Partial differential equation
There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers.Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability.[1] Among the many open questions are the existence and smoothness of solutions to the Navier–Stokes equations, named as one of the Millennium Prize Problems in 2000.Partial differential equations are ubiquitous in mathematically oriented scientific fields, such as physics and engineering.They also arise from many purely mathematical considerations, such as differential geometry and the calculus of variations; among other notable applications, they are the fundamental tool in the proof of the Poincaré conjecture from geometric topology.As such, it is usually acknowledged that there is no "universal theory" of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields.This is a reflection of the fact that they are not, in any immediate way, special cases of a "general solution formula" of the Laplace equation.The nature of this failure can be seen more concretely in the case of the following PDE: for a function v(x, y) of two variables, consider the equationTo understand it for any given equation, existence and uniqueness theorems are usually important organizational principles.In many introductory textbooks, the role of existence and uniqueness theorems for ODE can be somewhat opaque; the existence half is usually unnecessary, since one can directly check any proposed solution formula, while the uniqueness half is often only present in the background in order to ensure that a proposed solution formula is as general as possible.By contrast, for PDE, existence and uniqueness theorems are often the only means by which one can navigate through the plethora of different solutions at hand.For this reason, they are also fundamental when carrying out a purely numerical simulation, as one must have an understanding of what data is to be prescribed by the user and what is to be left to the computer to calculate.For instance, the following PDE, arising naturally in the field of differential geometry, illustrates an example where there is a simple and completely explicit solution formula, but with the free choice of only three numbers and not even one function.(This is separate from asymptotic homogenization, which studies the effects of high-frequency oscillations in the coefficients upon solutions to PDEs.)Assuming uxy = uyx, the general linear second-order PDE in two independent variables has the formJust as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant B2 − 4AC, the same can be done for a second-order PDE at a given point.If there are n independent variables x1, x2 , …, xn, a general linear partial differential equation of second order has the formHowever, the classification only depends on linearity of the second-order terms and is therefore applicable to semi- and quasilinear PDEs as well.The basic types also extend to hybrids such as the Euler–Tricomi equation; varying from elliptic to hyperbolic for different regions of the domain, as well as higher-order PDEs, but such knowledge is more specialized.The geometric interpretation of this condition is as follows: if data for u are prescribed on the surface S, then it may be possible to determine the normal derivative of u on S from the differential equation.This is possible for simple PDEs, which are called separable partial differential equations, and the domain is generally a rectangle (a product of intervals).More generally, applying the method to first-order PDEs in higher dimensions, one may find characteristic surfaces.An important example of this is Fourier analysis, which diagonalizes the heat equation using the eigenbasis of sinusoidal waves.Many interesting problems in science and engineering are solved in this way using computers, sometimes high performance supercomputers.Continuous group theory, Lie algebras and differential geometry are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform and finally finding exact analytic solutions to the PDE.Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version hp-FEM.[14][15] The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge–Kutta, etc.The meaning for this term may differ with context, and one of the most commonly used definitions is based on the notion of distributions.Regularity refers to the integrability and differentiability of weak solutions, which can often be represented by Sobolev spaces.
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