The first boundary condition is a thermodynamic term. In thermodynamics, the first boundary condition is expressed as: "When a large flat plate is treated as a one-dimensional problem, the temperature on one side of the flat plate remains unchanged."

The boundary temperature of a semi-infinite object in the direction of heat conduction must be of the first kind. The mathematical description is: t (x, 0) = f (x); T(0, t)=Ts, and the first boundary is the distribution of variables on the given boundary. The second boundary is the gradient value of the variable found on the given boundary. The third boundary is the functional relationship between the variable to be solved and the gradient value.

In mathematical and physical equations and special functions, the boundary conditions in solving definite solutions are divided into the first, second and third categories.

For example, in the problem of string vibration, there are usually three kinds of constraints on its endpoint (represented by x=a): one is the fixed end, that is, the endpoint of the string is always fixed during the vibration process, and the corresponding boundary condition is u (a, t)=0.

Second, the free end, that is, the chord, is free from the external force in the displacement direction at this end point, so that the tension in the displacement direction at this end point is zero, and the boundary condition is ux(a, t)=0.

Third, the elastic support end, that is, the string is supported by the elastic body at this end. Let the original position of the elastic support be u=0, then ux=a represents the strain of the elastic support. According to Hooke's law, the tension of the chord in the displacement direction at x=a can be obtained as (au/ax+ou)x=a=0.