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PDEs

Field: Differential Equations

Partial Differential Equations, involving functions of multiple independent variables and their partial derivatives.

Sequence of Expressions

Let Ω⊂Rn\Omega \subset \mathbb{R}^n be an open domain, and let u:Ω×[0,T]→Ru: \Omega \times [0, T] \to \mathbb{R} be the unknown scalar field. A general linear Partial Differential Equation (PDE) of order kk can be formally expressed as: ∑∣α∣≤kaα(x,t)∂∣α∣∂x1α1⋯∂xnαnu(x,t)+b(x,t)∂u∂t+c(x,t)u=f(x,t)\sum_{|\alpha| \le k} a_{\alpha}(x, t) \frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}} u(x, t) + b(x, t) \frac{\partial u}{\partial t} + c(x, t) u = f(x, t) where α=(α1,…,αn)\alpha = (\alpha_1, \dots, \alpha_n) is a multi-index, and aα,b,c,fa_{\alpha}, b, c, f are known coefficient functions. The study of such equations requires specifying initial and boundary conditions. Canonical examples include:\n\n1. **The Heat Equation (Parabolic Type):** Modeling diffusion processes, typically written as:\n∂u∂t=κ∇2u+g(x,t)\frac{\partial u}{\partial t} = \kappa \nabla^2 u + g(x, t) where κ>0\kappa > 0 is the thermal diffusivity and ∇2=∑i=1n∂2∂xi2\nabla^2 = \sum_{i=1}^n \frac{\partial^2}{\partial x_i^2}.\n\n2. **The Wave Equation (Hyperbolic Type):** Modeling wave propagation, given by:\n∂2u∂t2=c2∇2u+h(x,t)\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u + h(x, t) where cc is the wave speed.\n\n3. **The Laplace Equation (Elliptic Type):** Modeling steady-state phenomena (e.g., electrostatics), defined by the homogeneous equation:\n∇2u=0\nabla^2 u = 0
Definition

Definition

A partial differential equation is an equation that involves an unknown function of n≥2n\geq 2 variables and (some of) its partial derivatives. That is, for the unknown function u:U→R,u:U\rightarrow \mathbb {R} , of variables x=(x1,…,xn)x=(x_{1},\dots ,x_{n}) belonging to the open subset UU of Rn\mathbb {R} ^{n} , the kthk^{th} -order partial differential equation is defined as F[Dku,Dk−1u,…,Du,u,x]=0,F[D^{k}u,D^{k-1}u,\dots ,Du,u,x]=0, where F:Rnk×Rnk−1⋯×Rn×R×U→R,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 DD is the partial derivative operator. When writing PDEs, it is common to denote partial derivatives using subscripts. For example: ux=∂u∂x,uxx=∂2u∂x2,uxy=∂2u∂y ∂x=∂∂y(∂u∂x).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 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 Δu=u11+u22+⋯+unn.\Delta u=u_{11}+u_{22}+\cdots +u_{nn}. In the physics literature, the Laplace operator is often denoted by ∇^{2}; in the mathematics literature, ∇^{2}u may also denote the Hessian matrix of u. - ^Evans 1998, pp. 1–2.
Let Ω⊂Rn\Omega \subset \mathbb{R}^n be an open domain, and let u:Ω→Ru: \Omega \to \mathbb{R} be the unknown function. A Partial Differential Equation (PDE) is formally defined by an equation of the structure L(u)=f(x)\mathcal{L}(u) = f(x), where L\mathcal{L} is a linear differential operator of order kk, and f(x)f(x) is a known source function (or forcing term).\n\nFormally, the operator L\mathcal{L} is defined as:\nL(u)=∑∣α∣≤kaα(x)Dαu\mathcal{L}(u) = \sum_{|\alpha| \le k} a_{\alpha}(x) D^{\alpha} u \n\nHere:\n1. Dαu=∂∣α∣u∂x1α1⋯∂xnαnD^{\alpha} u = \frac{\partial^{|\alpha|} u}{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}} is the α\alpha-th partial derivative, where α=(α1,…,αn)\alpha = (\alpha_1, \dots, \alpha_n) is a multi-index with non-negative integer components αi\alpha_i. The order of the operator is k=max⁡α(∣α∣)k = \max_{\alpha} (|\alpha|) for which aα(x)≢0a_{\alpha}(x) \not\equiv 0.\n2. aα(x):Ω→Ra_{\alpha}(x): \Omega \to \mathbb{R} are the coefficient functions, assumed to be sufficiently smooth (e.g., aα∈Ck(Ω)a_{\alpha} \in C^k(\Omega)).\n3. The PDE is the equation L(u)=f(x)\mathcal{L}(u) = f(x), which seeks a solution u∈Ck(Ω)u \in C^k(\Omega) satisfying the equation pointwise on Ω\Omega. The classification of the PDE (e.g., elliptic, parabolic, hyperbolic) depends on the principal part of the operator L\mathcal{L} (i.e., the coefficients aαa_{\alpha} where ∣α∣=k|\alpha|=k).