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...[[hyperbolic partial differential equation|hyperbolic partial differential equations]], it is unstable unless artificial viscosity is included. The abbreviation ...rder convergence in space. For example, in one dimension, if the [[partial differential equation]] is ...

...tial algebraic equation (PDAE)''' set is an incomplete system of [[partial differential equation]]s that is closed with a set of [[algebraic equation]]s. ...y, \frac{\partial y_i}{\partial x_j}, \frac{\partial^2 y_i}{\partial x_j \partial x_k}, \ldots, \mathbf z \right),</math> ...

: <math>u(\mathbf{x}) = 0 \text{ for } \mathbf{x} \in \partial \Omega ...ath> is a [[function (mathematics)|function]] with positive values, <math>\partial \Omega</math> is a well-behaved [[boundary (topology)|boundary]] of the ope ...

...e coefficients and their [[derivative]]s. Consider a bivariate hyperbolic differential operator of the second order ''non-factorizable'' bivariate linear partial differential equations (LPDEs). ...

...mathematical study of [[Partial differential equation|partial differential equations]] (PDE) such as [[Laplace's equation]], about the [[integral#Terminology an ...ables as <math>x=(x_1,...,x_n)</math>. Representing the PDE as a partial [[differential operator]] <math>L</math> acting on an unknown function <math> ...

...rical analysis|numerical]] method for the solution of [[hyperbolic partial differential equation]]s based on [[finite difference]]s. The method can be described a Consider a one-dimensional, linear hyperbolic partial differential equation for <math>u(x,t)</math> of the form: ...

...ction of a [[vector bundle]] (specifically, the [[adjoint bundle]]). These equations are ...y are very closely related to the [[Ginzburg–Landau theory|Ginzburg–Landau equations]], when these are expressed in a general geometric setting. ...

...1928 in Osaka) is a retired Japanese mathematician who worked in [[partial differential equation]]s. He obtained his Ph.D. at the [[University of Tokyo]], under th ...-tokyo.ac.jp/record/39882/files/jfs130201.pdf }}</ref> studied the partial differential equation ...

...t methods|implicit scheme]], mainly used for solving non-linear hyperbolic equations. It is not used much nowadays. ...numerical stability.<ref name="An Implicit Finite-Difference Algorithm for Hyperbolic Systems in Conservation-Law Form" /> ...

...ty Press |location=Oxford |year=1959 }}</ref> namely a [[parabolic partial differential equation]] of the kind: <math display="block">\frac{\partial\theta}{\partial t}~=~\alpha~\nabla^2\theta ,</math> ...

{{Short description|Numerical method for solving differential equations}} ...mical reactions in fluid dynamics, and to solve multidimensional [[partial differential equation]]s by reducing them to a sum of one-dimensional problems. ...

{{short description|Special case of the Euler-Lagrange equations}} If <math>\frac{\partial L}{\partial x} = 0</math>, then the Euler–Lagrange equation reduces to the Beltrami ide ...

...he '''AKNS system''' is an [[integrable system]] of [[partial differential equations]], introduced by and named after [[Mark J. Ablowitz]], David J. Kaup, [[Ala The AKNS system is a pair of two partial [[differential equation]]s for two [[complex-valued function]]s ''p'' and ''q'' of 2 varia ...

...riate hyperbolic operator of the second order''' (see [[Hyperbolic partial differential equation]]), constructing two Laplace invariants. Each [[Laplace invariant] ...nd in<ref>R. Beals, E. Kartashova. Constructively factoring linear partial differential operators in two variables. [https://doi.org/10.1007%2Fs11232-005-0178-7 Th ...

...ential equation|parabolic]], or [[hyperbolic partial differential equation|hyperbolic]]. The first well-documented ...tp://www.efunda.com/formulae/fluids/navier_stokes.cfm |title=Navier-Stokes Equations in Fluid Mechanics |publisher=Efunda.com |date= |accessdate=2013-10-29}}</r ...

As the cantilever oscillates back and forth it arches into hyperbolic curves with the salient feature that a tangent to the end of the cantilever ...corresponding Lagrangian equations which need to be solved as a system of equations, ...

...iginal equations, reducing the order of—all or some of—the highest-order [[partial derivative]]s. But the validity of the assumptions which are made need to b <math display="block">\nabla^2 E - \frac{1}{ c^2} \frac{\partial^2E}{\partial t^2} = 0\, ,</math> ...

...uation|wave]] equations) and all the others are put together, an ordinary "differential" equation with respect to the variable which was put in evidence is obtaine ...e equation<ref name="abstract">{{cite web |last1=Dezin |first1=A.A. |title=Differential equation, abstract |url=http://www.encyclopediaofmath.org/index.php?title=D ...

...error on a range of M<0.2. For this range we have to apply [[Navier–Stokes equations#Incompressible flow|incompressible Navier Stokes]] and [[Euler equation]]s ...ailure for the compressible flow methods is the stiffness of the governing equations. ...

...r choice for use in higher-order numerical solvers for the [[Navier-Stokes Equations]]. ...uations for two unknowns and therefore stop checking for higher-order-term equations. ...