Random Integral Equations (Enhanced Edition‪)‬

Mathematical equations play a central role in all branches of the applied
mathematical sciences. In the formulation of an equation, or a system of
equations, used to represent a given physical system or phenomenon, coefficients
and/or parameters are often introduced which have some definite
physical interpretation. For example, in the theory of diffusion (or heat
conduction) we have the diffusion coefficient, in the theory of wave propagation
the propagation coefficient; and in the theory of elasticity the modulus
of elasticity. In concrete examples of all of the cases mentioned, the magnitudes
of the coefficients or parameters are experimentally determined. When solving
the associated mathematical equations, and in subsequent calculations, it is
usually the meun value of a set of experimental determinations that is used as
the value of the coefficient or parameter. In some instances this may provide
an adequate or reasonable description, but in many instances the variance
may be sufficiently large to warrant consideration. Hence, when we talk
about physical constants or parameters, etc., we are not, in many instances,
talking about constants at all, but random variables whose values are determined
by some probability distribution or law. The same thing can be said
about coefficients and nonhomogeneous terms (or forcing functions) of
equations, which may be random variables or functions.\