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+====== Numerical solutions of PDE ======
+<note warning>In general, the numerical solutions of partial differential equations are obtained by standard methods developped and implemented in specialized softwares :
+  * Finite element method
+  * Finite volume method
+  * Boundary element method
+  * Spectral method
+  * Multigrid methods
+  * ...
+These methods often make use of variational principles.
+</note>
+  * [[wp>Numerical_partial_differential_equations|Numerical partial differential equations]]
+However, the [[wp>Finite_difference_method|finite difference method]] can be more easliy applied to a lot of classical PDE. In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.
+One of the most usual problems concern the chemical and thermal diffusion problem, in either steady and unsteady conditions, for which same equations apply. The [[wp>Heat_equation|heat equation]] is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time :
+$$\frac{\partial T}{\partial t} = \kappa\left(\frac{\partial^2T}{\partial x^2}+\frac{\partial^2T}{\partial y^2}+\frac{\partial^2T}{\partial z^2}\right)$$
+  * Dimensional analysis, dimensionless numbers,...
+  * Space discretization
+    * second-order central finite differences, first and second derivatives
+  * Time discretization, Euler approximation, explicit and implicit schemes
+  * Laplace equation (elliptic partial differential equation)
+  * Heat equation (parabolic partial differential equation)
+  * Error propagation, explicit //vs// implicit schemes, semi-implicit methods,and their correspondant time complexity
+===== Applications =====
+  * Heat diffusion, in stationary, unsteady conditions, with heat source (or sink), and various geometry
+  * Chemical diffusion
+  * Heat and chemical diffusion in heterogeneous media
+  * ion exchange chromatography