Classification of Field Equations

• The wave equations (3.6) and (3.8) are special, and very, very simple, examples of a broader class of equations, which are   called the hyperbolic equations. They are called hyperbolic because the equation:
  
  x² - t² = a²
  
  
  describes a hyperbola. Hyperbolic equations almost always have some sort of waves as their solutions, but in general those waves may have very complex and nonlinear characteristics.
• In absence of charges and currents a static, i.e., time independent solutions to wave equations (3.6) and (3.8) satisfy:
  
   

  $$\nabla^2\boldsymbol{E}$$ =

  $$\nabla^2\boldsymbol{B}$$ = (3.14)

  
  
  This is called the Laplace equation   and is one of the simplest field equations.
• The Laplace equation describes many other systems, for example heat transfer.  
  • Heat always flows from a hotter to a cooler point, i.e., against temperature gradient
    

    $$\boldsymbol{Q} = -h \boldsymbol{\nabla} T$$ (3.15)

    
    
    where Q is the heat flux and T is the temperature field.
  • The divergence of Q, i.e., $\boldsymbol{\nabla}\cdot\boldsymbol{Q}$ describes production of heat in space, as $\boldsymbol{\nabla}\cdot\boldsymbol{E}$ describes production of electric field by charge density.
  • If heat is brought to a given point, then temperature at that point must increase so that
    

    $$\frac{\partial c_p T}{\partial t} = - \boldsymbol{\nabla}\cdot\boldsymbol{Q} = h \nabla^2 T$$ (3.16)

    
    
    (recall that convergence of field lines implies negative divergence $\boldsymbol{\nabla}\cdot\boldsymbol{Q}$) Assuming that c_p is a constant (we have already assumed this about h) we get:
    

    $$\frac{\partial T}{\partial t} - \kappa\nabla^2 T = 0$$ (3.17)

    
    
    where $\kappa = h/c_p$.
  • This is not a wave equation. This is a diffusion equation.  
  • A heat diffusion equation is a very simple example of a broader class of equations, which are called parabolic equations.   They are called parabolic because the equation:
    
    x² - t = a
    
    
    describes a parabola.
  • A static, i.e., time independent solution of the diffusion equation must satisfy the Laplace equation:
    

    $$\nabla^2 T = 0$$ (3.18)

    
    

• The Laplace equation is a very simple example of a broader class of equations, which are called elliptic equations.   They are called elliptic because
  
  x² + y² = a²
  
  
  describes a circle, which is a special case of an ellipse.
• Real life equations usually have both hyperbolic and diffusive terms. That is, their solutions have waves, but those waves eventually diffuse.
• Real life problems are usually described by many coupled equations, with various characteristics - the more realistic a model, the more complex its behaviour.
• In general:
  • Elliptic equations   describe static situations, e.g., static electromagnetic fields, static temperature distribution, static distribution of tensions in a geode.
  • Parabolic equations   describe diffusive systems, e.g., diffusion of heat, diffusion of impurities in semiconductors, diffusion of electrons across a pn-junction.
  • Hyperbolic equations   describe highly dynamic, vibrating systems, e.g., propagation of sound, propagation of electromagnetic waves, propagation of tectonic waves.
• Schrödinger equation   looks like a parabolic equation at first glance, but there is an i in front of $\partial/\partial t$. That i is responsible for the presence of waves (dispersive for unbound states).
• Elliptic, parabolic, and hyperbolic equations are specific cases of partial differential equations. Supercomputers are machines designed specifically for finding numerical solutions to partial differential equations. It is seldom necessary to use supercomputers for anything else. Some examples of non-PDE problems that are run on supercomputers:
  • particle simulations, but   these are usually coupled to PDEs in some ways;
  • neural   networks;
  • data   mining;
  • VR   visualisation - used for medical work sometimes.