Search results
Results from the WOW.Com Content Network
Today they provide a non-perturbative approach to quantum field theories and applications can be found in many fields of theoretical physics, such as solid-state physics and elementary particle physics. Schwinger also derived an equation for the two-particle irreducible Green functions, [2] which is nowadays referred to as the inhomogeneous ...
Another application of symmetry methods is to reduce systems of differential equations, finding equivalent systems of differential equations of simpler form. This is called reduction. In the literature, one can find the classical reduction process, [4] and the moving frame-based reduction process. [11] [12] [13] Also symmetry groups can be used ...
For instance, higher symmetries of a system of partial differential equations are the analog of vector fields on differentiable manifolds. The Euler operator, which associates to each variational problem the corresponding Euler–Lagrange equation, is the analog of the classical differential associating to a function on a variety its ...
The weak interaction fields Z, W ± satisfy the Proca equation. These equations can be solved exactly. One usually does so by considering first solutions that are periodic with some period L along each spatial axis; later taking the limit: L → ∞ will lift this periodicity restriction.
Ernst Hairer, Syvert Paul Nørsett and Gerhard Wanner, Solving ordinary differential equations I: Nonstiff problems, second edition, Springer Verlag, Berlin, 1993. ISBN 3-540-56670-8. Ernst Hairer and Gerhard Wanner, Solving ordinary differential equations II: Stiff and differential-algebraic problems, second edition, Springer Verlag, Berlin, 1996.
Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Symbolic computation engines usually recognize the majority of special functions.
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain "basis functions" (for example, as a Fourier series which is a sum of sinusoids) and then to choose the ...
Some solutions of a differential equation having a regular singular point with indicial roots = and .. In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form ″ + ′ + = with ′ and ″.