
14.2. Basic Problems of Mathematical Physics 1143
For simplicity, here we confine ourselves to the case where the coefficients a
ij
of equations
(14.2.1.1) and (14.2.1.5) have the special form
a
ij
(x, t) = a(x, t)δ
ij
, δ
ij
=
(
1 if i = j,
0 if i 6= j.
This situation is rather frequent in applications; such coefficients are used to describe vari-
ous phenomena (processes) in isotropic media.
First boundary value problem. The function w(x, t) takes prescribed values at the bo-
undary S of the domain:
w(x, t) = g
1
(x, t) for x ∈ S. (14.2.2.1)
Second boundary value problem. The derivative along the (outward) normal is pre-
scribed at the boundary S of the domain:
∂w