In mathematical terms, a partial differential equation (PDE) is any equation involving a function of more than one independent variable and at least one partial derivative of that function. The order of a PDE is the order of the highest order derivative that appears in the PDE. The principal part of a PDE is the collection of terms in the PDE containing derivatives of order equal to the order of the PDE. The following example illustrates these definitions and introduces the two most common notations for expressing partial derivatives.

If is a function of the two independent variables and , then

is a PDE of first order whose principal part is

Using subscript notation a more compact way to express this PDE is

in which case we would say that the principal part is .

A PDE in is classified as linear if all of the terms involving and any of its derivatives can be expressed as a linear combination in which the coefficients of the -terms are independent of . In a linear PDE, the coefficients can depend at most on the independent variables.

If is a function of the two independent variables and , then

is a linear (constant coefficient) PDE.

The PDE

is a linear (variable coefficient) PDE.

The PDE

is nonlinear.

The distinction between linear and nonlinear PDEs is extremely important in computational science. Many linear PDE problems can be solved exactly using techniques such as separation of variables, superposition, Fourier series, Laplace transform and Fourier transform. Exact solutions are valuable in a computational setting because they can be used to assist the computational scientist in the often difficult exercise of code validation. Generally, nonlinear PDEs do not yield to analytical solution approaches. Since most leading edge work in computational science involves nonlinear PDEs, a great deal of effort is directed toward obtaining numerical solutions. Whenever possible, computational scientists draw from the field of linear PDEs for guidance and insight in developing numerical methods for the more difficult nonlinear PDEs.