| 1 | # PDE Fundamentals v1 |
| 2 | |
| 3 | **Purpose:** concise reference for basic partial differential equations (PDEs) relevant to simulation (heat/wave/Poisson) and well-posed initial/boundary value problems. |
| 4 | |
| 5 | --- |
| 6 | |
| 7 | ## Core linear PDE classes |
| 8 | |
| 9 | - **Fact:** Many classical PDE models in physics and engineering fall into three main linear types: elliptic (e.g. Poisson), parabolic (e.g. heat), and hyperbolic (e.g. wave). |
| 10 | - **Heuristic:** when formulating or reading a PDE, first classify it (elliptic / parabolic / hyperbolic) — this strongly constrains appropriate boundary conditions, numerical schemes, and expectations about solution behaviour. |
| 11 | - **First adoption task:** for each new PDE problem in R&D, explicitly label its type in the problem spec (e.g. "2D heat (parabolic)", "Poisson (elliptic)", "1D wave (hyperbolic)"). |
| 12 | - **Success criterion:** no PDE is treated as a generic "equation" in design docs; type is always stated. |
| 13 | - **Confidence:** high |
| 14 | - *Source:* L.C. Evans — *Partial Differential Equations* (2nd ed., AMS, 2010) + Shearer & Levy — *Partial Differential Equations* (Princeton, 2015). |
| 15 | |
| 16 | --- |
| 17 | |
| 18 | ## Canonical examples |
| 19 | |
| 20 | - **Heat equation (parabolic):** |
| 21 | - Form: \( u_t = \alpha \Delta u + f(x,t) \) in a domain \(\Omega \subset \mathbb{R}^n\). |
| 22 | - Models diffusion / heat conduction; typical numerics use explicit or implicit time-stepping plus finite differences / finite elements in space. |
| 23 | - **Wave equation (hyperbolic):** |
| 24 | - Form: \( u_{tt} = c^2 \Delta u + f(x,t) \). |
| 25 | - Models vibrations / waves; energy conservation and finite propagation speed are key invariants. |
| 26 | - **Poisson / Laplace (elliptic):** |
| 27 | - Form: \( -\Delta u = f(x) \) (Poisson) or \( \Delta u = 0 \) (Laplace). |
| 28 | - Steady-state / potential problems; solutions are typically smooth inside the domain and controlled by boundary data. |
| 29 | |
| 30 | - **Heuristic:** for early R&D, standardise on these three as reference problems for validation and regression tests. |
| 31 | - **First adoption task:** define at least one canonical test case for each of heat, wave, and Poisson in 1D/2D (domain, IC/BC, analytic or high-quality reference solution). |
| 32 | - **Success criterion:** new numerical schemes and CA rules are always tested against at least one canonical problem before being used in more complex settings. |
| 33 | - **Confidence:** high |
| 34 | - *Source:* LeVeque — *Finite Difference Methods for Ordinary and Partial Differential Equations* (SIAM, 2007). |
| 35 | |
| 36 | --- |
| 37 | |
| 38 | ## Initial and boundary value problems |
| 39 | |
| 40 | - **Fact:** For parabolic and hyperbolic PDEs, one typically specifies an **initial condition** (IC) at \(t=0\) plus boundary conditions (BCs) on \(\partial \Omega\); for elliptic equations, only BCs on \(\partial \Omega\) are specified. |
| 41 | - **Heuristic:** always keep IC and BCs explicit and separate in problem specs; do not overload "source term" or coefficients to hide boundary effects. |
| 42 | - **First adoption task:** in the equation→CA IR, make IC and BC separate top-level fields, each with explicit type tags (Dirichlet / Neumann / mixed / periodic) and parameters. |
| 43 | - **Success criterion:** problem configs are unambiguous about which data belong to IC vs BC, and numerical code does not guess or infer them from context. |
| 44 | - **Confidence:** high |
| 45 | - *Source:* Shearer & Levy — *Partial Differential Equations: An Introduction to Theory and Applications* (Princeton, 2015). |
| 46 | |
| 47 | --- |
| 48 | |
| 49 | ## Boundary condition types |
| 50 | |
| 51 | - **Dirichlet BC:** prescribe the value of the solution on the boundary (e.g. fixed temperature). |
| 52 | - **Neumann BC:** prescribe the normal derivative (e.g. insulated/no-flux boundary in heat problems). |
| 53 | - **Robin / mixed BC:** linear combination of value and normal derivative. |
| 54 | - **Periodic BC:** identify opposite sides of the domain; often best for avoiding artificial reflections. |
| 55 | |
| 56 | - **Heuristic:** choose BC type based on the physical meaning first, then on numerical convenience; when in doubt, start with simpler BCs (Dirichlet/Neumann) for validation before moving to more realistic mixed/periodic cases. |
| 57 | - **First adoption task:** for each canonical test problem, document the BC type per side (e.g. left/right Dirichlet, top/bottom Neumann) and why it makes sense physically. |
| 58 | - **Success criterion:** BC choices can be explained in one sentence of physical reasoning; no "because it was easier to code" defaults in core tests. |
| 59 | - **Confidence:** medium |
| 60 | - *Source:* standard PDE texts (Evans; Shearer & Levy). |
| 61 | |
| 62 | --- |
| 63 | |
| 64 | ## Well-posedness (at a high level) |
| 65 | |
| 66 | - **Fact:** Well-posed problems (in the sense of Hadamard) require existence, uniqueness, and continuous dependence of the solution on input data; many pathological PDE/BC combinations violate one of these. |
| 67 | - **Heuristic:** for R&D prototypes, stick to well-known well-posed setups (classical heat/wave/Poisson test problems) and treat exotic boundary or domain configurations as *experiments*, not baselines. |
| 68 | - **First adoption task:** when defining a new problem type, check a trusted PDE text (Evans / Shearer & Levy) to see whether the chosen IC/BC/domain combination is standard and well-posed. |
| 69 | - **Success criterion:** no core regression tests rely on intentionally ill-posed PDE setups. |
| 70 | - **Confidence:** medium |
| 71 | - *Source:* Evans — *Partial Differential Equations* (2nd ed.). |
| 72 | |