Math: Differential Equations

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Analytic Solution for the Burgers Equation
http://home.usit.net/~cmdaven/burgers.htm
Provides the general analytic solution for the Burgers equation in the form of a 4-D commutative hypercomplex function. The solution exhibits the main dynamic features in a Burgers medium: propagation of disturbances, shock waves, propagating state change fronts, and solitons. A page is included to explain the hypercomplex mathematics.

Analytical solution for the Korteweg-de Vries equation
http://home.usit.net/~cmdaven/korteweg.htm
Provides the general analytic solution for the KdV equation. In one function, the result models traveling wavetrains, solitary spikes (solitons), and sech-form long waves.

arXiv Front: AP Analysis of PDEs
http://front.math.ucdavis.edu/math.AP
PDEs section of the mathematics e-print arXiv.

Bifurcations, Equilibria, and Phase Lines: Modern Topics in Differential Equation Courses
http://math.bu.edu/DYSYS/ode-bif/ode-bif.html
Online course material

Computational PDEs Unit
http://www.scs.leeds.ac.uk/cpde/
School of Computing, University of Leeds. Research details, publications, software and resources.

Difference Method for Numerical Approximation to Applied Differential Equations.
http://www.geocities.com/b_ward.rm/na.html
This page explains how to use the difference formula of differentials to approximate the differential equations for applied systems. This method is used when analytical techniques are unavailable or cause computers to spit out garbage. This difference method is very similar to the Runge-Kata and Newton's method.

Differential Equations in Banach Algebras
http://www.gwfa.de/math/Fuchs_1996.pdf
Fuchsian Singularities of Linear Ordinary Differential Equations in Banach Algebras. By Gerald Albrecht in Wuppertal.

Differential Equations in Industry and Commerce
http://www.maths.ox.ac.uk/ociam/TMR/
European TMR network coordinated at the Oxford Centre for Industrial and Applied Mathematics.

Elliptic Problems with Concentrated Loading
http://www.sci.hkbu.edu.hk/msc/full/billy/billy.html
A web text on the background to the extrapolation method for the numerical solution of elliptic boundary value problems by Kwok Sui-Yuen Billy.

Finding Green's Functions for ODEs
http://www.mathphysics.com/pde/green/g15.html
A brief but technical overview of methods of finding Green's functions. By Evans M. Harrell II and James V. Herod.

GetDP (a General environment for the treatment of Discrete Problems)
http://www.geuz.org/getdp/
A scientific software environment for the numerical solution of integro-differential equations, open to the coupling of physical problems (electromagnetic, acoustic, thermal, mechanical, ...) as well as of numerical methods (finite element methods, boundary element and integral methods, ...).

Green's Function Theory
http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node59.html
A set of lecture notes on Green's functions and their applications.

Introduction to Green's Functions
http://www.boulder.nist.gov/div853/greenfn/tutorial.html
Green's functions play an important role in the solution of linear ordinary and partial differential equations, and are a key component to the development of boundary integral equation methods.

Linear Mathematics in Infinite Dimensions
http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node2.html
A set of lecture notes on the mathematical framework that underlies linear systems arising in physics, engineering and applied mathematics.

Math Unit III: More on the derivative and differential equations
http://dept.physics.upenn.edu/courses/gladney/mathphys/subsection3_1_1.html
Exact definition of derivation and calculating the relationship of derivatives of related functions.

MathPages: Calculus and DiffEq Notes
http://www.mathpages.com/home/icalculu.htm
Kevin Brown's compilation of postings including many topics in differential equations.

MGNet
http://www.mgnet.org/
Information related to multigrid, multilevel, multiscale, aggregation, defect correction, and domain decomposition methods.

Navier-Stokes Type Equations
http://www.coolissues.com/mathematics/Navier-Stokes/nstokes.htm
Explicit solutions provided for this particular type of equation and their relations to the heat equation, Burger's equation, and Euler's equation.

Nonlinear Differential Equations at Glasgow
http://www.maths.gla.ac.uk/~ca/
The site describes research activities of the differential equations group in the mathematics department at the university of Glasgow, UK, and provides some resources of a general nature.

Numerical Methods for Partial Differential Equations
http://www.fusion.kth.se/courses/pde
Methods such as finite differences, finite elements, fast Fourier transforms, Monte-Carlo and Lagrangian schemes are discussed in 1D to solve a variety of problems including the advection, diffusion, Black-Scholes, Burger, Korteweg-DeVries and the Schroedinger equations.

ODEcalc
http://www.digitalcalculus.com/demos/ODEcalc/
An ordinary differential equation (ODE) calculator. State your equation and boundary or initial value conditions and it solves your problem. Plots solution, y, and derivative, ydot, versus x. Solves nth order ODE as IVP and BVP.

Partial Differential Equations
http://www.sst.ph.ic.ac.uk/angus/Lectures/compphys/node24.html
An overview of partial differential equations and their physical applications.

PDEase2D 3.0
http://www.scientek.com/macsyma/pdmain.htm
Solves partial differential equations numerically by finite element analysis for use in such problems as heat transfer, reaction diffusion, solid and fluid mechanics, electromagnetics, groundwater flow, and quantum mechanics.

PRIDE
http://www.amsta.leeds.ac.uk/Applied/CAGD.dir/PRIDE/index.htm
Products by Rapid Integrated Detailed Engineering. An application of PDEs in engineering design.

Stripf's Homepage
http://home.t-online.de/home/wstripf/
A Java Applet to illustrate and solve initial value problems. Uses different numerical methods (e.g. Runge-Kutta) that can be compared to each other.

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