Here is a summary of the different types of boundary conditions that are commonly used in partial differential equation (PDE) problems:

1. Dirichlet boundary conditions: These boundary conditions specify a fixed value for a variable at the boundary. They are often used to prescribe the value of the solution at the boundary in a boundary value problem. In general, Dirichlet boundary conditions are specified as follows:

u = g \ \ \ \ on \ the \ boundary

where $u$ is the variable (such as temperature or velocity) and $g$ is the fixed value specified by the boundary condition.

2. Neumann boundary conditions: These boundary conditions specify the derivative of a variable at the boundary. They are often used to prescribe the flux of a variable at the boundary in a boundary value problem. In general, Neumann boundary conditions are specified as follows:

{du \over dn} = h \ \ \ \ \ on \ the \ boundary

where $u$ is the variable (such as temperature or velocity) and ${du/dn}$ is the normal derivative of $u$ at the boundary, and $h$ is the fixed value specified by the boundary condition.

3. Mixed boundary conditions: These boundary conditions specify a combination of different types of boundary conditions, such as Dirichlet, Neumann, and periodic conditions. They are often used to model physical phenomena that involve both fixed values and fluxes at the boundary and are periodic in nature. In general, mixed boundary conditions are specified as follows:

u = g + h_1 {du \over dn} + h_2 (u - u(x + p, y + q)) \ \ \ \ \ on \ the \ boundary

where $u$ is the variable (such as temperature or velocity), $h$ and $h_1$ and $h_2$ are fixed coefficients, ${du \over dn}$ is the normal derivative of $u$ at the boundary, $p$ and $q$ are the periods of the boundary condition in the $x$ and $y$ directions, respectively, and $u(x+p,y+q)$ is the value of $u$ at the point on the opposite side of the boundary. This equation expresses the mixed boundary condition as a combination of a Dirichlet condition ($g$), a Neumann condition $(h_1 {du \over dn})$, and a periodic condition $(h_2(u-u(x+p,y+q)))$.

4. Periodic boundary conditions: These boundary conditions specify that the variable is periodic over the boundary. They are often used to model physical phenomena that are periodic in nature, such as vibrations or oscillations. In general, periodic boundary conditions are specified as follows:

u(x + p, y + q) = u(x, y)  \ \ \ \ \ on \ the \ boundary

where $u$ is the variable (such as temperature or velocity), $p$ and $q$ are the periods of the boundary condition in the $x$ and $y$ directions, respectively, and $u(x,y)$ is the value of $u$ at the point on the boundary.

5. Robin boundary conditions: These boundary conditions specify a combination of a Dirichlet condition and a Neumann condition at the boundary. They are often used to model physical phenomena that involve both fixed values and fluxes at the boundary. In general, Robin boundary conditions are specified as follows:

u + h{du \over dn} = g \ \ \ \ \ on \ the \ boundary

where $u$ is the variable (such as temperature or velocity), ${du /over dn }$ is the normal derivative of $u$ at the boundary, $h$ and $g$ are fixed coefficients, and $u$ and ${du \over dn}$ are evaluated at the boundary. This equation expresses the Robin boundary condition as a combination of a Dirichlet condition ($g$ ) and a Neumann condition$(h {du \over dn} )$.