## ABSTRACT

Different engineering disciplines solve different types of problems in their respective fields. For mechanical engineers, they may need to solve the temperature change within a solid when it is heated by the interior heat sources or due to a rise or decrease of its boundary temperatures. For electrical engineers, they may need to find the voltages at all circuit joints of a computer chip board. Temperature and voltage are the variables in their respective fields. Hence, they are called field variables. It is easy to understand that the value of the field variable is space-dependent and time-dependent. That is to say, that we are interested to know the spatial and temporal changes of the field variable. Let us denote the field variable as ϕ. and let the independent variables be X_{i} which could be the time t, or, the space coordinates as x, y, and z. In order not to overly complicate the discussion, we introduce the general two-dimensional partial differential equation which governs the field variable in the form of:
()
A
(
x
1
,
x
2
)
∂
2
ϕ
∂
x
1
2
+
B
(
x
1
,
x
2
)
∂
2
ϕ
∂
x
1
∂
x
2
+
C
(
x
1
,
x
2
)
∂
2
ϕ
∂
x
2
2
=
F
(
x
1
,
x
2
,
ϕ
,
∂
ϕ
∂
x
1
,
∂
ϕ
∂
x
2
)
https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429126482/4d207c6c-f89a-41ff-855c-8afc9f1275a4/content/eq431.tif"/>