Partial Differential Equation


Partial Differential Equations (PDE) are another mathematical language required for expressing multiphysics in addition to tensors. A general form of a second-order PDE for the function $u\left(x_{1} ,x_{2} ,\cdot \cdot \cdot ,x_{n} \right)$ is \[F\left(\frac{\partial ^{2} u}{\partial x_{1} \partial x_{1} } ,...,\frac{\partial ^{2} u}{\partial x_{1} \partial x_{n} } ,...,\frac{\partial ^{2} u}{\partial x_{n} \partial x_{n} } ,\frac{\partial u}{\partial x_{1} } ,...,\frac{\partial u}{\partial x_{n} } ,x_{1} ,...x_{n} \right)=0,\] where $x_{i} $'s are general coordinates including both spatial and temporal coordinates here. It is noted that $x_{i} $'s are general coordinates in the context of mathematics but are spatial coordinates in the context of physics. Assuming $\frac{\partial u^{2} }{\partial x\partial y} =\frac{\partial u^{2} }{\partial y\partial x} $, the general form of a second-order PDE with two independent variables are as follows \[Au_{xx} +2Bu_{xy} +Cu_{yy} +...{\rm (lower\; order\; terms)}=0,\] where the coefficients A, B and C may depend upon x and y. If $A^{2} +B^{2} +C^{2} >0$ over a region of the xy plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section by replacing $\partial x$ with x and likewise for other variables: \[Ax^{2} +2Bxy+Cy^{2} +...=0.\] Similar to the way of classifying conic sections and quadratic forms into parabolic, hyperbolic, and elliptic functions based on the discriminant $B^{2} -AC$, we can category a second-order PDE into the three types using the discriminant as well.

Elliptic PDEs:

$B^{2} -AC <0$

e.g. $\frac{\partial ^{2} u}{\partial x^{2} } +\frac{\partial ^{2} u}{\partial y^{2} } =0$, in which A=1, B=0, C=1, and $B^{2} -AC$ =-1

Parabolic PDEs:

$B^{2} -AC=0$,

e.g., $\frac{\partial u}{\partial t} -\frac{\partial ^{2} u}{\partial x^{2} } =0$, in which A=1, B=0, C=0, and $B^{2} -AC$ =0

Hyperbolic PDEs:

$B^{2} -AC>0$

e.g., $\frac{\partial ^{2} u}{\partial t^{2} } -\frac{\partial ^{2} u}{\partial x^{2} } =0$, in which A=1, B=0, C=-1, and $B^{2} -AC$ =1

The above classification method may not be directly used when there are more than two independent variables. In this case, we first need to reduce the quadratic form of the second-order PDE, i.e., \[\sum a_{ij} \frac{\partial ^{2} u}{\partial x_{i} \partial x_{j} } +{\rm lower\; order\; terms}=0\] into the canonical form, \[\sum c_{i} \frac{\partial ^{2} u}{\partial x_{i}^{2} } +{\rm lower\; order\; terms}=0\] by means of an appropriate linear nondegenerate transformation (Polyanin, 1997). The following criterion can be used:

  • The PDE is elliptic if all the $c_{i} $'s are nonzero and have the same sign.
  • The PDE is parabolic if at least one of the $c_{i} $'s is zero.
  • The PDE is hyperbolic if all the $c_{i} $'s are nonzero and have different signs.
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    Common PDEs in Engineering Applications

    Linear Equations

    Laplace's equations: $\Delta u=0$. This equation is widely used to describe the electric, gravitational, and fluid potentials in the equilibrium state.

    Helmholtz's equation (involves eigenvalues):$-\Delta u=\lambda u$. This equation a time-independent form of the wave equation, resulting from applying the technique of separation of variables to reduce the complexity of the analysis. It is widely used for applications such as electromagnetic radiation, seismology, and acoustics.

    First-order linear transport equation: $u_{t} +{\rm c}\nabla u=0$. This is the general time-dependent transport equation, e.g. for mass, momentum, and energy, with a diffusive term.

    Heat or diffusion equation: $u_{t} -\Delta u=0$. This is a simplified version of the above linear transport equation.

    Schrodinger's equation: $iu_{t} +\Delta u\; =0$. This is the time-dependent version of the Schrodinger's equation for quantum mechanical systems.

    Wave equation: $u_{tt} -{\rm c}^{2} \Delta u=0$. The wave equation can be used to describe both mechanical waves and electromagnetic waves.

    Telegraph equation:$u_{tt} +du_{t} -u_{xx} =0$. The telegraph equation is proposed to formulate changes of the voltage and current on an electrical transmission line with distance and time.


    Nonlinear Equations

    Eikonal equation: $\left|\nabla u\right|\cdot f\left(x\right)=1$. This equation is proposed to solve problems of wave propagation, when the wave equation is approximated using the WKB (Wentzel-Kramers-Brillouin) theory. Derivable from Maxwell's equations, the equation provides a link between physical (wave) optics and geometric (ray) optics.

    Nonlinear Poisson equation:$-\nabla u=f\left(u\right)$. This equation is similar to the Laplace equation but with a source term that relies on the dependent variable $u$. It turns into the linear Poisson equation if the function $f$ is not dependent on $u$.

    Burgers' equation:$u_{t} +uu_{x} =0$. This equation provides a mathematical description for some very special problems in fluid dynamics.

    Minimal surface equation: $\nabla \cdot [\frac{\nabla u}{\left(1+|\nabla u|^{2}\right)^{1/2}}]=0$. This equation is the key equation in the minimal surface theory. The theory is used to solve the variational problem of finding the surface of least area stretched across a given closed contour.

    Monge-Ampére equation: $\det \left(\nabla \nabla u\right)=f$. The Monge-Ampére equation arises in several problems in Riemannian geometry and conformal geometry. One of the simplest of applications is the prescribed Gauss curvature.

    Korteweg-deVries equation (KdV): $u_{t} +f\left(u\right)u_{x} +u_{xxx} =0$. This equation presents a mathematical model of waves on shallow water surfaces.

    Reaction-diffusion equation: $u_{t} -\Delta u=f\left(u\right)$. This is a transport equation with a diffusive term and nonlinear source term.


    System of Partial Differential Equations

    Evolution equation of linear elasticity: $u_{tt} -\mu \Delta u-\left(\lambda +\mu \right)\nabla \left(\nabla \cdot u\right)=0$. This the governing equation of the linear stress-strain problems.

    System of conservation laws: $u_{t} +\nabla \cdot F\left(u\right)=0$. This is the general form the conservation equation with multiple scalar quantities of the dependent variables.

    Maxwell's equations in vacuum: \[\nabla \times E=-B_{t} \] \[\nabla \times B=\mu _{0} \varepsilon _{0} E_{t} \] \[\nabla \cdot B=0\] \[\nabla \cdot E=0\] Maxwell's equation for describing electromagnetic fields in vacuum.

    Reaction-diffusion system: \[v_{t} -\Delta v=f\left(v\right)\] \[u_{t} +u\cdot \nabla u=-\nabla p\] \[\nabla \cdot u=0\] The equation systems consider both the fluid movement and the solute balance involving both transient transport and chemical reactions in the fluid.

    Euler's equations for incompressible and inviscid fluid: \[u_{t} +u\cdot \nabla u=-\nabla p\] \[\nabla \cdot u=0\] A simplified version of the Navier-Stokes equations for impressible and inviscid fluids.

    Navier-Stokes equations for incompressible viscous fluid: \[u_{t} +u\cdot \nabla u=-\nabla p+\Delta u\] \[\nabla \cdot u=0\] The Navier-Stokes equations for incompressible viscous fluids.


    Generic PDE

    It is interesting to mention that the above equations can be summarized into a generic form. For example, the generic PDE used by COMSOL is a good generalization of common PDEs: \[e_{\alpha } \frac{\partial ^{2} u}{\partial t^{2} } +d_{a} \frac{\partial u}{\partial t} -\frac{\partial }{\partial x_{i} } \left(c\frac{\partial u}{\partial x_{i} } +\alpha u-\gamma \right)+\beta \frac{\partial u}{\partial x_{i} } +au=0.\]

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