Partial Differential Equations


While searching for a quantitative description of physical phenomena, the engineer or the physicist establishes generally a system of ordinary or partial differential equations valid in a certain region (or domain) and imposes on this system suitable boundary and initial conditions.

The general form of a second order PDE is :
a(.)Uxx+b(.)Uxy+c(.)Uyy+d(.)Ux+ e(.)Uy+f(.)U+g(.) = 0
where Ux means the partial derivative of U with respect to x, Uxx means the second derivative of U with respect to x2, etc.
a(.), b(.), c(.), ... are functions that can depend on x, y, TIME, etc.

Depending on the coefficients PDEs can be clasified in :

The two firsts types (hyperbolic and parabolic) defines an initial value problem, the third defines a boundary problem.

Boundary conditions can be clasified in three kinds :
The conditions are called homogeneous when g is zero.

Resolution methods

To solve numerically those equations it is necessary to recast the problem into a purely algebraic form, involving only the basic arithmetic operations. To achieve this, various forms of discretization of the continuum problem can be used. Some of the most popular methods are :

There are a lot of methods to discretize the domain, we can clasify them into :


We have prepared several examples of the solution of PDEs :

Example pages:
1-d Heat Equation
2-d steady state Heat Equation
2-d Heat Equation
1-d non diffusive transport equation
1-d diffusive transport equation
2-d Non diffusive transport equation
Mesh generation with OOCSMP
Moving grids
Application pages:
Heating of two beams
Heating of two moving beams
Solving the equation Ut+Uxx+Uxy+Uyx=0
Solving the equation Ut+Uxx+Uxy+Uyx=0 using MGEN
Heat 1d using several outputs
Solving the Heat equation with a CA
Comparing a CA with the FEM
Gordon's sine equation
Using MGEN to modify the PDE and the conditions

Other courses
Other pages

Last modified 2/1/2000 by Juan de Lara (, need help for using this courses?.

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