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Elliptic Partial Differential Equation
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, where , , , , , , and are functions of and and where u_x=\frac, u_=\frac and similarly for u_,u_y,u_. A PDE written in this form is elliptic if :B^2-AC, applying the chain rule once gives :u_=u_\xi \xi_x+u_\eta \eta_x and u_=u_\xi \xi_y+u_\eta \eta_y, a second application gives :u_=u_ _x+u_ _x+2u_\xi_x\eta_x+u_\xi_+u_\eta_, :u_=u_ _y+u_ _y+2u_\xi_y\eta_y+u_\xi_+u_\eta_, and :u_=u_ \xi_x\xi_y+u_ \eta_x\eta_y+u_(\xi_x\eta_y+\xi_y\eta_x)+u_\xi_+u_\eta_. We can replace our PDE in x and y with an equivalent equation in \xi and \eta :au_ + 2bu_ + cu_ \text= 0,\, where :a=A^2+2B\xi_x\xi_y+C^2, :b=2A\xi_x\eta_x+2B(\xi_x\eta_y+\xi_y\eta_x) +2C\xi_y\eta_y , and :c=A^2+2B\eta_x\eta_y+C^2. To transform our PDE into the desired canonical f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Linear Differential Equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x) where and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable . Such an equation is an ordinary differential equation (ODE). A ''linear differential equation'' may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives. A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. 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. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Hyperbolic Partial Differential Equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation. In one spatial dimension, this is : \frac = c^2 \frac The equation has the property that, if ''u'' and its first time derivative are arbitrarily specified initial data on the line (with sufficient smoothness properties), then there exists a solution for all time ''t''. The solutions of hyperbolic equations are "wave-like". If a disturbance is made in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Parabolic Partial Differential Equation
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, including heat conduction, particle diffusion, and pricing of derivative investment instruments. Definition To define the simplest kind of parabolic PDE, consider a real-valued function u(x, y) of two independent real variables, x and y. A second-order, linear, constant-coefficient PDE for u takes the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + F = 0, and this PDE is classified as being ''parabolic'' if the coefficients satisfy the condition :B^2 - AC = 0. Usually x represents one-dimensional position and y represents time, and the PDE is solved subject to prescribed initial and boundary conditions. The name "parabolic" is used because the assumption on the coefficients is the same as the condition for the analytic geometry equation A x^2 + 2B xy + C y^2 + D x + E y + F = 0 to define a planar parabola. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ellipse
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity e, a number ranging from e = 0 (the limiting case of a circle) to e = 1 (the limiting case of infinite elongation, no longer an ellipse but a parabola). An ellipse has a simple algebraic solution for its area, but only approximations for its perimeter (also known as circumference), for which integration is required to obtain an exact solution. Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is: : \frac+\frac = 1 . Assuming a \ge b, the foci are (\pm c, 0) for c = \sqrt. The standard parametric equation is: : (x,y) = (a\cos(t),b\sin(t)) \quad \text \quad 0\leq t\leq 2\pi. Ellip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Laplace Equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nabla \cdot \nabla = \nabla^2 is the Laplace operator,The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, \Delta x = x_1 - x_2. Its use to represent the Laplacian should not be confused with this use. \nabla \cdot is the divergence operator (also symbolized "div"), \nabla is the gradient operator (also symbolized "grad"), and f (x, y, z) is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function, h(x, y, z), we have \Delta f = h. This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Poisson Equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and physicist Siméon Denis Poisson. Statement of the equation Poisson's equation is \Delta\varphi = f where \Delta is the Laplace operator, and f and \varphi are real or complex-valued functions on a manifold. Usually, f is given and \varphi is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as and so Poisson's equation is frequently written as \nabla^2 \varphi = f. In three-dimensional Cartesian coordinates, it takes the form \left( \frac + \frac + \frac \right)\va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Method Of Characteristics
In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation. The method is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hypersurface. Characteristics of first-order partial differential equation For a first-order PDE (partial differential equation), the method of characteristics discovers curves (called characteristic curves or just characteristics) along which the PDE becomes an ordinary differential equation (ODE). Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE. For the sake of simplicity, we confine our attention to the case of a function of two independent variables ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Cauchy Problem
A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain. A Cauchy problem can be an initial value problem or a boundary value problem (for this case see also Cauchy boundary condition). It is named after Augustin-Louis Cauchy. Formal statement For a partial differential equation defined on R''n+1'' and a smooth manifold ''S'' ⊂ R''n+1'' of dimension ''n'' (''S'' is called the Cauchy surface), the Cauchy problem consists of finding the unknown functions u_1,\dots,u_N of the differential equation with respect to the independent variables t,x_1,\dots,x_n that satisfiesPetrovskii, I. G. (1954). Lectures on partial differential equations. Interscience Publishers, Inc, Translated by A. Shenitzer, (Dover publications, 1991) \begin&\frac = F_i\left(t,x_1,\dots,x_n,u_1,\dots,u_N,\dots,\frac,\dots\right) \\ &\text i,j = 1,2,\dots,N;\, k_0+k_1+\dots+k_n=k\leq n_j;\, k_0 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Heat Equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. As the prototypical parabolic partial differential equation, the heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations. The heat equation can also be considered on Riemannian manifolds, leading to many geometric applications. Following work of Subbaramiah Minakshisundaram and Åke Pleijel, the heat equation is closely related with spectral geometry. A seminal nonlinear variant of the heat equation was introduced to differential geometry by James Eells and Joseph Sampson in 1964, inspiring the introduction of the Ricci flow by Richard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Elliptic Operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions. Elliptic operators are typical of potential theory, and they appear frequently in electrostatics and continuum mechanics. Elliptic regularity implies that their solutions tend to be smooth functions (if the coefficients in the operator are smooth). Steady-state solutions to hyperbolic and parabolic equations generally solve elliptic equations. Definitions Let L be linear differential operator of order ''m'' on a domain \Omega in R''n'' given by Lu = \sum_ a_\alpha(x)\partial^\alpha u where \alpha = (\alpha_1, \dots, \alpha_n) denotes a multi-index, and \partial^\alpha u = \partial^_1 \cdots \partial_n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Hyperbolic Partial Differential Equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation. In one spatial dimension, this is : \frac = c^2 \frac The equation has the property that, if ''u'' and its first time derivative are arbitrarily specified initial data on the line (with sufficient smoothness properties), then there exists a solution for all time ''t''. The solutions of hyperbolic equations are "wave-like". If a disturbance is made in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |