Introduction - Heat Conduction: Mathematical Models and Analytical Solutions - page 12

ematical physics contain partial derivatives of physical variables with respect to
time and space coordinates. Therefore, the fundamental relation between physical
variables in the equations of mathematical physics is local in space and time. The
relation between physical variables in their solutions is global in space and time.
Thus solving equations of mathematical physics under supplementary conditions is
actually a mathematical method to find the global relations for a process from the
local relations at a time instant .

1.2.2 Approaches of Developing Equations
of Mathematical Physics

In this subsection, we discuss the methods of developing equations of mathematical
physics from physical laws. As the equations represent physical relations at a point
in space and at an instant in time, we should pay attention to instant states of physical
points of the system.

Approach 1.

The equations are obtained by direct application of physical laws to a physical point
and an instant in time. Since physical laws are, in general, applicable for a mate-
rial system and a process, the physical point here refers to an infinitesimal special
region; the time instant refers to an infinitesimal temporal period. Thus the one-
dimensional point x stands actually for the region ( x , x + Δ x )( Δ x → 0); the time

instant t denotes the time period from t to t 0). Using this approach we

often use physical laws such as Newton's laws of motion and the conservation of
mass, momentum and energy.

+ Δ t ( Δ t →

Approach 2.

Based on fundamental physical laws, this approach first develops relations among
different physical variables for an arbitrary material system and process. Note that
physical quantities for the system and the process are normally integrals. Using this
approach, the physical laws first lead to equations in a form in which some integrals
are equal to zero. Both the continuity of the integrands and the localization theorem
are then used to conclude that the integrands must be zero. This leads to the desired
differential equation governing the local and instant relations among variables.
The one-dimensional localization theorem states that

f ( x ) ≡ 0 , x ∈ [ a , b ] ,

x + Δ x

if f ( x ) ∈ C [ a , b ] and for any ( x , x + Δ x ) ∈ [ a , b ] , f ( ξ ) d ξ = 0.

x