Partial Differential Equation

partial differential equation, in mathematics, equation relating a function of several variables to its partial derivatives. A partial derivative of a function of several variables expresses how fast the function changes when one of its variables is changed, the others being held constant (compare ordinary differential equation). The partial derivative of a function is again a function, and, if f(x, y) denotes the original function of the variables x and y, the partial derivative with respect to x—i.e., when only x is allowed to vary—is typically written as f_x(x, y) or ∂f/∂x. The operation of finding a partial derivative can be applied to a function that is itself a partial derivative of another function to get what is called a second-order partial derivative. For example, taking the partial derivative of f_x(x, y) with respect to y produces a new function f_xy(x, y), or ∂²f/∂y∂x. The order and degree of partial differential equations are defined the same as for ordinary differential equations.

In general, partial differential equations are difficult to solve, but techniques have been developed for simpler classes of equations called linear, and for classes known loosely as “almost” linear, in which all derivatives of an order higher than one occur to the first power and their coefficients involve only the independent variables.

Many physically important partial differential equations are second-order and linear. For example:

• u_xx + u_yy = 0 (two-dimensional Laplace equation)
• u_xx = u_t (one-dimensional heat equation)
• u_xx − u_yy = 0 (one-dimensional wave equation)

The behaviour of such an equation depends heavily on the coefficients a, b, and c of au_xx + bu_xy + cu_yy. They are called elliptic, parabolic, or hyperbolic equations according as b² − 4ac < 0, b² − 4ac = 0, or b² − 4ac > 0, respectively. Thus, the Laplace equation is elliptic, the heat equation is parabolic, and the wave equation is hyperbolic.