* Indice de la p gina de PDEs
* AUTHOR Juan de Lara
* EMAIL Juan.Lara@ii.uam.es
* DATE 2/1/2000
TITLE Partial Differential Equations
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SECTION "2","Introduction"
DESCRIPTION While searching for a quantitative description of physical
DESCRIPTION phenomena, the engineer or the physicist establishes generally
DESCRIPTION a system of ordinary or partial differential equations valid in
DESCRIPTION a certain region (or domain) and imposes on this system suitable
DESCRIPTION boundary and initial conditions.\n\n
DESCRIPTION The general form of a second order PDE is :\n
DESCRIPTION \ITALIC{a(.)U\SUB{xx}+b(.)U\SUB{xy}+c(.)U\SUB{yy}+d(.)U\SUB{x}+
DESCRIPTION e(.)U\SUB{y}+f(.)U+g(.) = 0}\n
DESCRIPTION where \ITALIC{U\SUB{x}} means the partial derivative of U with
DESCRIPTION respect to \ITALIC{x}, \ITALIC{U\SUB{xx}} means the second
DESCRIPTION derivative of U with respect to \ITALIC{x\SUP{2}}, etc.\n
DESCRIPTION \ITALIC{a(.), b(.), c(.), ...} are functions that can depend
DESCRIPTION on \ITALIC{x, y, TIME}, etc.\n\n
DESCRIPTION Depending on the coefficients PDEs can be clasified in :\n
DESCRIPTION \ITEM{\BOLD{Hyperbolic :} when \ITALIC{b\SUP{2}-ac > 0}, a
DESCRIPTION typical example is the wave equation.}
DESCRIPTION \ITEM{\BOLD{Parabolic :} when \ITALIC{b\SUP{2}-ac = 0}, a
DESCRIPTION typical example is the diffusion equation.}
DESCRIPTION \ITEM{\BOLD{Eliptic :} when \ITALIC{b\SUP{2}-ac < 0}, a
DESCRIPTION typical example is the Poisson equation.}
DESCRIPTION The two firsts types (hyperbolic and parabolic) defines an
DESCRIPTION initial value problem, the third defines a boundary problem.\n\n
DESCRIPTION Boundary conditions can be clasified in three kinds :\n
DESCRIPTION \ITEM{\BOLD{Dirichlet :} if we specify the function that we are
DESCRIPTION searching in the boundary, that is : \ITALIC{u = g(\BOLD{x},t)}.}
DESCRIPTION \ITEM{\BOLD{Neumann :} if the derivative of the function is
DESCRIPTION specified : \ITALIC{u\SUB{n} = g(\BOLD{x},t)}.}
DESCRIPTION \ITEM{\BOLD{Robin :} if the condition is specified in the form:
DESCRIPTION \ITALIC{u\SUB{n}+a(\BOLD{x},t)u = g(\BOLD{x},t)}.}
DESCRIPTION The conditions are called homogeneous when g is zero.
SECTION "2","Resolution methods"
DESCRIPTION To solve numerically those equations it is necessary to recast
DESCRIPTION the problem into a purely algebraic form, involving only
DESCRIPTION the basic arithmetic operations.
DESCRIPTION To achieve this, various forms of discretization of the continuum
DESCRIPTION problem can be used. Some of the most popular methods are :\n
DESCRIPTION \ITEM{\LINK{"FDM.html"\BOLD{The finite difference method :}} replaces each derivative
DESCRIPTION by a discretization, usually truncated Taylor series. There are
DESCRIPTION a lot of schemes, depending on the chosen discretization. The scheme
DESCRIPTION can be \BOLD{explicit} - if there's no need to solve a system of equations,
DESCRIPTION just to walk the grid nodes - or \BOLD{implicit} if we have to
DESCRIPTION solve a system of equations for each row of the grid.}
DESCRIPTION \ITEM{\LINK{"FEM.html"\BOLD{The finite elements method :}} Consists in aproximating
DESCRIPTION the function in small pieces of the domain called finite elements.
DESCRIPTION Once the approximation for each element is determined, the solution
DESCRIPTION for the whole domain is obtained assembling the solution for each
DESCRIPTION element.}
DESCRIPTION \ITEM{\LINK{"CA.html"\BOLD{Cellular automatons (CA):}} Cellular
DESCRIPTION automatons are discrete systems of lattice sites having various
DESCRIPTION initial values. These sites evolve in discrete time steps, each
DESCRIPTION site assumes a new value based on the values of some local
DESCRIPTION neighborhood of sites and a finite number of previous time steps.}
DESCRIPTION There are a lot of methods to discretize the domain, we can clasify
DESCRIPTION them into :\n
DESCRIPTION \ITEM{\BOLD{Structured}}
DESCRIPTION \ITEM{\BOLD{Non structured}}
SECTION "2","Examples"
DESCRIPTION We have prepared several examples of the solution of PDEs :\n
TABLE "Example pages:", [4;2], [C,80],
"\CENTER{\LINK{"Heat.html"1-d Heat Equation}}",
"\CENTER{\LINK{"h2d.html"2-d steady state Heat Equation}}",
"\CENTER{\LINK{"dh2d.html"2-d Heat Equation}}",
"\CENTER{\LINK{"tr1d.html"1-d non diffusive transport equation}}",
"\CENTER{\LINK{"trdiff.html"1-d diffusive transport equation}}",
"\CENTER{\LINK{"transp2d.html"2-d Non diffusive transport equation}}",
"\CENTER{\LINK{"grid.html"Mesh generation with OOCSMP}}",
"\CENTER{\LINK{"mgrid.html"Moving grids}}"
TABLE "Application pages:", [3;2], [C,80],
"\CENTER{\LINK{"LBeam1.html"Heating of two beams}}",
"\CENTER{\LINK{"LBeam1.html"Heating of two moving beams}}",
"\CENTER{\LINK{"pde1.html"Solving the equation U\SUB{t}+U\SUB{xx}+U\SUB{xy}+U\SUB{yx}=0}}",
"\CENTER{\LINK{"pde_dyn.html"Solving the equation U\SUB{t}+U\SUB{xx}+U\SUB{xy}+U\SUB{yx}=0 using MGEN}}",
"\CENTER{\LINK{"heat1da.html"Heat 1d using several outputs}}",
"\CENTER{\LINK{"CA.html"Solving the Heat equation with a CA}}"
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