Bpw/sturm lcpp

doc/sphinx/source/examples/Sturm-Liouville/Bessel.md

@@ -0,0 +1,54 @@
+# Bessel Sturm-Liouville Problem
+
+This example solves the Bessel differential equation, which is a classic Sturm-Liouville problem:
+
+$$ x^2 u''  + x u' + (x^2 - \nu^2) u = 0, \quad 0 < x 1 $$
+
+with Dirichlet boundary conditions:
+$$ u(0) = 0, \quad u(1) = J_\nu(1) $$
+
+The exact solution to this problem is the Bessel function of the first kind of order $\nu$, denoted as $J_\nu(x)$. For $\nu = 3$, the solution is $ J_3(x) = \frac{1}{\pi}\int^{\pi}_0 \cos(3\tau) - x\sin(\tau)\,d\tau $.
+
+## Mathematical Background
+
+Bessel's differential equation is a special case of the Sturm-Liouville problem, which has the general form:
+
+$$\frac{d}{dx}\left(p(x)\frac{du}{dx}\right) + q(x)u + \lambda r(x)u = 0$$
+
+For Bessel's equation, we have:
+
+- $p(x) = x^2$
+- $q(x) = x$
+- $r(x) = \frac{1}{x}$
+- $\lambda = -\nu^2$
+
+## Discretization
+
+The equation is discretized using mimetic finite difference operators. The spatial derivative operators are constructed with a specified order of accuracy $k$.
+
+The discrete system is:
+
+$$A u = b$$
+
+where:
+
+- $A = x^2 L + x I_{FC} G + (x^2 - \nu^2) I$
+- $L$ is the mimetic Laplacian
+- $G$ is the mimetic gradient
+- $I_{FC}$ is the interpolation operator from faces to centers
+- $I$ is the identity matrix
+
+Boundary conditions are applied using `RobinBC`.
+
+---
+
+This example is implemented in:
+
+- [MATLAB/OCTAVE](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab_octave/sturmLiouvilleBessel.m)
+- [C++](https://github.com/csrc-sdsu/mole/blob/main/examples/cpp/sturmLiouvilleBessel.cpp)
+
+## Results
+
+The numerical solution closely matches the exact solution, which is the Bessel function of the first kind of order $\nu$:  
+
+$J_\nu(x) = \frac{1}{\pi}\int^{\pi}_0 \cos(\nu\tau) - x\sin(\tau)\,d\tau$.

doc/sphinx/source/examples/Sturm-Liouville/Chebyshev.md

@@ -16,6 +16,7 @@ Chebyshev's differential equation is a special case of the Sturm-Liouville probl
 $$\frac{d}{dx}\left(p(x)\frac{du}{dx}\right) + q(x)u + \lambda r(x)u = 0$$
 
 For Chebyshev's equation, we have:
+
 - $p(x) = 1-x^2$
 - $q(x) = 0$
 - $r(x) = 1$
@@ -30,23 +31,28 @@ The discrete system is:
 $$A u = b$$
 
 where:
-- $A = (1-x^2) L - x I G + n^2 I$
+
+- $A = (1-x^2) L - x I_{FC} G + n^2 I$
 - $L$ is the mimetic Laplacian
 - $G$ is the mimetic gradient
-- $I$ is the interpolation operator from faces to centers
+- $I_{FC}$ is the interpolation operator from faces to centers
+- $I$ is the identity matrix
 
 Boundary conditions are applied using the `addScalarBC1D` function.
 
 ---
 
 This example is implemented in:
+
 - [MATLAB/OCTAVE](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab_octave/sturmLiouvilleChebyshev.m)
+- [C++](https://github.com/csrc-sdsu/mole/blob/main/examples/cpp/sturmLiouvilleChebyshev.cpp)
 
 ## Results
 
-The numerical solution closely matches the exact solution, which is the Chebyshev polynomial $T_2(x) = 2x^2 - 1$. 
+The numerical solution closely matches the exact solution, which is the Chebyshev polynomial $T_2(x) = 2x^2 - 1$.
 
 Chebyshev polynomials are important in numerical analysis and approximation theory because they:
+
 1. Minimize the maximum error in polynomial approximation
 2. Have roots that are optimal interpolation points (Chebyshev nodes)
-3. Are closely related to the Fourier cosine series 
+3. Are closely related to the Fourier cosine series

doc/sphinx/source/examples/Sturm-Liouville/Helmholtz.md

@@ -0,0 +1,56 @@
+# Helmholtz Sturm-Liouville Problem
+
+This example solves the Helmholtz differential equation, which is a classic Sturm-Liouville problem:
+
+$$ u'' + u = 0, \quad 0 < x < 3 $$
+or
+$$ u'' + \mu^2 u = 0, \quad 0 < x < 1 $$
+
+with boundary conditions:
+$$ u(0) = 0, \quad u(3) = \sin(3) $$
+or
+$$ u'(0) = 0, \quad u(1) + u'(1) = \cos(\mu) - \mu \sin(\mu) $$
+
+The exact solution to this problem is $\sin(x)$ or $\cos(\mu x)$.
+
+## Mathematical Background
+
+Helmholtz's differential equation is a special case of the Sturm-Liouville problem, which has the general form:
+
+$$\frac{d}{dx}\left(p(x)\frac{du}{dx}\right) + q(x)u + \lambda r(x)u = 0$$
+
+For Helmholtz's equation, we have:
+
+- $p(x) = 1$
+- $q(x) = 0$
+- $r(x) = 1$
+- $\lambda = \mu^2$
+
+## Discretization
+
+The equation is discretized using mimetic finite difference operators. The spatial derivative operators are constructed with a specified order of accuracy $k$.
+
+The discrete system is:
+
+$$A u = b$$
+
+where:
+
+- $A = L + I$
+- $L$ is the mimetic Laplacian
+- $I$ is the identity matrix
+
+Boundary conditions are applied using `RobinBC` or `MixedBC`.
+
+---
+
+These examples are implemented in:
+
+- [MATLAB/OCTAVE](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab_octave/sturmLiouvilleHelmholtzDirichletDirichlet.m)
+- [C++](https://github.com/csrc-sdsu/mole/blob/main/examples/cpp/sturmLiouvilleHelmholtzDirichletDirichlet.cpp)
+- [MATLAB/OCTAVE](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab_octave/sturmLiouvilleHelmholtzDirichletRobin.m)
+- [C++](https://github.com/csrc-sdsu/mole/blob/main/examples/cpp/sturmLiouvilleHelmholtzDirichletRobin.cpp)
+
+## Results
+
+The numerical solutions closely match the exact solutions, which are $\sin(x)$ or $\cos(\mu x)$.

doc/sphinx/source/examples/Sturm-Liouville/Hermite.md

@@ -0,0 +1,52 @@
+# Hermite Sturm-Liouville Problem
+
+This example solves the Hermite differential equation, which is a classic Sturm-Liouville problem:
+
+$$ u'' - 2 u' + 2 m u = 0, \quad -1 < x < 1 $$
+
+with Dirichlet boundary conditions:
+$$ u(-1) = H_m(-1), \quad u(1) = H_m(1) $$
+
+The exact solution to this problem is the Hermite polynomial of degree $m$, denoted as $H_m(x)$. For $m = 4$, the solution is $H_4(x) = e^{x^2} \frac{d^4}{dx^4}e^{-x^2}$.
+
+## Mathematical Background
+
+Hermite's differential equation is a special case of the Sturm-Liouville problem, which has the general form:
+
+$$\frac{d}{dx}\left(p(x)\frac{du}{dx}\right) + q(x)u + \lambda r(x)u = 0$$
+
+For Hermite's equation, we have:
+
+- $p(x) = e^{-x^2}$
+- $q(x) = 0$
+- $r(x) = e^{-x^2}$
+- $\lambda = m^2$
+
+## Discretization
+
+The equation is discretized using mimetic finite difference operators. The spatial derivative operatores are constructed with a specified order of accuracy $k$.
+
+The discrete system is:
+
+$$A u = b$$
+
+where
+
+- $A = L - 2 x I_{FC} G + 2 m I$
+- $L$ is the mimetic Laplacian
+- $G$ is the mimetic gradient
+- $I_{FC}$ is the interpolation operator from faces to centers
+- $I$ is the identity matrix
+
+Boundary conditions are applied using `RobinBC`.
+
+---
+
+This example is implemented in:
+
+- [MATLAB/OCTAVE](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab_octave/sturmLiouvilleHermite.m)
+- [C++](https://github.com/csrc-sdsu/mole/blob/main/examples/cpp/sturmLiouvilleHermite.cpp)
+
+## Results
+
+The numerical solution closely matches the exact solution, which is the Hermite polynomial $H_4(x) = e^{x^2} \frac{d^4}{dx^4}e^{-x^2}$.

doc/sphinx/source/examples/Sturm-Liouville/Laguerre.md

@@ -0,0 +1,52 @@
+# Laguerre Sturm-Liouville Problem
+
+This example solves the Laguerre differential equation, which is a classic Sturm-Liouville problem:
+
+$$x u'' + (1 - x) u' + n u = 0, \quad 0 < x < 2$$
+
+with Dirichlet boundary conditions:
+$$ u(0) = L_n(2), \quad u(2) = L_n(2) $$
+
+The exact solution to this problem is the Laguerre polynomial of degree $n$, denoted as $L_n(x)$. For $n = 4$, the solution is $L_4(x) = \frac{e^x}{4!} \frac{d^4}{dx^4} (e^{-x}x^4) $.
+
+## Mathematical Background
+
+Laguerre's differential equation is a special case of the Sturm-Liouville problem, which has the general form:
+
+$$\frac{d}{dx}\left(p(x)\frac{du}{dx}\right) + q(x)u + \lambda r(x)u = 0$$
+
+For Laguerre's equation, we have:
+
+- $p(x) = xe^{-x}$
+- $q(x) = 0$
+- $r(x) = e^{-x}$
+- $\lambda = n$
+
+## Discretization
+
+The equation is discretized using mimetic finite difference operators. The spatial derivative operators are constructed with a specified order of accuracy $k$.
+
+The discrete system is:
+
+$$A u = b$$
+
+where:
+
+- $A = x L + (1 - x) I_{FC} G + n I$
+- $L$ is the mimetic Laplacian
+- $G$ is the mimetic gradient
+- $I_{FC}$ is the interpolation operator from faces to centers
+- $I$ is the identity matrix
+
+Boundary conditions are applied using `RobinBC`.
+
+---
+
+This example is implemented in:
+
+- [MATLAB/OCTAVE](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab_octave/sturmLiouvilleLaguerre.m)
+- [C++](https://github.com/csrc-sdsu/mole/blob/main/examples/cpp/sturmLiouvilleLaguerre.cpp)
+
+## Results
+
+The numerical solution closely matches the exact solution, which is the Laguerre polynomial $L_4(x) = \frac{e^x}{4!} \frac{d^4}{dx^4} (e^{-x}x^4) $.

doc/sphinx/source/examples/Sturm-Liouville/Legendre.md

@@ -0,0 +1,52 @@
+# Legendre Sturm-Liouville Problem
+
+This example solves the Legendre differential equation, which is a classic Sturm-Liouville problem:
+
+$$ (1 - x^2) u'' - 2 x u' + n (n + 1) u = 0, \quad -1 < x < 1 $$
+
+with Dirichlet boundary conditions:
+$$ u(-1) = -1, \quad u(1) = 1 $$
+
+The exact solutoin to this problem is the Legendre polynomial of degree $n$, denoted as $L_n(x)$. For $n = 4$, the solution is $L_4(x) = \frac{1}{8}(35x^4 -30x^2 + 3)$.
+
+## Mathematical Background
+
+Legendre's differential equation is a special case of the Sturm-Liouville problem, which has the general form:
+
+$$\frac{d}{dx}\left(p(x)\frac{du}{dx}\right) + q(x)u + \lambda r(x)u = 0$$
+
+For Legendre's equation, we have:
+
+- $p(x) = 1-x^2$
+- $q(x) = 0$
+- $r(x) = 1$
+- $\lambda = n(n+1)$
+
+## Discretization
+
+The equation is discretized using mimetic finite difference operators. The spatial derivative operators are constructed with a specified order of accuracy $k$.
+
+The discrete system is:
+
+$$A u = b$$
+
+where:
+
+- $A = (1 - x^2) L - 2 x I_{FC} G + n(n+1) I$
+- $L$ is the mimetic Laplacian
+- $G$ is the mimetic gradient
+- $I_{FC}$ is the interpolation operator from faces to centers
+- $I$ is the identity matrix
+
+Boundary conditions are applied using `RobinBC`.
+
+---
+
+This example is implemented in:
+
+- [MATLAB/OCTAVE](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab_octave/sturmLiouvilleLegendre.m)
+- [C++](https://github.com/csrc-sdsu/mole/blob/main/examples/cpp/sturmLiouvilleLegendre.cpp)
+
+## Results
+
+The numerical solution closely matches the exact solution, which is the Legendre polynomial $L_4(x) = \frac{1}{8}(35x^4 -30x^2 + 3)$.

doc/sphinx/source/examples/Sturm-Liouville/index.md

@@ -6,5 +6,10 @@ Sturm-Liouville theory deals with a class of second-order linear differential eq
 :maxdepth: 1
 :caption: Contents
 
+Bessel
 Chebyshev
-``` 
+Helmholtz
+Hermite
+Laguerre
+Legendre
+```