*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*.

*Laplace Equation*

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 .