Partial Differential Equations (PDE)

https://en.wikipedia.org/wiki/Partial_differential_equation

https://en.wikipedia.org/wiki/Dirichlet_boundary_condition

https://en.wikipedia.org/wiki/Neumann_boundary_condition

https://en.wikipedia.org/wiki/Finite_element_method

https://en.wikipedia.org/wiki/Finite_difference_method

https://en.wikipedia.org/wiki/Finite_volume_method

INTRODUCTION

A partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. It is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers.

There are existence and uniqueness theorems for differential equations where the domain of the unknown function is regarded as part of the problem. There are several possible boundary conditions for the unknown function.

The 'Dirichlet boundary condition' specifies the values that a solution needs to take along the boundary of the domain.

The Neumann boundary condition specifies the values of the derivative applied at the boundary of the domain.

An unknown function u(x, y),
u_x, u_y - first partial derivatives,
u_xx, u_xy, u_yy - second partial derivatives.

u_xx + u_yy = 0 is Laplace's equation,

where u = 0 on the boundary (Dirichlet problem).

NUMERICAL SOLUTIONS

The three most widely used numerical methods to solve PDEs are:
(1) the finite element method (FEM),
(2) the finite difference method (FDM),
(3) the finite volume method (FVM).