Merge pull request #57 from kanav99/docstrings

Fix latex in docstrings

Author: ChrisRackauckas

src/dde/ddetst.jl

@@ -45,19 +45,19 @@ const prob_dde_DDETST_A1 =
 Delay differential equation model of chronic granulocytic leukemia, given by
 
 ```math
-\begin{align*}
-  u_1'(t) &= \frac{1.1}{1 + \sqrt{10} u_1(t - 20)^{5/4}} - \frac{10 u_1(t)}{1 + 40 u_2(t)}, \\
-  u_2'(t) &= \frac{100 u_1(t)}{1 + 40 u_2(t)} - 2.43 u_2(t),
-\end{align*}
+u_1'(t) = \frac{1.1}{1 + \sqrt{10} u_1(t - 20)^{5/4}} - \frac{10 u_1(t)}{1 + 40 u_2(t)},
+```
+```math
+u_2'(t) = \frac{100 u_1(t)}{1 + 40 u_2(t)} - 2.43 u_2(t),
 ```
 
 for ``t \in [0, 100]`` and history function
 
 ```math
-\begin{align*}
-  \phi_1(t) &= 1.05767027/3, \\
-  \phi_2(t) &= 1.030713491/3,
-\end{align*}
+\phi_1(t) = 1.05767027/3,
+```
+```math
+\phi_2(t) = 1.030713491/3,
 ```
 
 for ``t \leq 0``.
@@ -246,19 +246,19 @@ const prob_dde_DDETST_C1 =
 Delay differential equation
 
 ```math
-\begin{align*}
-  u_1'(t) &= - 2 u_1(t - u_2(t)), \\
-  u_₂'(t) &= \frac{|u_1(t - u_2(t))| - |u_1(t)|}{1 + |u_1(t - u_2(t))|},
-\end{align*}
+u_1'(t) = - 2 u_1(t - u_2(t)),
+```
+```math
+u_₂'(t) = \frac{|u_1(t - u_2(t))| - |u_1(t)|}{1 + |u_1(t - u_2(t))|},
 ```
 
 for ``t \in [0, 40]`` with history function
 
 ```math
-\begin{align*}
-  \phi_1(t) &= 1, \\
-  \phi_2(t) &= 0.5,
-\end{align*}
+\phi_1(t) = 1,
+```
+```math
+\phi_2(t) = 0.5,
 ```
 
 for ``t \leq 0``.
@@ -305,11 +305,13 @@ const prob_dde_DDETST_C2 =
 Delay differential equation model of hematopoiesis, given by
 
 ```math
-\begin{align*}
-  u_1'(t) &= \hat{s}_0 u_2(t - T_1) - \gamma u_1(t) - Q,\\
-  u_2'(t) &= f(u_1(t)) - k u_2(t),\\
-  u_3'(t) &= 1 - \frac{Q \exp(\gamma u_3(t))}{\hat{s}_0 u_2(t - T_1 - u_3(t))},
-\end{align*}
+u_1'(t) = \hat{s}_0 u_2(t - T_1) - \gamma u_1(t) - Q,
+```
+```math
+u_2'(t) = f(u_1(t)) - k u_2(t),
+```
+```math
+u_3'(t) = 1 - \frac{Q \exp(\gamma u_3(t))}{\hat{s}_0 u_2(t - T_1 - u_3(t))},
 ```
 
 for ``t \in [0, 300]`` with history function ``\phi_1(0) = 3.325``, ``\phi_3(0) = 120``, and
@@ -367,17 +369,19 @@ Delay differential equation model of hematopoiesis, given by the same delay diff
 equation as [`prob_dde_DDETST_C3`](@ref)
 
 ```math
-\begin{align*}
-  u_1'(t) &= \hat{s}_0 u_2(t - T_1) - \gamma u_1(t) - Q,\\
-  u_2'(t) &= f(u_1(t)) - k u_2(t),\\
-  u_3'(t) &= 1 - \frac{Q \exp(\gamma u_3(t))}{\hat{s}_0 u_2(t - T_1 - u_3(t))},
-\end{align*}
+u_1'(t) = \hat{s}_0 u_2(t - T_1) - \gamma u_1(t) - Q,
+```
+```math
+u_2'(t) = f(u_1(t)) - k u_2(t),
+```
+```math
+u_3'(t) = 1 - \frac{Q \exp(\gamma u_3(t))}{\hat{s}_0 u_2(t - T_1 - u_3(t))},
 ```
 
 for ``t \in [0, 100]`` with history function
 ``\phi_1(0) = 3.5``, ``\phi_3(0) = 50``, and ``\phi_2(t) = 10`` for ``t \leq 0``, where
 ``f(y) = a / (1 + K y^r)``, ``\hat{s}_0 = 0.00372``, ``T_1 = 3``, ``\gamma = 0.1``,
-``Q = 0.00178``, ``k = 6.65`, ``a = 15600``, ``K = 0.0382``, and ``r = 6.96``.
+``Q = 0.00178``, ``k = 6.65``, ``a = 15600``, ``K = 0.0382``, and ``r = 6.96``.
 
 # References
 
@@ -421,19 +425,19 @@ const prob_dde_DDETST_C4 =
 Delay differential equation
 
 ```math
-\begin{align*}
-  u_1'(t) &= u_2(t), \\
-  u_2'(t) &= - u_2(\exp(1 - u_2(t))) u_2(t)^2 \exp(1 - u_2(t)),
-\end{align*}
+u_1'(t) = u_2(t), 
+```
+```math
+u_2'(t) = - u_2(\exp(1 - u_2(t))) u_2(t)^2 \exp(1 - u_2(t)),
 ```
 
 for ``t \in [0.1, 5]`` with history function
 
 ```math
-\begin{align*}
-  \phi_1(t) &= \log t, \\
-  \phi_2(t) &= 1 / t,
-\end{align*}
+\phi_1(t) = \log t, 
+```
+```math
+\phi_2(t) = 1 / t,
 ```
 
 for ``t \in (0, 0.1]``.
@@ -443,10 +447,10 @@ for ``t \in (0, 0.1]``.
 The analytical solution for ``t \in [0.1, 5]`` is
 
 ```math
-\begin{align*}
-  u_1(t) &= \log t, \\
-  u_2(t) &= 1 / t.
-\end{align*}
+u_1(t) = \log t, 
+```
+```math
+u_2(t) = 1 / t.
 ```
 
 # References
@@ -498,23 +502,31 @@ const prob_dde_DDETST_D1 =
 Delay differential equation model of antigen antibody dynamics with fading memory, given by
 
 ```math
-\begin{align*}
-  u_1'(t) &= - r_1 u_1(t) u_2(t) + r_2 u_3(t), \\
-  u_2'(t) &= - r_1 u_1(t) u_2(t) + \alpha r_1 u_1(t - u_4(t)) u_2(t - u_4(t)), \\
-  u_3'(t) &= r_1 u_1(t) u_2(t) - r_2 u_3(t), \\
-  u_4'(t) &= 1 + \frac{3 \delta - u_1(t) u_2(t) - u_3(t)}{u_1(t - u_4(t)) u_2(t - u_4(t)) + u_3(t - u_4(t))} \exp(\delta u_4(t)),
-\end{align*}
+u_1'(t) = - r_1 u_1(t) u_2(t) + r_2 u_3(t), 
+```
+```math
+u_2'(t) = - r_1 u_1(t) u_2(t) + \alpha r_1 u_1(t - u_4(t)) u_2(t - u_4(t)),
+```
+```math
+u_3'(t) = r_1 u_1(t) u_2(t) - r_2 u_3(t), 
+```
+```math
+u_4'(t) = 1 + \frac{3 \delta - u_1(t) u_2(t) - u_3(t)}{u_1(t - u_4(t)) u_2(t - u_4(t)) + u_3(t - u_4(t))} \exp(\delta u_4(t)),
 ```
 
 for ``t \in [0, 40]`` with history function
 
 ```math
-\begin{align*}
-  \phi_1(t) &= 5, \\
-  \phi_2(t) &= 0.1, \\
-  \phi_3(t) &= 0, \\
-  \phi_4(t) &= 0,
-\end{align*}
+\phi_1(t) = 5, 
+```
+```math
+\phi_2(t) = 0.1, 
+```
+```math
+\phi_3(t) = 0, 
+```
+```math
+\phi_4(t) = 0,
 ```
 
 for ``t \leq 0``, where ``r_1 = 0.02``, ``r_2 = 0.005``, ``\alpha = 3``, and ``\delta = 0.01``.
@@ -598,19 +610,19 @@ const prob_dde_DDETST_E1 =
 Delay differential equation model of a logistic Gauss-type predator-prey system, given by
 
 ```math
-\begin{align*}
-  u_1'(t) &= u_1(t) (1 - u_1(t - \tau) - \rho u_1'(t - \tau)) - \frac{u_2(t) u_1(t)^2}{u_1(t)^2 + 1}, \\
-  u_2'(t) &= u_2(t) \left(\frac{u_1(t)^2}{u_1(t)^2 + 1} - \alpha\right),
-\end{align*}
+u_1'(t) = u_1(t) (1 - u_1(t - \tau) - \rho u_1'(t - \tau)) - \frac{u_2(t) u_1(t)^2}{u_1(t)^2 + 1}, 
+```
+```math
+u_2'(t) = u_2(t) \left(\frac{u_1(t)^2}{u_1(t)^2 + 1} - \alpha\right),
 ```
 
 for ``t \in [0, 2]`` with history function
 
 ```math
-\begin{align*}
-  \phi_1(t) &= 0.33 - t / 10, \\
-  \phi_2(t) &= 2.22 + t / 10,
-\end{align*}
+\phi_1(t) = 0.33 - t / 10, 
+```
+```math
+\phi_2(t) = 2.22 + t / 10,
 ```
 
 for ``t \leq 0``, where ``\alpha = 0.1``, ``\rho = 2.9``, and ``\tau = 0.42``.
@@ -742,19 +754,23 @@ for ``t \in [0.25, 0.499]`` with history function ``\phi(t) = \exp(-t^2)`` and
 
 The analytical solution for ``t \in [0.25, 0.499]`` is
 
-``math
+```math
 u(t) = u_i(t) = \exp(-4^i t^2 + B_i t + C_i) / 2^i + K_i
 ```
 
 if ``t \in [x_i, x_{i + 1}]``, where
 
 ```math
-\begin{align*}
-  x_i &= (1 - 2^{-i}) / 2, \\
-  B_i &= 2 (4^{i-1} + B_{i-1}), \\
-  C_i &= - 4^{i-2} - B_{i-1} / 2 + C_{i-1}, \\
-  K_i &= - \exp(-4^i x_i^2 + B_i x_i + C_i) / 2^i + u_{i-1}(x_i),
-\end{align*}
+x_i = (1 - 2^{-i}) / 2, 
+```
+```math
+B_i = 2 (4^{i-1} + B_{i-1}), 
+```
+```math
+C_i = - 4^{i-2} - B_{i-1} / 2 + C_{i-1}, 
+```
+```math
+K_i = - \exp(-4^i x_i^2 + B_i x_i + C_i) / 2^i + u_{i-1}(x_i),
 ```
 
 and ``B_0 = C_0 = K_0 = 0``.

src/dde/radar5.jl

@@ -11,19 +11,19 @@ http://www.unige.ch/~hairer/radar5-v2.1.tar
 Delay differential equation model from chemical kinetics, given by
 
 ```math
-\begin{align*}
-  u_1'(t) &= - k_1 A u_2(t) - k_2 u_1(t) u_2(t - \tau) + k_3 B u_1(t) - 2 k_4 u_1(t)^2, \\
-  u_2'(t) &= - k_1 A u_2(t) - k_2 u_1(t) u_2(t - \tau) + f k_3 B u_1(t),
-\end{align*}
+  u_1'(t) = - k_1 A u_2(t) - k_2 u_1(t) u_2(t - \tau) + k_3 B u_1(t) - 2 k_4 u_1(t)^2, 
+```
+```math
+  u_2'(t) = - k_1 A u_2(t) - k_2 u_1(t) u_2(t - \tau) + f k_3 B u_1(t),
 ```
 
 for ``t \in [0, 100.5]`` with history function
 
 ```math
-\begin{align*}
-  \phi_1(t) &= 1e-10, \\
-  \phi_2(t) &= 1e-5,
-\end{align*}
+  \phi_1(t) = 1e-10, 
+```
+```math
+  \phi_2(t) = 1e-5,
 ```
 
 for ``t \leq 0``, where ``k_1 = 1.34``, ``k_2 = 1.6e9``, ``k_3 = 8000``, ``k_4 = 4e7``,
@@ -74,11 +74,13 @@ Delay differential equation model of a chemical reaction with steady state solut
 by
 
 ```math
-\begin{align*}
-  u_1'(t) &= - a u_1(t) + b u_2(t - \tau) u_3(t), \\
-  u_2'(t) &= a u_1(t) - b u_2(t - \tau) u_3(t) - c u_2(t)^2, \\
-  u_3'(t) &= c u_2(t)^2,
-\end{align*}
+  u_1'(t) = - a u_1(t) + b u_2(t - \tau) u_3(t), 
+```
+```math
+  u_2'(t) = a u_1(t) - b u_2(t - \tau) u_3(t) - c u_2(t)^2, 
+```
+```math
+  u_3'(t) = c u_2(t)^2,
 ```
 
 for ``t \in [0, 10e10]`` with history function ``\phi_1(0) = 1``, ``\phi_2(t) = 0`` for
@@ -130,27 +132,42 @@ const prob_dde_RADAR5_robertson =
 Delay differential equation model of antibody production, given by
 
 ```math
-\begin{align*}
-  u_1'(t) &= - r u_1(t) u_2(t) - s u_1(t) u_4(t), \\
-  u_2'(t) &= - r u_1(t) u_2(t) + \alpha r u_1(u_5(t)) u_2(u_5(t)) [t \geq t_0], \\
-  u_3'(t) &= r u_1(t) u_2(t), \\
-  u_4'(t) &= - s u_1(t) u_4(t) - \gamma u_4(t) + \beta r u_1(u_6(t)) u_2(u_6(t)) [t > t_1], \\
-  u_5'(t) &= [t \geq t_0] \frac{u_1(t) u_2(t) + u_3(t)}{u_1(u_5(t)) u_2(u_5(t)) + u_3(u_5(t))}, \\
-  u_6'(t) &= [t \geq t_1] \frac{1e-12 + u_2(t) + u_3(t)}{1e-12 + u_2(u_6(t)) + u_3(u_6(t))},
-\end{align*}
+  u_1'(t) = - r u_1(t) u_2(t) - s u_1(t) u_4(t), 
+```
+```math
+  u_2'(t) = - r u_1(t) u_2(t) + \alpha r u_1(u_5(t)) u_2(u_5(t)) [t \geq t_0], 
+```
+```math
+  u_3'(t) = r u_1(t) u_2(t), 
+```
+```math
+  u_4'(t) = - s u_1(t) u_4(t) - \gamma u_4(t) + \beta r u_1(u_6(t)) u_2(u_6(t)) [t > t_1], 
+```
+```math
+  u_5'(t) = [t \geq t_0] \frac{u_1(t) u_2(t) + u_3(t)}{u_1(u_5(t)) u_2(u_5(t)) + u_3(u_5(t))}, 
+```
+```math
+  u_6'(t) = [t \geq t_1] \frac{1e-12 + u_2(t) + u_3(t)}{1e-12 + u_2(u_6(t)) + u_3(u_6(t))},
 ```
 
 for ``t \in [0, 300]`` with history function
 
 ```math
-\begin{align*}
-  \phi_1(t) &= \phi_0, \\
-  \phi_2(t) &= 1e-15, \\
-  \phi_3(t) &= 0, \\
-  \phi_4(t) &= 0, \\
-  \phi_5(t) &= 0, \\
-  \phi_6(t) &= 0,
-\end{align*}
+  \phi_1(t) = \phi_0, 
+```
+```math
+  \phi_2(t) = 1e-15, 
+```
+```math
+  \phi_3(t) = 0, 
+```
+```math
+  \phi_4(t) = 0, 
+```
+```math  \phi_5(t) = 0, 
+```
+```math
+  \phi_6(t) = 0,
 ```
 
 for ``t \leq 0``, where ``\alpha = 1.8``, ``\beta = 20``, ``\gamma = 0.002``, ``r = 5e4``,