https://www.chenk.top/en/tags/finite-difference/Recent content in Finite Difference on Chen Kai BlogHugoenSun, 21 Jan 2024 09:00:00 +0000https://www.chenk.top/en/ode/13-pde-introduction/Sun, 21 Jan 2024 09:00:00 +0000https://www.chenk.top/en/ode/13-pde-introduction/<p><strong>Once a quantity depends on more than one variable, the ODE world splinters into a vastly richer one: partial differential equations.</strong> Heat in a metal rod is a function of position <em>and</em> time; a vibrating string moves in space <em>and</em> time; a steady electrostatic potential lives in three spatial dimensions. ODE techniques become tools, not solutions — separation of variables turns one PDE into a <em>family</em> of ODEs, the eigenvalues of those ODEs become the spectrum of the operator, and superposition stitches everything back together.</p>https://www.chenk.top/en/ode/12-boundary-value-problems/Thu, 04 Jan 2024 09:00:00 +0000https://www.chenk.top/en/ode/12-boundary-value-problems/<p>An initial value problem hands you a starting state and asks you to march forward. A boundary value problem (BVP) hands you partial information at two different points and asks you to find a path that fits both. The change is small in wording, large in consequence: BVPs can have a unique solution, no solution at all, or infinitely many. They demand a fundamentally different toolkit — one that is iterative, global, and intimately connected to linear algebra.</p>