One-dimensional Heat Equation
Consider the classical example of heat equation describing the temperature distribution in a thin long rod with thermally insulated sidewalls
In this case, we disregard temperature variations across the rod and assume that the temperature T is a function of the coordinate x and time t . The solution domain consists of the space interval [0, x] and the time interval, which can be finite [t(0), t(end)] or extended to infinity [t(0),∞). Different kinds of boundary conditions are possible. The situation when the ends of the rod are kept at constant temperature corresponds to the Dirichlet boundary conditions
Constant heat flux at the ends is described by the Neumann boundary conditions:
Periodic boundary conditions are also possible:
Initial temperature distribution u(x) is used for the initial conditions:
The complete PDE problem is of marching type and includes the PDE , the computational domain, one boundary condition, such as on each end, and the initial condition.
The Laplace equation can be obtained as an equation that describes a steady-state temperature distribution in a domain . For example, let us assume that we consider a heat transfer problem in a body with fixed boundary temperature or fixed boundary heat flux and are not interested in transients. We only want to know the final equilibrium distribution of temperature. Setting the time-derivative of temperature to zero transforms.
If internal heat sources are present within space, their intensity being defined by the function f (x), the final steady-state temperature distribution is a solution of the Poisson equation
Another situation described by the Laplace equation is the irrotational flow, in which velocity is a gradient of a scalar potential V = ∇φ(x). If the fluid is incompressible, the continuity equation becomes
It should be stressed that, in this case, the Laplace equation does not imply the steady-state character of the process. An important example of a similar situation is the behavior of pressure in incompressible flows. The pressure field satisfies a Poisson equation
with a time-dependent right-hand side.
One-dimensional Wave Equation
The displacement from the line y = 0 is defined as y = u(x, t ) . The solution domain includes the space interval [0, L] and the time interval [t(0), t(end)]. The boundary conditions at x = 0 and x = L can be,
For example, those of a fixed end
or a freely moving end
The initial conditions must include both the shape and velocity of the string at t = t(0):
The PDE problem is of transient type. A correct formulation includes the equation , the solution domain, one boundary condition at each boundary, and the two initial conditions .