Improve problem formulation descriptions

pina/_src/problem/zoo/acoustic_wave_problem.py

@@ -28,8 +28,50 @@ def initial_condition(input_, output_):
 
 class AcousticWaveProblem(TimeDependentProblem, SpatialProblem):
     r"""
-    Implementation of the acoustic wave problem in the spatial interval
-    :math:`[0, 1]` and temporal interval :math:`[0, 1]`.
+    Implementation of the one-dimensional acoustic wave problem on the
+    space-time domain :math:`\Omega\times T = [0, 1] \times [0, 1]`.
+
+    The problem is governed by the acoustic wave equation
+
+    .. math::
+
+        \frac{\partial^2 u}{\partial t^2}
+        =
+        c^2 \frac{\partial^2 u}{\partial x^2},
+
+    where :math:`u = u(x, t)` is the solution field and :math:`c > 0` is the
+    wave propagation speed.
+
+    Homogeneous Dirichlet boundary conditions are imposed at the spatial
+    boundaries:
+
+    .. math::
+
+        u(0, t) = u(1, t) = 0, \qquad t \in [0, 1].
+
+    The initial displacement is prescribed as
+
+    .. math::
+
+        u(x, 0) = \sin(\pi x) + \frac{1}{2}\sin(4\pi x),
+        \qquad x \in [0, 1],
+
+    together with zero initial velocity:
+
+    .. math::
+
+        \frac{\partial u}{\partial t}(x, 0) = 0,
+        \qquad x \in [0, 1].
+
+    The analytical solution is given by
+
+    .. math::
+
+        u(x, t)
+        =
+        \sin(\pi x)\cos(c\pi t)
+        +
+        \frac{1}{2}\sin(4\pi x)\cos(4c\pi t).
 
     .. seealso::
 

pina/_src/problem/zoo/advection_problem.py

@@ -24,9 +24,39 @@ def initial_condition(input_, output_):
 
 class AdvectionProblem(SpatialProblem, TimeDependentProblem):
     r"""
-    Implementation of the advection problem in the spatial interval
-    :math:`[0, 2 \pi]` and temporal interval :math:`[0, 1]` with periodic
-    boundary conditions.
+    Implementation of the one-dimensional advection problem on the space-time
+    domain :math:`\Omega\times T = [0, 2\pi] \times [0, 1]`.
+
+    The problem is governed by the linear advection equation
+
+    .. math::
+
+        \frac{\partial u}{\partial t}
+        +
+        c \frac{\partial u}{\partial x}
+        =
+        0,
+
+    where :math:`u = u(x, t)` is the solution field and :math:`c` is the
+    advection velocity.
+
+    Periodic boundary conditions are imposed at the spatial boundaries:
+
+    .. math::
+
+        u(0, t) = u(2\pi, t), \qquad t \in [0, 1].
+
+    The initial condition is prescribed as
+
+    .. math::
+
+        u(x, 0) = \sin(x), \qquad x \in [0, 2\pi].
+
+    The analytical solution is given by
+
+    .. math::
+
+        u(x, t) = \sin(x - ct).
 
     .. seealso::
 

pina/_src/problem/zoo/allen_cahn_problem.py

@@ -26,9 +26,35 @@ def initial_condition(input_, output_):
 
 class AllenCahnProblem(TimeDependentProblem, SpatialProblem):
     r"""
-    Implementation of the Allen Cahn problem in the spatial interval
-    :math:`[-1, 1]` and temporal interval :math:`[0, 1]` with periodic
-    boundary conditions.
+    Implementation of the one-dimensional Allen-Cahn problem on the space-time
+    domain :math:`\Omega\times T = [-1, 1] \times [0, 1]`.
+
+    The problem is governed by the Allen-Cahn equation
+
+    .. math::
+
+        \frac{\partial u}{\partial t}
+        -
+        \alpha \frac{\partial^2 u}{\partial x^2}
+        +
+        \beta \left(u^3 - u\right)
+        =
+        0,
+
+    where :math:`u = u(x, t)` is the solution field, :math:`\alpha` is the
+    diffusion coefficient, and :math:`\beta` is the reaction coefficient.
+
+    Periodic boundary conditions are imposed at the spatial boundaries:
+
+    .. math::
+
+        u(-1, t) = u(1, t), \qquad t \in [0, 1].
+
+    The initial condition is prescribed as
+
+    .. math::
+
+        u(x, 0) = x^2 \cos(\pi x), \qquad x \in [-1, 1].
 
     .. seealso::
 

pina/_src/problem/zoo/diffusion_reaction_problem.py

@@ -35,8 +35,63 @@ def initial_condition(input_, output_):
 
 class DiffusionReactionProblem(TimeDependentProblem, SpatialProblem):
     r"""
-    Implementation of the diffusion-reaction problem in the spatial interval
-    :math:`[-\pi, \pi]` and temporal interval :math:`[0, 1]`.
+    Implementation of the one-dimensional diffusion-reaction problem on the
+    space-time domain :math:`\Omega\times T = [-\pi, \pi] \times [0, 1]`.
+
+    The problem is governed by the forced diffusion-reaction equation
+
+    .. math::
+
+        \frac{\partial u}{\partial t}
+        -
+        \alpha \frac{\partial^2 u}{\partial x^2}
+        =
+        f(x, t),
+
+    where :math:`u = u(x, t)` is the solution field, :math:`\alpha` is the
+    diffusion coefficient, and :math:`f(x, t)` is a forcing term.
+
+    Homogeneous Dirichlet boundary conditions are imposed at the spatial
+    boundaries:
+
+    .. math::
+
+        u(-\pi, t) = u(\pi, t) = 0, \qquad t \in [0, 1].
+
+    The initial condition is prescribed as
+
+    .. math::
+
+        u(x, 0)
+        =
+        \sin(x)
+        +
+        \frac{1}{2}\sin(2x)
+        +
+        \frac{1}{3}\sin(3x)
+        +
+        \frac{1}{4}\sin(4x)
+        +
+        \frac{1}{8}\sin(8x).
+
+    The analytical solution is given by
+
+    .. math::
+
+        u(x, t)
+        =
+        e^{-t}
+        \left(
+        \sin(x)
+        +
+        \frac{1}{2}\sin(2x)
+        +
+        \frac{1}{3}\sin(3x)
+        +
+        \frac{1}{4}\sin(4x)
+        +
+        \frac{1}{8}\sin(8x)
+        \right).
 
     .. seealso::
 

pina/_src/problem/zoo/helmholtz_problem.py

@@ -11,8 +11,45 @@
 
 class HelmholtzProblem(SpatialProblem):
     r"""
-    Implementation of the Helmholtz problem in the square domain
-    :math:`[-1, 1] \times [-1, 1]`.
+    Implementation of the two-dimensional Helmholtz problem on the square domain
+    :math:`\Omega = [-1, 1] \times [-1, 1]`.
+
+    The problem is governed by the forced Helmholtz equation
+
+    .. math::
+
+        \Delta u + k u = f(x, y),
+
+    where :math:`u = u(x, y)` is the solution field, :math:`k` is the squared
+    wavenumber, and :math:`f(x, y)` is a forcing term.
+
+    Homogeneous Dirichlet boundary conditions are imposed on the boundary of
+    the domain:
+
+    .. math::
+
+        u(x, y) = 0, \qquad (x, y) \in \partial \Omega.
+
+    The analytical solution is given by
+
+    .. math::
+
+        u(x, y)
+        =
+        \sin(\alpha_x \pi x)
+        \sin(\alpha_y \pi y),
+
+    with forcing term
+
+    .. math::
+
+        f(x, y)
+        =
+        \left[
+        k - (\alpha_x^2 + \alpha_y^2)\pi^2
+        \right]
+        \sin(\alpha_x \pi x)
+        \sin(\alpha_y \pi y).
 
     .. seealso::
 

pina/_src/problem/zoo/inverse_poisson_problem.py

@@ -74,9 +74,33 @@ def laplace_equation(input_, output_, params_):
 
 class InversePoisson2DSquareProblem(SpatialProblem, InverseProblem):
     r"""
-    Implementation of the inverse 2-dimensional Poisson problem in the square
-    domain :math:`[0, 1] \times [0, 1]`, with unknown parameter domain
-    :math:`[-1, 1] \times [-1, 1]`.
+    Implementation of the inverse two-dimensional Poisson problem on the square
+    domain :math:`\Omega = [-2, 2] \times [-2, 2]`, with unknown parameter
+    domain :math:`\Theta = [-1, 1] \times [-1, 1]`.
+
+    The problem is governed by the parameterized Poisson equation
+
+    .. math::
+
+        \Delta u
+        =
+        \exp\left(
+        -2(x - \mu_1)^2
+        -2(y - \mu_2)^2
+        \right),
+
+    where :math:`u = u(x, y)` is the solution field and :math:`\mu_1, \mu_2` are
+    unknown parameters controlling the forcing term.
+
+    Homogeneous Dirichlet boundary conditions are imposed on the boundary of the
+    domain:
+
+    .. math::
+
+        u(x, y) = 0, \qquad (x, y) \in \partial \Omega.
+
+    The inverse problem aims to infer the unknown parameters :math:`\mu_1` and
+    :math:`\mu_2` from solution data.
 
     The `"data"` condition is added only if the required files are downloaded
     successfully.

pina/_src/problem/zoo/poisson_problem.py

@@ -26,8 +26,40 @@ def forcing_term(input_):
 
 class Poisson2DSquareProblem(SpatialProblem):
     r"""
-    Implementation of the 2-dimensional Poisson problem in the square domain
-    :math:`[0, 1] \times [0, 1]`.
+    Implementation of the two-dimensional Poisson problem on the square domain
+    :math:`\Omega = [0, 1] \times [0, 1]`.
+
+    The problem is governed by the Poisson equation
+
+    .. math::
+
+        \Delta u = f(x, y),
+
+    where :math:`u = u(x, y)` is the solution field and :math:`f(x, y)` is the
+    forcing term.
+
+    Homogeneous Dirichlet boundary conditions are imposed on the boundary of the
+    domain:
+
+    .. math::
+
+        u(x, y) = 0, \qquad (x, y) \in \partial \Omega.
+
+    The forcing term is given by
+
+    .. math::
+
+        f(x, y)
+        =
+        2\pi^2 \sin(\pi x)\sin(\pi y).
+
+    The analytical solution is given by
+
+    .. math::
+
+        u(x, y)
+        =
+        -\sin(\pi x)\sin(\pi y).
 
     :Example: