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Introduction and examples

Basic concepts of PDEs and classic examples like the Heat, Wave, and Laplace equations.
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The statement of the theorem

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