From now, let’s talk about various partial differential equations.

At first, I will talk about Laplace equation and Poisson equation.

The Laplace equation is a second-order partial differential equation that arises in many physical systems, such as heat conduction, electrostatics, and fluid dynamics. It can be written in the following general form:

∇^2 u = 0

where $u$ is the unknown function (the “solution” to the equation) and $∇^2$ is the Laplace operator (the sum of the second-order partial derivatives of $u$ with respect to each coordinate). The equation is usually solved for u, subject to certain boundary conditions on the domain where the equation is defined.

In two dimensions, the Laplace equation can be written as:

u_{xx} + u_{yy} = 0

where $u_{xx}$ and $u_{yy}$ are the second-order partial derivatives of $u$ with respect to the $x$ and $y$ coordinates, respectively. In three dimensions, the equation becomes:

u_{xx} + u_{yy} + u_{zz} = 0

where $u_{zz}$ is the second-order partial derivative of $u$ with respect to the $z$ coordinate.

The Laplace equation is often used to model systems that are in a state of equilibrium, where the forces or fluxes acting on the system are balanced. It can be solved using various numerical methods, such as finite difference methods or finite element methods. The finite element method involves discretizing the domain into smaller elements, and approximating the solution u as a linear or higher-order polynomial within each element. The solution is then obtained by solving a system of linear equations representing the discretized form of the Laplace equation and the boundary conditions.

The Laplace equation and the Poisson equation are closely related, as the Poisson equation is a generalization of the Laplace equation.

The Poisson equation is a second-order partial differential equation that arises in many physical systems, such as heat conduction, electrostatics, and fluid dynamics. It can be written in the following general form:

∇^2 u = f

where u is the unknown function (the “solution” to the equation), $∇^2$ is the Laplace operator (the sum of the second-order partial derivatives of $u$ with respect to each coordinate), and $f$ is a given function (the “source term”). The equation is usually solved for $u$, subject to certain boundary conditions on the domain where the equation is defined.

In two dimensions, the Poisson equation can be written as:

u_{xx} + u_{yy} = f

where $u_{xx}$ and $u_{yy}$ are the second-order partial derivatives of u with respect to the $x$ and $y$ coordinates, respectively. In three dimensions, the equation becomes:

u_{xx} + u_{yy} + u_{zz} = f

where $u_{zz}$ is the second-order partial derivative of u with respect to the z coordinate.

The Poisson equation can be solved using various numerical methods, such as finite difference methods or finite element methods. The finite element method involves discretizing the domain into smaller elements, and approximating the solution u as a linear or higher-order polynomial within each element. The solution is then obtained by solving a system of linear equations representing the discretized form of the Poisson equation and the boundary conditions.