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100-716: PDE may refer to: Mathematics [ edit ] Partial differential equation , differential equation involving partial derivatives (of a function of multiple variables) Life sciences [ edit ] Phosphodiesterase , enzyme important in intracellular communication Pug dog encephalitis Organizations [ edit ] The European Democratic Party (esp. in Spanish, French or Italian languages) The Pennsylvania Department of Education Technology and engineering [ edit ] Program development environment Pulse detonation engine , proposed substitute to

200-462: A {\displaystyle \rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=-\nabla p+\nabla \cdot {\boldsymbol {\tau }}+\rho \,\mathbf {a} } where In this form, it is apparent that in the assumption of an inviscid fluid – no deviatoric stress – Cauchy equations reduce to the Euler equations . Assuming conservation of mass , with the known properties of divergence and gradient we can use

300-644: A . {\displaystyle \left({\frac {\partial }{\partial t}}+\mathbf {u} \cdot \nabla -\nu \,\nabla ^{2}-({\tfrac {1}{3}}\nu +\xi )\,\nabla (\nabla \cdot )\right)\mathbf {u} =-{\frac {1}{\rho }}\nabla p+\mathbf {a} .} The convective acceleration term can also be written as u ⋅ ∇ u = ( ∇ × u ) × u + 1 2 ∇ u 2 , {\displaystyle \mathbf {u} \cdot \nabla \mathbf {u} =(\nabla \times \mathbf {u} )\times \mathbf {u} +{\tfrac {1}{2}}\nabla \mathbf {u} ^{2},} where

400-446: A . {\displaystyle \rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} \right)=-\nabla p+\nabla \cdot \left\{\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]\right\}+\nabla [\zeta (\nabla \cdot \mathbf {u} )]+\rho \mathbf {a} .} in index notation,

500-520: A . {\displaystyle \rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} \right)=-\nabla p+\nabla \cdot \left\{\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]\right\}+\rho \mathbf {a} .} If the dynamic μ and bulk ζ {\displaystyle \zeta } viscosities are assumed to be uniform in space,

600-459: A . {\displaystyle {\frac {D\mathbf {u} }{Dt}}=-{\frac {1}{\rho }}\nabla p+\nu \,\nabla ^{2}\mathbf {u} +({\tfrac {1}{3}}\nu +\xi )\,\nabla (\nabla \cdot \mathbf {u} )+\mathbf {a} .} where D D t {\textstyle {\frac {\mathrm {D} }{\mathrm {D} t}}} is the material derivative . ν = μ ρ {\displaystyle \nu ={\frac {\mu }{\rho }}}

700-409: A . {\displaystyle {\frac {\partial }{\partial t}}(\rho \mathbf {u} )+\nabla \cdot \left(\rho \mathbf {u} \otimes \mathbf {u} +[p-\zeta (\nabla \cdot \mathbf {u} )]\mathbf {I} -\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]\right)=\rho \mathbf {a} .} Apart from its dependence of pressure and temperature,

800-578: A 2 ( x , y ) u x y + a 3 ( x , y ) u y x + a 4 ( x , y ) u y y + a 5 ( x , y ) u x + a 6 ( x , y ) u y + a 7 ( x , y ) u = f ( x , y ) {\displaystyle a_{1}(x,y)u_{xx}+a_{2}(x,y)u_{xy}+a_{3}(x,y)u_{yx}+a_{4}(x,y)u_{yy}+a_{5}(x,y)u_{x}+a_{6}(x,y)u_{y}+a_{7}(x,y)u=f(x,y)} where

900-530: A 3 ( u x , u y , u , x , y ) u y x + a 4 ( u x , u y , u , x , y ) u y y + f ( u x , u y , u , x , y ) = 0 {\displaystyle a_{1}(u_{x},u_{y},u,x,y)u_{xx}+a_{2}(u_{x},u_{y},u,x,y)u_{xy}+a_{3}(u_{x},u_{y},u,x,y)u_{yx}+a_{4}(u_{x},u_{y},u,x,y)u_{yy}+f(u_{x},u_{y},u,x,y)=0} Many of

1000-601: A i . {\displaystyle \rho \left({\frac {\partial u_{i}}{\partial t}}+u_{k}{\frac {\partial u_{i}}{\partial x_{k}}}\right)=-{\frac {\partial p}{\partial x_{i}}}+{\frac {\partial }{\partial x_{k}}}\left[\mu \left({\frac {\partial u_{i}}{\partial x_{k}}}+{\frac {\partial u_{k}}{\partial x_{i}}}-{\frac {2}{3}}\delta _{ik}{\frac {\partial u_{l}}{\partial x_{l}}}\right)\right]+{\frac {\partial }{\partial x_{i}}}\left(\zeta {\frac {\partial u_{l}}{\partial x_{l}}}\right)+\rho a_{i}.} The corresponding equation in conservation form can be obtained by considering that, given

1100-466: A i and f are functions of the independent variables x and y only. (Often the mixed-partial derivatives u xy and u yx will be equated, but this is not required for the discussion of linearity.) If the a i are constants (independent of x and y ) then the PDE is called linear with constant coefficients . If f is zero everywhere then the linear PDE is homogeneous , otherwise it

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1200-929: A Fourier series is appropriate, but an integral of solutions such as a Fourier integral is generally required for infinite domains. The solution for a point source for the heat equation given above is an example of the use of a Fourier integral. Often a PDE can be reduced to a simpler form with a known solution by a suitable change of variables . For example, the Black–Scholes equation ∂ V ∂ t + 1 2 σ 2 S 2 ∂ 2 V ∂ S 2 + r S ∂ V ∂ S − r V = 0 {\displaystyle {\frac {\partial V}{\partial t}}+{\tfrac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0}

1300-1082: A hypersurface S is given in the implicit form φ ( x 1 , x 2 , … , x n ) = 0 , {\displaystyle \varphi (x_{1},x_{2},\ldots ,x_{n})=0,} where φ has a non-zero gradient, then S is a characteristic surface for the operator L at a given point if the characteristic form vanishes: Q ( ∂ φ ∂ x 1 , … , ∂ φ ∂ x n ) = det [ ∑ ν = 1 n A ν ∂ φ ∂ x ν ] = 0. {\displaystyle Q\left({\frac {\partial \varphi }{\partial x_{1}}},\ldots ,{\frac {\partial \varphi }{\partial x_{n}}}\right)=\det \left[\sum _{\nu =1}^{n}A_{\nu }{\frac {\partial \varphi }{\partial x_{\nu }}}\right]=0.} The geometric interpretation of this condition

1400-408: A parabolic equation and therefore have better analytic properties, at the expense of having less mathematical structure (e.g. they are never completely integrable ). The Navier–Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents , water flow in a pipe and air flow around

1500-433: A quasilinear PDE the highest order derivatives likewise appear only as linear terms, but with coefficients possibly functions of the unknown and lower-order derivatives: a 1 ( u x , u y , u , x , y ) u x x + a 2 ( u x , u y , u , x , y ) u x y +

1600-637: A wing . The Navier–Stokes equations, in their full and simplified forms, help with the design of aircraft and cars, the study of blood flow , the design of power stations , the analysis of pollution , and many other problems. Coupled with Maxwell's equations , they can be used to model and study magnetohydrodynamics . The Navier–Stokes equations are also of great interest in a purely mathematical sense. Despite their wide range of practical uses, it has not yet been proven whether smooth solutions always exist in three dimensions—i.e., whether they are infinitely differentiable (or even just bounded) at all points in

1700-410: A flow with respect to space. While individual fluid particles indeed experience time-dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle. Remark: here, the deviatoric stress tensor is denoted τ {\textstyle {\boldsymbol {\tau }}} as it was in the general continuum equations and in

1800-572: A general second order semi-linear PDE in two variables is a 1 ( x , y ) u x x + a 2 ( x , y ) u x y + a 3 ( x , y ) u y x + a 4 ( x , y ) u y y + f ( u x , u y , u , x , y ) = 0 {\displaystyle a_{1}(x,y)u_{xx}+a_{2}(x,y)u_{xy}+a_{3}(x,y)u_{yx}+a_{4}(x,y)u_{yy}+f(u_{x},u_{y},u,x,y)=0} In

1900-484: A guide to appropriate initial- and boundary conditions and to the smoothness of the solutions. Assuming u xy = u yx , the general linear second-order PDE in two independent variables has the form A u x x + 2 B u x y + C u y y + ⋯ (lower order terms) = 0 , {\displaystyle Au_{xx}+2Bu_{xy}+Cu_{yy}+\cdots {\mbox{(lower order terms)}}=0,} where

2000-728: 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 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 . For instance, they are foundational in

2100-442: A more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups , be referred, to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact . A general approach to solving PDEs uses

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2200-460: A second-order PDE at a given point. However, the discriminant in a PDE is given by B − AC due to the convention of the xy term being 2 B rather than B ; formally, the discriminant (of the associated quadratic form) is (2 B ) − 4 AC = 4( B − AC ) , with the factor of 4 dropped for simplicity. If there are n independent variables x 1 , x 2 , …, x n , a general linear partial differential equation of second order has

2300-521: A solution of that PDE in the same function space. There are no generally applicable analytical methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as the Cauchy–Kowalevski theorem ) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of analysis ). Nevertheless, some techniques can be used for several types of equations. The h -principle

2400-521: A subclass of partial differential equations, corresponding to functions of a single variable. Stochastic partial differential equations and nonlocal equations are, as of 2020, particularly widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, include elliptic and parabolic partial differential equations, fluid mechanics , Boltzmann equations , and dispersive partial differential equations . A function u ( x , y , z ) of three variables

2500-532: A vector whose direction and magnitude are those of the velocity of the fluid at that point in space and at that moment in time. It is usually studied in three spatial dimensions and one time dimension, although two (spatial) dimensional and steady-state cases are often used as models, and higher-dimensional analogues are studied in both pure and applied mathematics. Once the velocity field is calculated, other quantities of interest such as pressure or temperature may be found using dynamical equations and relations. This

2600-525: Is ∇ ( ∇ ⋅ u ) {\textstyle \nabla \left(\nabla \cdot \mathbf {u} \right)} , one finally arrives to the compressible Navier–Stokes momentum equation: D u D t = − 1 ρ ∇ p + ν ∇ 2 u + ( 1 3 ν + ξ ) ∇ ( ∇ ⋅ u ) +

2700-444: Is inhomogeneous . (This is separate from asymptotic homogenization , which studies the effects of high-frequency oscillations in the coefficients upon solutions to PDEs.) Nearest to linear PDEs are semi-linear PDEs, where only the highest order derivatives appear as linear terms, with coefficients that are functions of the independent variables. The lower order derivatives and the unknown function may appear arbitrarily. For example,

2800-443: Is nonlinear , owing to the square roots and the squares. A linear PDE is one such that, if it is homogeneous, the sum of any two solutions is also a solution, and any constant multiple of any solution is also a solution. A partial differential equation is an equation that involves an unknown function of n ≥ 2 {\displaystyle n\geq 2} variables and (some of) its partial derivatives. That is, for

2900-593: Is " harmonic " or "a solution of the Laplace equation " if it satisfies the condition ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 = 0. {\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}=0.} Such functions were widely studied in

3000-429: Is an ordinary differential equation if in one variable – these are in turn easier to solve. This is possible for simple PDEs, which are called separable partial differential equations , and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to diagonal matrices – thinking of "the value for fixed x " as a coordinate, each coordinate can be understood separately. This generalizes to

3100-408: 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. If the data on S and the differential equation determine the normal derivative of u on S , then S is non-characteristic. If the data on S and the differential equation do not determine the normal derivative of u on S , then

PDE - Misplaced Pages Continue

3200-436: Is considered to be a correction to an equation with a known solution. Alternatives are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods . Many interesting problems in science and engineering are solved in this way using computers , sometimes high performance supercomputers . From 1870 Sophus Lie 's work put the theory of differential equations on

3300-399: Is different from what one normally sees in classical mechanics , where solutions are typically trajectories of position of a particle or deflection of a continuum . Studying velocity instead of position makes more sense for a fluid, although for visualization purposes one can compute various trajectories . In particular, the streamlines of a vector field, interpreted as flow velocity, are

3400-418: Is done by a Fourier transform ), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a homogeneous polynomial , here a quadratic form ) being most significant for the classification. Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant B − 4 AC , the same can be done for

3500-423: Is impossible to meaningfully formulate the results. That is, the domain of the unknown function must be regarded as part of the structure of the PDE itself. The following provides two classic examples of such existence and uniqueness theorems. Even though the two PDE in question are so similar, there is a striking difference in behavior: for the first PDE, one has the free prescription of a single function, while for

3600-547: Is not. It may be surprising that the two examples of harmonic functions are of such strikingly different form. 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. This is in striking contrast to the case of ordinary differential equations (ODEs) roughly similar to the Laplace equation, with the aim of many introductory textbooks being to find algorithms leading to general solution formulas. For

3700-530: Is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation like x − 3 x + 2 = 0 . However, it is usually impossible to write down explicit formulae 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

3800-748: Is proportional to the shear viscosity: σ ′ = τ = μ [ ∇ u + ( ∇ u ) T − 2 3 ( ∇ ⋅ u ) I ] {\displaystyle {\boldsymbol {\sigma }}'={\boldsymbol {\tau }}=\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]} Both bulk viscosity ζ {\textstyle \zeta } and dynamic viscosity μ {\textstyle \mu } need not be constant – in general, they depend on two thermodynamics variables if

3900-1140: Is reducible to the heat equation ∂ u ∂ τ = ∂ 2 u ∂ x 2 {\displaystyle {\frac {\partial u}{\partial \tau }}={\frac {\partial ^{2}u}{\partial x^{2}}}} by the change of variables V ( S , t ) = v ( x , τ ) , x = ln ⁡ ( S ) , τ = 1 2 σ 2 ( T − t ) , v ( x , τ ) = e − α x − β τ u ( x , τ ) . {\displaystyle {\begin{aligned}V(S,t)&=v(x,\tau ),\\[5px]x&=\ln \left(S\right),\\[5px]\tau &={\tfrac {1}{2}}\sigma ^{2}(T-t),\\[5px]v(x,\tau )&=e^{-\alpha x-\beta \tau }u(x,\tau ).\end{aligned}}} Inhomogeneous equations can often be solved (for constant coefficient PDEs, always be solved) by finding

4000-704: Is the identity tensor , and tr ⁡ ( ε ) {\textstyle \operatorname {tr} ({\boldsymbol {\varepsilon }})} is the trace of the rate-of-strain tensor. So this decomposition can be explicitly defined as: σ = − p I + λ ( ∇ ⋅ u ) I + μ ( ∇ u + ( ∇ u ) T ) . {\displaystyle {\boldsymbol {\sigma }}=-p\mathbf {I} +\lambda (\nabla \cdot \mathbf {u} )\mathbf {I} +\mu \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }\right).} Since

4100-415: Is the outer product of the flow velocity ( u {\displaystyle \mathbf {u} } ): u ⊗ u = u u T {\displaystyle \mathbf {u} \otimes \mathbf {u} =\mathbf {u} \mathbf {u} ^{\mathrm {T} }} The left side of the equation describes acceleration, and may be composed of time-dependent and convective components (also

PDE - Misplaced Pages Continue

4200-469: Is the interaction of two waves in phase being combined to result in a greater amplitude, for example sin x + sin x = 2 sin x . The same principle can be observed in PDEs where the solutions may be real or complex and additive. If u 1 and u 2 are solutions of linear PDE in some function space R , then u = c 1 u 1 + c 2 u 2 with any constants c 1 and c 2 are also

4300-399: Is the most powerful method to solve underdetermined equations. The Riquier–Janet theory is an effective method for obtaining information about many analytic overdetermined systems. The method of characteristics can be used in some very special cases to solve nonlinear partial differential equations. In some cases, a PDE can be solved via perturbation analysis in which the solution

4400-831: Is the partial derivative operator. When writing PDEs, it is common to denote partial derivatives using subscripts. For example: u x = ∂ u ∂ x , u x x = ∂ 2 u ∂ x 2 , u x y = ∂ 2 u ∂ y ∂ x = ∂ ∂ y ( ∂ u ∂ x ) . {\displaystyle u_{x}={\frac {\partial u}{\partial x}},\quad u_{xx}={\frac {\partial ^{2}u}{\partial x^{2}}},\quad u_{xy}={\frac {\partial ^{2}u}{\partial y\,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial x}}\right).} In

4500-784: Is the shear kinematic viscosity and ξ = ζ ρ {\displaystyle \xi ={\frac {\zeta }{\rho }}} is the bulk kinematic viscosity. The left-hand side changes in the conservation form of the Navier–Stokes momentum equation. By bringing the operator on the flow velocity on the left side, on also has: ( ∂ ∂ t + u ⋅ ∇ − ν ∇ 2 − ( 1 3 ν + ξ ) ∇ ( ∇ ⋅ ) ) u = − 1 ρ ∇ p +

4600-535: Is three: tr ⁡ ( I ) = 3. {\displaystyle \operatorname {tr} ({\boldsymbol {I}})=3.} the trace of the stress tensor in three dimensions becomes: tr ⁡ ( σ ) = − 3 p + ( 3 λ + 2 μ ) ∇ ⋅ u . {\displaystyle \operatorname {tr} ({\boldsymbol {\sigma }})=-3p+(3\lambda +2\mu )\nabla \cdot \mathbf {u} .} So by alternatively decomposing

4700-400: Is usually neglected most of the time (that is whenever we are not dealing with processes such as sound absorption and attenuation of shock waves, where second viscosity coefficient becomes important) by explicitly assuming ζ = 0 {\textstyle \zeta =0} . The assumption of setting ζ = 0 {\textstyle \zeta =0} is called as

4800-794: The k t h {\displaystyle k^{th}} -order partial differential equation is defined as F [ D k u , D k − 1 u , … , D u , u , x ] = 0 , {\displaystyle F[D^{k}u,D^{k-1}u,\dots ,Du,u,x]=0,} where F : R n k × R n k − 1 ⋯ × R n × R × U → R , {\displaystyle F:\mathbb {R} ^{n^{k}}\times \mathbb {R} ^{n^{k-1}}\dots \times \mathbb {R} ^{n}\times \mathbb {R} \times U\rightarrow \mathbb {R} ,} and D {\displaystyle D}

4900-591: The Cauchy stress tensor σ {\textstyle {\boldsymbol {\sigma }}} to be the sum of a viscosity term τ {\textstyle {\boldsymbol {\tau }}} (the deviatoric stress ) and a pressure term − p I {\textstyle -p\mathbf {I} } (volumetric stress), we arrive at: ρ D u D t = − ∇ p + ∇ ⋅ τ + ρ

5000-526: 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 classification of partial differential equations can be extended to systems of first-order equations, where the unknown u is now a vector with m components, and the coefficient matrices A ν are m by m matrices for ν = 1, 2, …, n . The partial differential equation takes

5100-899: The Stokes hypothesis . The validity of Stokes hypothesis can be demonstrated for monoatomic gas both experimentally and from the kinetic theory; for other gases and liquids, Stokes hypothesis is generally incorrect. With the Stokes hypothesis, the Navier–Stokes equations become ρ D u D t = ρ ( ∂ u ∂ t + ( u ⋅ ∇ ) u ) = − ∇ p + ∇ ⋅ { μ [ ∇ u + ( ∇ u ) T − 2 3 ( ∇ ⋅ u ) I ] } + ρ

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5200-955: The bulk viscosity ζ {\textstyle \zeta } , ζ ≡ λ + 2 3 μ , {\displaystyle \zeta \equiv \lambda +{\tfrac {2}{3}}\mu ,} we arrive to the linear constitutive equation in the form usually employed in thermal hydraulics : σ = − [ p − ζ ( ∇ ⋅ u ) ] I + μ [ ∇ u + ( ∇ u ) T − 2 3 ( ∇ ⋅ u ) I ] {\displaystyle {\boldsymbol {\sigma }}=-[p-\zeta (\nabla \cdot \mathbf {u} )]\mathbf {I} +\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]} which can also be arranged in

5300-417: The deviatoric stress tensor σ ′ {\displaystyle {\boldsymbol {\sigma }}'} is still coincident with the shear stress tensor τ {\displaystyle {\boldsymbol {\tau }}} (i.e. the deviatoric stress in a Newtonian fluid has no normal stress components), and it has a compressibility term in addition to the incompressible case, which

5400-423: The domain . This is called the Navier–Stokes existence and smoothness problem. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$ 1 million prize for a solution or a counterexample. The solution of the equations is a flow velocity . It is a vector field —to every point in a fluid, at any moment in a time interval, it gives

5500-466: The fundamental solution (the solution for a point source P ( D ) u = δ {\displaystyle P(D)u=\delta } ), then taking the convolution with the boundary conditions to get the solution. This is analogous in signal processing to understanding a filter by its impulse response . The superposition principle applies to any linear system, including linear systems of PDEs. A common visualization of this concept

5600-664: The incompressible flow section . The compressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor: σ ( ε ) = − p I + λ tr ⁡ ( ε ) I + 2 μ ε {\displaystyle {\boldsymbol {\sigma }}({\boldsymbol {\varepsilon }})=-p\mathbf {I} +\lambda \operatorname {tr} ({\boldsymbol {\varepsilon }})\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}} where I {\textstyle \mathbf {I} }

5700-445: The method of characteristics , and is also used in integral transforms . The characteristic surface in n = 2 - dimensional space is called a characteristic curve . In special cases, one can find characteristic curves on which the first-order PDE reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the method of characteristics . More generally, applying

5800-410: The trace of the rate-of-strain tensor in three dimensions is the divergence (i.e. rate of expansion) of the flow: tr ⁡ ( ε ) = ∇ ⋅ u . {\displaystyle \operatorname {tr} ({\boldsymbol {\varepsilon }})=\nabla \cdot \mathbf {u} .} Given this relation, and since the trace of the identity tensor in three dimensions

5900-925: The 19th century due to their relevance for classical mechanics , for example the equilibrium temperature distribution of a homogeneous solid is a harmonic function. If explicitly given a function, it is usually a matter of straightforward computation to check whether or not it is harmonic. For instance u ( x , y , z ) = 1 x 2 − 2 x + y 2 + z 2 + 1 {\displaystyle u(x,y,z)={\frac {1}{\sqrt {x^{2}-2x+y^{2}+z^{2}+1}}}} and u ( x , y , z ) = 2 x 2 − y 2 − z 2 {\displaystyle u(x,y,z)=2x^{2}-y^{2}-z^{2}} are both harmonic while u ( x , y , z ) = sin ⁡ ( x y ) + z {\displaystyle u(x,y,z)=\sin(xy)+z}

6000-512: The Laplace equation, as for a large number of partial differential equations, such solution formulas fail to exist. 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 equation ∂ 2 v ∂ x ∂ y = 0. {\displaystyle {\frac {\partial ^{2}v}{\partial x\partial y}}=0.} It can be directly checked that any function v of

6100-508: The Lyapunov method, are independent of small physical parameters as compared to the well known perturbation theory , thus giving these methods greater flexibility and solution generality. The three most widely used numerical methods to solve PDEs are the finite element method (FEM), finite volume methods (FVM) and finite difference methods (FDM), as well other kind of methods called meshfree methods , which were made to solve problems where

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6200-454: The PDE. Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines. The Adomian decomposition method , the Lyapunov artificial small parameter method, and his homotopy perturbation method are all special cases of the more general homotopy analysis method . These are series expansion methods, and except for

6300-737: The aforementioned methods are limited. The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version hp-FEM . Other hybrid versions of FEM and Meshfree methods include the generalized finite element method (GFEM), extended finite element method (XFEM), spectral finite element method (SFEM), meshfree finite element method , discontinuous Galerkin finite element method (DGFEM), element-free Galerkin method (EFGM), interpolating element-free Galerkin method (IEFGM), etc. Navier%E2%80%93Stokes equations The Navier–Stokes equations ( / n æ v ˈ j eɪ s t oʊ k s / nav- YAY STOHKS ) are partial differential equations which describe

6400-457: The assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing viscous flow . The difference between them and the closely related Euler equations is that Navier–Stokes equations take viscosity into account while the Euler equations model only inviscid flow . As a result, the Navier–Stokes are

6500-401: The boundary conditions, then it is the solution (this also applies to ODEs). We assume as an ansatz that the dependence of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter, and then see if this can be made to solve the problem. In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which

6600-494: The coefficients A , B , C ... may depend upon x and y . If A + B + C > 0 over a region of the xy -plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section: A x 2 + 2 B x y + C y 2 + ⋯ = 0. {\displaystyle Ax^{2}+2Bxy+Cy^{2}+\cdots =0.} More precisely, replacing ∂ x by X , and likewise for other variables (formally this

6700-431: The compressible case the pressure is no more proportional to the isotropic stress term, since there is the additional bulk viscosity term: p = − 1 3 tr ⁡ ( σ ) + ζ ( ∇ ⋅ u ) {\displaystyle p=-{\frac {1}{3}}\operatorname {tr} ({\boldsymbol {\sigma }})+\zeta (\nabla \cdot \mathbf {u} )} and

6800-622: The conservation form of the equations of motion. This is often written: ∂ ∂ t ( ρ u ) + ∇ ⋅ ( ρ u ⊗ u ) = − ∇ p + ∇ ⋅ τ + ρ a {\displaystyle {\frac {\partial }{\partial t}}(\rho \,\mathbf {u} )+\nabla \cdot (\rho \,\mathbf {u} \otimes \mathbf {u} )=-\nabla p+\nabla \cdot {\boldsymbol {\tau }}+\rho \,\mathbf {a} } where ⊗ {\textstyle \otimes }

6900-407: The deviatoric (shear) stress tensor in terms of viscosity and the fluid velocity gradient, and assuming constant viscosity, the above Cauchy equations will lead to the Navier–Stokes equations below. A significant feature of the Cauchy equation and consequently all other continuum equations (including Euler and Navier–Stokes) is the presence of convective acceleration: the effect of acceleration of

7000-705: The effect of the volume viscosity ζ {\textstyle \zeta } is that the mechanical pressure is not equivalent to the thermodynamic pressure : as demonstrated below. ∇ ⋅ ( ∇ ⋅ u ) I = ∇ ( ∇ ⋅ u ) , {\displaystyle \nabla \cdot (\nabla \cdot \mathbf {u} )\mathbf {I} =\nabla (\nabla \cdot \mathbf {u} ),} p ¯ ≡ p − ζ ∇ ⋅ u , {\displaystyle {\bar {p}}\equiv p-\zeta \,\nabla \cdot \mathbf {u} ,} However, this difference

7100-418: The effects of non-inertial coordinates if present). The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces (such as gravity). All non-relativistic balance equations, such as the Navier–Stokes equations, can be derived by beginning with the Cauchy equations and specifying the stress tensor through a constitutive relation . By expressing

7200-919: The equation can be written as ρ ( ∂ u i ∂ t + u k ∂ u i ∂ x k ) = − ∂ p ∂ x i + ∂ ∂ x k [ μ ( ∂ u i ∂ x k + ∂ u k ∂ x i − 2 3 δ i k ∂ u l ∂ x l ) ] + ∂ ∂ x i ( ζ ∂ u l ∂ x l ) + ρ

7300-478: The equations in convective form can be simplified further. By computing the divergence of the stress tensor, since the divergence of tensor ∇ u {\textstyle \nabla \mathbf {u} } is ∇ 2 u {\textstyle \nabla ^{2}\mathbf {u} } and the divergence of tensor ( ∇ u ) T {\textstyle \left(\nabla \mathbf {u} \right)^{\mathrm {T} }}

7400-888: The fluid contains a single chemical species, say for example, pressure and temperature. Any equation that makes explicit one of these transport coefficient in the conservation variables is called an equation of state . The most general of the Navier–Stokes equations become ρ D u D t = ρ ( ∂ u ∂ t + ( u ⋅ ∇ ) u ) = − ∇ p + ∇ ⋅ { μ [ ∇ u + ( ∇ u ) T − 2 3 ( ∇ ⋅ u ) I ] } + ∇ [ ζ ( ∇ ⋅ u ) ] + ρ

7500-403: The form L u = ∑ ν = 1 n A ν ∂ u ∂ x ν + B = 0 , {\displaystyle Lu=\sum _{\nu =1}^{n}A_{\nu }{\frac {\partial u}{\partial x_{\nu }}}+B=0,} where the coefficient matrices A ν and the vector B may depend upon x and u . If

7600-496: The form L u = ∑ i = 1 n ∑ j = 1 n a i , j ∂ 2 u ∂ x i ∂ x j + lower-order terms = 0. {\displaystyle Lu=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}\quad +{\text{lower-order terms}}=0.} The classification depends upon

7700-582: The form v ( x , y ) = f ( x ) + g ( y ) , for any single-variable functions f and g whatsoever, will satisfy this condition. This is far beyond the choices available in ODE solution formulas, which typically allow the free choice of some numbers. In the study of PDEs, one generally has the free choice of functions. The nature of this choice varies from PDE to PDE. To understand it for any given equation, existence and uniqueness theorems are usually important organizational principles. In many introductory textbooks,

7800-563: The fundamental PDEs in physics are quasilinear, such as the Einstein equations of general relativity and the Navier–Stokes equations describing fluid motion. A PDE without any linearity properties is called fully nonlinear , and possesses nonlinearities on one or more of the highest-order derivatives. An example is the Monge–Ampère equation , which arises in differential geometry . The elliptic/parabolic/hyperbolic classification provides

7900-609: The fundamental tool in the proof of the Poincaré conjecture from geometric topology . Partly due to this variety of sources, there is a wide spectrum of different types of partial differential equations, and methods have been developed for dealing with many of the individual equations which arise. As such, it is usually acknowledged that there is no "general theory" of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields. Ordinary differential equations can be viewed as

8000-548: The general situation that u is a function of n variables, then u i denotes the first partial derivative relative to the i -th input, u ij denotes the second partial derivative relative to the i -th and j -th inputs, and so on. The Greek letter Δ denotes the Laplace operator ; if u is a function of n variables, then Δ u = u 11 + u 22 + ⋯ + u n n . {\displaystyle \Delta u=u_{11}+u_{22}+\cdots +u_{nn}.} In

8100-1012: The mass continuity equation , the left side is equivalent to: ρ D u D t = ∂ ∂ t ( ρ u ) + ∇ ⋅ ( ρ u ⊗ u ) {\displaystyle \rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}={\frac {\partial }{\partial t}}(\rho \mathbf {u} )+\nabla \cdot (\rho \mathbf {u} \otimes \mathbf {u} )} To give finally: ∂ ∂ t ( ρ u ) + ∇ ⋅ ( ρ u ⊗ u + [ p − ζ ( ∇ ⋅ u ) ] I − μ [ ∇ u + ( ∇ u ) T − 2 3 ( ∇ ⋅ u ) I ] ) = ρ

8200-1574: The mass continuity equation, which represents the mass per unit volume of a homogenous fluid with respect to space and time (i.e., material derivative D D t {\displaystyle {\frac {\mathbf {D} }{\mathbf {Dt} }}} ) of any finite volume ( V ) to represent the change of velocity in fluid media: D m D t = ∭ V ( D ρ D t + ρ ( ∇ ⋅ u ) ) d V D ρ D t + ρ ( ∇ ⋅ u ) = ∂ ρ ∂ t + ( ∇ ρ ) ⋅ u + ρ ( ∇ ⋅ u ) = ∂ ρ ∂ t + ∇ ⋅ ( ρ u ) = 0 {\displaystyle {\begin{aligned}{\frac {\mathbf {D} m}{\mathbf {Dt} }}&={\iiint \limits _{V}}\left({{\frac {\mathbf {D} \rho }{\mathbf {Dt} }}+\rho (\nabla \cdot \mathbf {u} )}\right)dV\\{\frac {\mathbf {D} \rho }{\mathbf {Dt} }}+\rho (\nabla \cdot {\mathbf {u} })&={\frac {\partial \rho }{\partial t}}+({\nabla \rho })\cdot {\mathbf {u} }+{\rho }(\nabla \cdot \mathbf {u} )={\frac {\partial \rho }{\partial t}}+\nabla \cdot ({\rho \mathbf {u} })=0\end{aligned}}} where to arrive at

8300-446: The method to first-order PDEs in higher dimensions, one may find characteristic surfaces. An integral transform may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator. An important example of this is Fourier analysis , which diagonalizes the heat equation using the eigenbasis of sinusoidal waves. If the domain is finite or periodic, an infinite sum of solutions such as

8400-421: The modern scientific understanding of sound , heat , diffusion , electrostatics , electrodynamics , thermodynamics , fluid dynamics , elasticity , general relativity , and quantum mechanics ( Schrödinger equation , Pauli equation etc.). They also arise from many purely mathematical considerations, such as differential geometry and the calculus of variations ; among other notable applications, they are

8500-685: The motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes . They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes). The Navier–Stokes equations mathematically express momentum balance for Newtonian fluids and make use of conservation of mass . They are sometimes accompanied by an equation of state relating pressure , temperature and density . They arise from applying Isaac Newton's second law to fluid motion , together with

8600-567: The other usual form: σ = − p I + μ ( ∇ u + ( ∇ u ) T ) + ( ζ − 2 3 μ ) ( ∇ ⋅ u ) I . {\displaystyle {\boldsymbol {\sigma }}=-p\mathbf {I} +\mu \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }\right)+\left(\zeta -{\frac {2}{3}}\mu \right)(\nabla \cdot \mathbf {u} )\mathbf {I} .} Note that in

8700-697: The paths along which a massless fluid particle would travel. These paths are the integral curves whose derivative at each point is equal to the vector field, and they can represent visually the behavior of the vector field at a point in time. The Navier–Stokes momentum equation can be derived as a particular form of the Cauchy momentum equation , whose general convective form is: D u D t = 1 ρ ∇ ⋅ σ + f . {\displaystyle {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}={\frac {1}{\rho }}\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {f} .} By setting

8800-449: The physics literature, the Laplace operator is often denoted by ∇ ; in the mathematics literature, ∇ u may also denote the Hessian matrix of u . A PDE is called linear if it is linear in the unknown and its derivatives. For example, for a function u of x and y , a second order linear PDE is of the form a 1 ( x , y ) u x x +

8900-473: 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. To discuss such existence and uniqueness theorems, it is necessary to be precise about the domain of the "unknown function". Otherwise, speaking only in terms such as "a function of two variables", it

9000-442: 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

9100-404: The second PDE, one has the free prescription of two functions. Even more phenomena are possible. 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. In contrast to the earlier examples, this PDE

9200-521: The second viscosity coefficient also depends on the process, that is to say, the second viscosity coefficient is not just a material property. Example: in the case of a sound wave with a definitive frequency that alternatively compresses and expands a fluid element, the second viscosity coefficient depends on the frequency of the wave. This dependence is called the dispersion . In some cases, the second viscosity ζ {\textstyle \zeta } can be assumed to be constant in which case,

9300-503: The signature of the eigenvalues of the coefficient matrix a i , j . The theory of elliptic, parabolic, and hyperbolic equations have been studied for centuries, largely centered around or based upon the standard examples of the Laplace equation , the heat equation , and the wave equation . However, 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

9400-843: The stress tensor into isotropic and deviatoric parts, as usual in fluid dynamics: σ = − [ p − ( λ + 2 3 μ ) ( ∇ ⋅ u ) ] I + μ ( ∇ u + ( ∇ u ) T − 2 3 ( ∇ ⋅ u ) I ) {\displaystyle {\boldsymbol {\sigma }}=-\left[p-\left(\lambda +{\tfrac {2}{3}}\mu \right)\left(\nabla \cdot \mathbf {u} \right)\right]\mathbf {I} +\mu \left(\nabla \mathbf {u} +\left(\nabla \mathbf {u} \right)^{\mathrm {T} }-{\tfrac {2}{3}}\left(\nabla \cdot \mathbf {u} \right)\mathbf {I} \right)} Introducing

9500-409: The surface is characteristic , and the differential equation restricts the data on S : the differential equation is internal to S . Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a feature of solutions to differential equations: if one can find any solution that solves the equation and satisfies

9600-451: The symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions ( Lie theory ). 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

9700-467: The title PDE . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=PDE&oldid=1251540672 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Partial differential equation The function

9800-480: The traditional jet engine Philosophy [ edit ] Principle of double effect Places [ edit ] Punta del Este Other modes of abbreviation [ edit ] Polydichloric euthimal , fictional substance Pde (or Pde.), " Parade " (when serving as part of the proper name of a street or other way) Present Day English Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

9900-440: The unknown function u : U → R , {\displaystyle u:U\rightarrow \mathbb {R} ,} of variables x = ( x 1 , … , x n ) {\displaystyle x=(x_{1},\dots ,x_{n})} belonging to the open subset U {\displaystyle U} of R n {\displaystyle \mathbb {R} ^{n}} ,

10000-617: The vector ( ∇ × u ) × u {\textstyle (\nabla \times \mathbf {u} )\times \mathbf {u} } is known as the Lamb vector . For the special case of an incompressible flow , the pressure constrains the flow so that the volume of fluid elements is constant: isochoric flow resulting in a solenoidal velocity field with ∇ ⋅ u = 0 {\textstyle \nabla \cdot \mathbf {u} =0} . The incompressible momentum Navier–Stokes equation results from

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