNeumannRightRobin.m)
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Additional MATLAB/ OCTAVE variants of this example with different boundary conditions:
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- [Homogeneous Dirichlet](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DHomogeneousDirichlet.m)
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- [Non-Homogeneous Dirichlet](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DNonHomogeneousDirichlet.m)
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- [Left Dirichlet, Right Neumann](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftDirichletRightNeumann.m)
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- [Left Dirichlet, Right Robin](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftDirichletRightRobin.m)
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- [Left Neumann, Right Neumann](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftNeumannRightNeumann.m)
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- [Left Robin, Right Robin](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftRobinRightRobin.m)
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- [Periodic Boundary Conditions](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DPeriodicBC.m)
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- [Non-Periodic Boundary Conditions](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DNonPeriodicBC.m)

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

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### Elliptic1D Pure Robin Boundary Conditions
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Solves the 1D Poisson boundary value problem with pure Robin boundary conditions.
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$$
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-\nabla^2 u(x) = \pi^2 \sin(\pi x)
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$$
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with $x\in[0,1]$.
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The boundary conditions are given by
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$$
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au + b\frac{du}{dx} = g
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$$
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with the left hand side boundary condition (Robin) satisfying
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$$
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-200u(0) + 1\frac{du(0)}{dx} = 10
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$$
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and the right hand boundary condition (Robin) satisfying
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$$
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400u(1) + 1\frac{du(1)}{dx} = 15
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$$
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This corresponds to the call to addScalarBC1D of `addScalarBC1D(A,b,k,m,dx,dc,nc,v)`, where `dc`, `nc`, and `vc` are vectors which hold the coefficients for $a$, $b$, and $g$ in the above system of equations. $a=[-200,400]$, $b=[1,1]$ and $g=[10,15]$.
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Substituting these values in gives:
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$$
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-200u(0) + \frac{du(0)}{dx} = 10
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$$
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$$
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400u(1) + \frac{du(1)}{dx} = 15
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$$
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The exact solution is:
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$$
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u(x) = \sin(\pi x) + \frac{35 - \pi}{403}x + \frac{402\pi - 3995}{80600}
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$$
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The example is taken from [this paper](https://www.scirp.org/journal/paperinformation?paperid=50586
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)
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---
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This example is implemented in:
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- [MATLAB/ OCTAVE](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftNRobinRightRobin.m)
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Additional MATLAB/ OCTAVE variants of this example with different boundary conditions:
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- [Homogeneous Dirichlet](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DHomogeneousDirichlet.m)
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- [Non-Homogeneous Dirichlet](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DNonHomogeneousDirichlet.m)
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- [Left Dirichlet, Right Neumann](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftDirichletRightNeumann.m)
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- [Left Dirichlet, Right Robin](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftDirichletRightRobin.m)
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- [Left Neumann, Right Neumann](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftNeumannRightNeumann.m)
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- [Left Neumann, Right Robin](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DLeftNeumannRightRobin.m)
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- [Periodic Boundary Conditions](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DPeriodicBC.m)
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- [Non-Periodic Boundary Conditions](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/elliptic1DNonPeriodicBC.m)

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