Ad
related to: parabolic equations examples
Search results
Results From The WOW.Com Content Network
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena in, i.a., engineering science, quantum mechanics and financial mathematics. Examples include the heat equation, time-dependent Schrödinger equation and the Black–Scholes ...
For a parametric equation of a parabola in general position see § As the affine image of the unit parabola. The implicit equation of a parabola is defined by an irreducible polynomial of degree two: + + + + + =, such that =, or, equivalently, such that + + is the square of a linear polynomial.
In this equation, the origin is the midpoint of the horizontal range of the projectile, and if the ground is flat, the parabolic arc is plotted in the range . This expression can be obtained by transforming the Cartesian equation as stated above by y = r sin ϕ {\displaystyle y=r\sin \phi } and x = r cos ϕ {\displaystyle x=r\cos \phi } .
This justifies Laplace equation as an example of this type. [6] B 2 − AC = 0 (parabolic partial differential equation): Equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases.
While a parabolic arch may resemble a catenary arch, a parabola is a quadratic function while a catenary is the hyperbolic cosine, cosh(x), a sum of two exponential functions. One parabola is f(x) = x 2 + 3x − 1, and hyperbolic cosine is cosh(x) = e x + e −x / 2 . The curves are unrelated.
The heat equation is the prototypical example of a parabolic partial differential equation. Using the Laplace operator, the heat equation can be simplified, and generalized to similar equations over spaces of arbitrary number of dimensions, as = =,
[2] [3] It is also used to numerically solve parabolic and elliptic partial differential equations, and is a classic method used for modeling heat conduction and solving the diffusion equation in two or more dimensions. [4] It is an example of an operator splitting method. [5]
The convection–diffusion equation is a parabolic partial differential equation that combines the diffusion and convection equations. It describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection .