Collected Lectures on Partial Differential Equations (PDE)

Mon, 07 Jul 2025 00:00:00 +0000

📝 Notes on Partial Differential Equations - John K. Hunter (University of California at Davis)
📝 Partial Differential Equations: Lecture Notes - Erich Miersemann (Leipzig University)
📝 Linear Methods of Applied Mathematics - E. Harrell, J. Herod (Georgia Tech)

Some popular partial differential equations (PDEs)

Thu, 27 Jun 2024 23:14:15 +0800

Single PDEs #

Linear equations #

Laplace’s equation

$$ \begin{equation} \Delta u = \sum_{i=1}^{n} u_{x_i x_i} = 0. \end{equation} $$

Helmholtz’s (or eigenvalue) equation

$$ \begin{equation} -\Delta u = \lambda u. \end{equation} $$

Linear transport equation

$$ \begin{equation} u_t + \sum_{i=1}^{n} b^i u_{x_i} = 0. \end{equation} $$

Liouville’s equation

$$ \begin{equation} u_t + \sum_{i=1}^{n} (b^i u)_{x_i} = 0. \end{equation} $$

Heat (or diffusion) equation

$$ \begin{equation} u_t - \Delta u = 0. \end{equation} $$

Schrödinger’s equation

$$ \begin{equation} i u_t + \Delta u = 0. \end{equation} $$

Kolmogorov’s equation

$$ \begin{equation} u_t - \sum_{i,j=1}^{n} a^{ij} u_{x_i x_j} + \sum_{i=1}^{n} b^i u_{x_i} = 0. \end{equation} $$

Fokker–Planck equation

$$ \begin{equation} u _t - \sum _{i,j = 1}^{n} (a^{ij} u) _{x_i x_j} - \sum _{i=1}^{n} (b^i u) _{x_i} = 0. \end{equation} $$

Wave equation

$$ \begin{equation} u_{tt} - \Delta k = 0. \end{equation} $$

Klein–Gordon equation

$$ \begin{equation} u_{tt} - \Delta u + m^2 u = 0. \end{equation} $$

Telegraph equation

$$ \begin{equation} u_{tt} + 2\delta u_t - u_{xx} = 0. \end{equation} $$

General wave equation

$$ \begin{equation} u_t - \sum_{i,j=1}^{n} a^{ij} u_{x_i x_j} + \sum_{i=1}^{n} b^i u_{x_i} = 0. \end{equation} $$

Airy’s equation

$$ \begin{equation} u_t + u_{xxx} = 0. \end{equation} $$

Beam equation

$$ \begin{equation} u_t + u_{xxxx} = 0. \end{equation} $$

Nonlinear equations #

Eikonal equation

$$ \begin{equation} |Du| = 1. \end{equation} $$

Nonlinear Poisson equation

$$ \begin{equation} -\Delta u = f(u). \end{equation} $$

$p$-Laplacian equation

$$ \begin{equation} \operatorname{div}(|Du|^{p-2} Du) = 0. \end{equation} $$

Minimal surface equation

$$ \begin{equation} \operatorname{div} \left( \frac{Du}{\sqrt{1 + |Du|^2}} \right) = 0. \end{equation} $$

Monge–Ampère equation

$$ \begin{equation} \det(D^2 u) = f. \end{equation} $$

Hamilton–Jacobi equation

$$ \begin{equation} u_t + H(Du, x) = 0. \end{equation} $$

Scalar conservation law

$$ \begin{equation} u_t + \operatorname{div} F(u) = 0. \end{equation} $$

Inviscid Burgers’ equation

$$ \begin{equation} u_t + u u_x = 0. \end{equation} $$

Scalar reaction-diffusion equation

$$ \begin{equation} u_t - \Delta u = f(u). \end{equation} $$

Porous medium equation

$$ \begin{equation} u_t - \Delta(u^m) = 0. \end{equation} $$

Nonlinear wave equation

$$ \begin{equation} u_{tt} - \Delta u + f(u) = 0. \end{equation} $$

Korteweg–deVries (KdV) equation

$$ \begin{equation} u_t + u u_x + u_{xxx} = 0. \end{equation} $$

Nonlinear Schrödinger equation

$$ \begin{equation} i u_t + \Delta u = f(|u|^2) u. \end{equation} $$

Systems of PDEs #

Linear systems #

Equilibrium equations of linear elasticity

$$ \begin{equation} \mu \Delta u + (\lambda + \mu) D(\operatorname{div} u) = 0. \end{equation} $$

Evolution equations of linear elasticity

$$ \begin{equation} u_{tt} - \mu \Delta u - (\lambda + \mu) D(\operatorname{div} u) = 0. \end{equation} $$

Maxwell’s equations

$$ \begin{equation} \begin{cases} E_t = \operatorname{curl} B \\ B_t = -\operatorname{curl} E \\ \operatorname{div} B = \operatorname{div} E = 0. \end{cases} \end{equation} $$

Nonlinear systems #

System of conservation laws