Partial Differential Equations

Introduction

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(.)U_xx+b(.)U_xy+c(.)U_yy+d(.)U_x+ e(.)U_y+f(.)U+g(.) = 0
where U_x means the partial derivative of U with respect to x, U_xx 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 :

Hyperbolic : when b²-ac > 0, a typical example is the wave equation.

Parabolic : when b²-ac = 0, a typical example is the diffusion equation.

Eliptic : when b²-ac < 0, a typical example is the Poisson equation.

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 :

Dirichlet : if we specify the function that we are searching in the boundary, that is : u = g(x,t).

Neumann : if the derivative of the function is specified : u_n = g(x,t).

Robin : if the condition is specified in the form: u_n+a(x,t)u = g(x,t).

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 :

The finite difference method : replaces each derivative by a discretization, usually truncated Taylor series. There are a lot of schemes, depending on the chosen discretization. The scheme can be explicit - if there's no need to solve a system of equations, just to walk the grid nodes - or implicit if we have to solve a system of equations for each row of the grid.

The finite elements method : Consists in aproximating the function in small pieces of the domain called finite elements. Once the approximation for each element is determined, the solution for the whole domain is obtained assembling the solution for each element.

Cellular automatons (CA): Cellular automatons are discrete systems of lattice sites having various initial values. These sites evolve in discrete time steps, each site assumes a new value based on the values of some local neighborhood of sites and a finite number of previous time steps.

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

Structured

Non structured

Examples

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 U_t+U_xx+U_xy+U_yx=0	Solving the equation U_t+U_xx+U_xy+U_yx=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
Gravitation	Ecology	Electronics	PDEs	Other pages

Last modified 2/1/2000 by Juan de Lara ( Juan.Lara@ii.uam.es, http://www.ii.uam.es/~jlara) need help for using this courses?.

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