Editing Partial differential equation (section)

== Numerical solutions ==
The three most widely used [[Numerical partial differential equations|numerical methods to solve PDEs]] are the [[finite element analysis|finite element method]] (FEM), [[finite volume method]]s (FVM) and [[finite difference method]]s (FDM), as well other kind of methods called [[meshfree methods]], which were made to solve problems where the aforementioned methods are limited. The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version [[hp-FEM]]. Other hybrid versions of FEM and Meshfree methods include the generalized finite element method (GFEM), [[extended finite element method]] (XFEM), [[Spectral element method|spectral finite element method]] (SFEM), [[Meshfree methods|meshfree finite element method]], [[Discontinuous Galerkin method|discontinuous Galerkin finite element method]] (DGFEM), [[element-free Galerkin method]] (EFGM), [[interpolating element-free Galerkin method]] (IEFGM), etc.

=== Finite element method ===
{{main|Finite element method}}
The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations.<ref>{{cite book |first=P. |last=Solin |title=Partial Differential Equations and the Finite Element Method |publisher=J. Wiley & Sons |location=Hoboken, New Jersey |year=2005 |isbn=0-471-72070-4 }}</ref><ref>{{cite book |first1=P. |last1=Solin |first2=K. |last2=Segeth |name-list-style=amp |first3=I. |last3=Dolezel |title=Higher-Order Finite Element Methods |publisher=Chapman & Hall/CRC Press |location=Boca Raton |year=2003 |isbn=1-58488-438-X }}</ref> The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge–Kutta, etc.

===Finite difference method===
{{main|Finite difference method}}
Finite-difference methods are numerical methods for approximating the solutions to differential equations using [[finite difference]] equations to approximate derivatives.

=== Finite volume method ===
{{main|Finite volume method}}
Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using the [[divergence theorem]]. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods conserve mass by design.

=== Neural networks ===
{{Excerpt|Deep learning|Partial differential equations}}