Math 2.0

Partial differential equations Definition

Order of a PDE

Builds on Partial differential equationMulti-index notationOrder of a differential equation

Definition

Let F(Dku,…,Du,u,x)=0F\left(D^k u, \dots, D u, u, x\right) = 0 be a partial differential equation in an open set Ω⊆ℝn\Omega \subseteq \R^n. Here DjuD^j u is the collection of the partial derivatives ∂αu\partial^\alpha u with ∣α∣=j\abs{\alpha} = j, in multi-index notation. Its order is the largest integer mm such that FF actually depends on some derivative ∂αu\partial^\alpha u with ∣α∣=m\abs{\alpha} = m. For a system of equations, the order is the largest order of a derivative that occurs in some equation of the system.

Remarks

Example: the minimal surface equation in two variables involves ∂x2u\partial_x^2 u, ∂x∂yu\partial_x \partial_y u and ∂y2u\partial_y^2 u but no third derivatives, so it has order 22. Non-example of a misreading: (∂xu)2+(∂yu)2=1(\partial_x u)^2 + (\partial_y u)^2 = 1 has order 11, not 22, because the square is a power of a first derivative and not a second derivative. This is the same notion as the order of a differential equation, applied to equations in several variables. A mixed derivative ∂αu\partial^\alpha u counts with its total order ∣α∣=α1+⋯+αn\abs{\alpha} = \alpha_1 + \dots + \alpha_n.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptOrder of a differential equation, concept, no evidence levelOrder of a differenti…conceptPartial differential equation, concept, no evidence levelPartial differential …conceptFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, definition, no evidence levelPartial derivativedefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionMulti-index notation, concept, no evidence levelMulti-index notationconceptOrder of a PDE, definition, no evidence levelOrder of a PDEdefinition
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  1. Euclidean space object
  2. Function concept
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  8. Higher partial derivatives definition
  9. Multi-index notation concept
  10. Order of a PDE definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Partial_differential_equation
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