Some popular partial differential equations (PDEs)

Single PDEs #

Linear equations #

1. Laplace’s equation

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

2. Helmholtz’s (or eigenvalue) equation

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

3. Linear transport equation

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

4. Liouville’s equation

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

5. Heat (or diffusion) equation

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

6. Schrödinger’s equation

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

7. Kolmogorov’s equation

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

8. Fokker–Planck equation

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

9. Wave equation

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

10. Klein–Gordon equation

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

11. Telegraph equation

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

12. General wave equation

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

13. Airy’s equation

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

14. Beam equation

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

Nonlinear equations #

1. Eikonal equation

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

2. Nonlinear Poisson equation

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

3. $p$-Laplacian equation

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

4. Minimal surface equation

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

5. Monge–Ampère equation

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

6. Hamilton–Jacobi equation

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

7. Scalar conservation law

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

8. Inviscid Burgers’ equation

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

9. Scalar reaction-diffusion equation

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

10. Porous medium equation

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

11. Nonlinear wave equation

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

12. Korteweg–deVries (KdV) equation

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

13. Nonlinear Schrödinger equation

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

Systems of PDEs #

Linear systems #

1. Equilibrium equations of linear elasticity

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

2. Evolution equations of linear elasticity

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

3. 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 #

1. System of conservation laws

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

2. Reaction-diffusion system

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

3. Euler’s equations for incompressible, inviscid flow

$$ \begin{equation} \begin{cases} u_t + u \cdot Du = -Dp \ \operatorname{div} u = 0. \end{cases} \end{equation} $$

4. Navier–Stokes equations for incompressible, viscous flow