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Differential.png|thumb|right|An illustration of a differential equation. The arrows show how the differential equation locally influences a state, while the lines display how specific solutions are determined by starting conditions (red dots).]] In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. Many of the fundamental laws of physics, chemistry, biology and economics can be formulated as differential equations. The mathematical theory of differential equations has developed together with the sciences where the equations originate and where the results find application. Diverse scientific fields often give rise to identical problems in differential equations. In such cases, the mathematical theory can unify otherwise quite distinct scientific fields. A celebrated example is Fourier's theory of the conduction of heat in terms of sums of trigonometric functions Fourier series, which finds application in the propagation of sound, the propagation of electric and magnetic fields, radio waves, optics, elasticity, spectral analysis of radiation, and other scientific fields.
The order of a differential equation is that of the highest derivative that it contains. For instance, a first-order differential equation contains only first derivatives.
Mathematicians typically study weak solutions (relying on weak derivatives), which are types of solutions that do not have to be differentiable everywhere. This extension is often necessary for solutions to exist, and it also results in more physically reasonable properties of solutions, such as shocks in hyperbolic (or wave) equations.
More on [ Differential equation ]
Calculus :: Math
Mathematical Physics :: Physics
Videos for Elementary Differential Equations with Linear Algebra http://ow.ly/NgOQ
AnkammaRao (AnkammaRao) Fri, 18 Dec 2009 05:51:18 -0000
Videos for Elementary Differential Equations with Linear Algebra http://ow.ly/NgOQ
Ahh differential equations. How I miss eigenvectors and eigenvalues
Raptor464 (Phil Martin) Fri, 18 Dec 2009 04:22:50 -0000
Ahh differential equations. How I miss eigenvectors and eigenvalues
Around the Research of Vladimir Maz'ya II: Partial Differential Equations (International Mathematical Series) - Springer. http://j.mp/7i2cW9
new_textbook (New Textbooks) Fri, 18 Dec 2009 04:20:11 -0000
Around the Research of Vladimir Maz'ya II: Partial Differential Equations (International Mathematical Series) - Springer. http://j.mp/7i2cW9
#ebookdownload MIT Open Course: Differential Equations http://bit.ly/5PEaDH #ebookfree
freeebookslink (free ebook) Fri, 18 Dec 2009 03:17:43 -0000
#ebookdownload MIT Open Course: Differential Equations http://bit.ly/5PEaDH #ebookfree
Around the Research of Vladimir Maz'ya II: Partial Differential Equations (International Mathematical Series). http://tinyurl.com/yffbjba
math_book (Mathematics Books) Fri, 18 Dec 2009 03:17:00 -0000
Around the Research of Vladimir Maz'ya II: Partial Differential Equations (International Mathematical Series). http://tinyurl.com/yffbjba
Damn...that was the hardest differential equations test this semester. Packing! Work & then off to CT for the weekend! :)
Dalyssa (Dalyssa Hernandez) Fri, 18 Dec 2009 02:14:01 -0000
Damn...that was the hardest differential equations test this semester. Packing! Work & then off to CT for the weekend! :)
Analytic Solution for the Burgers Equation - 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.
Meta Description: [ The general analytical solution for the
Burgers equation is presented in terms of 4-D commutative hypercomplex mathematics.
The one general solution (4-D characteristic function) is found to implicitly
model planar shock waves, propagating diffusion fronts, and solitons. ]
Analytical solution for the Korteweg-de Vries equation - 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.
Meta Description: [ The general analytical solution for the Korteweg-
de Vries equation is presented in terms of 4-D commutative hypercomplex mathematics.
The one general solution (4-D characteristic function) is found to implicitly
model fluid behavior, traveling waves, shock waves, periodic potential wells, ... ]
arXiv Front: AP Analysis of PDEs - PDEs section of the mathematics e-print arXiv.
Bifurcations, Equilibria, and Phase Lines: Modern Topics in Differential Equation Courses - Online course material
C*ODE*E Archive - Consortium of ODE Experiments at Harvey Mudd College. Newsletter, graphics, links.
Computational PDEs Unit - School of Computing, University of Leeds. Research details, publications, software and resources.
Differential Equations in Banach Algebras - Fuchsian Singularities of Linear Ordinary Differential Equations in Banach Algebras. By Gerald Albrecht in Wuppertal.
Differential Equations in Industry and Commerce - European TMR network coordinated at the Oxford Centre for Industrial and Applied Mathematics.
Meta Description: [ TMR Network for Differential Equations in Industry and Commerce ]
Elliptic Problems with Concentrated Loading - 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 - 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) - 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 - A set of lecture notes on Green's functions and their applications.
Meta Description: [ Sturm-Liouville Theory ]
Introduction to Green's Functions - 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 - A set of lecture notes on the mathematical framework that underlies linear systems arising in physics, engineering and applied mathematics.
Meta Description: [ Contents ]
Math Unit III: More on the derivative and differential equations - Exact definition of derivation and calculating the relationship of derivatives of related functions.
MathPages: Calculus and DiffEq Notes - Kevin Brown's compilation of postings including many topics in differential equations.
MGNet - Information related to multigrid, multilevel, multiscale, aggregation, defect correction, and domain decomposition methods.
Navier-Stokes Type Equations - 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 - 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.
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Numerical Methods for Partial Differential Equations - 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.
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Osaka University - PDE-Analysis research group. Organises the East Asia Symposium on PDE.
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Partial Differential Equations - An overview of partial differential equations and their physical applications.
Meta Description: [ Partial Differential Equations ]
PDEase2D 3.0 - 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 - Products by Rapid Integrated Detailed Engineering. An application of PDEs in engineering design.
Table of Laplace Transforms - This page contains an extensive table of Laplace transforms. Laplace transforms are used to solve certain differential equations.
The Animated Telegraph equation - This demonstration illustrates the behaviour of solutions of the telegraph equation
Meta Description: [ The Telegraph Equation (animated) ]
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The Polar Representation Theorem - An article covering n-dimensional time-dependent linear Hamiltonian systems. By Jorge Rezende from the University of Lisbon. In PDF format.
The World of Mathematical Equations - Gives solutions to different types of ordinary differential equations, including linear and nonlinear functions. Many pages use PDF.
Meta Description: [ The EqWorld website presents extensive information on ordinary differential,
partial differential, integral, functional, and other mathematical equations. ]
Weak and Variational Forms of Poisson's Equation - A set of lecture notes on Poisson's equation. [PDF Format]
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