ABSTRACT
Modelling of many physical problems in chemical engineering and allied disciplines leads to partial differential equations (PDEs) when the quantity to be determined depends on two or more independent variables. A PDE is a relation between an unknown dependent variable and more than one independent variable and their derivatives of different orders with respect to the independent variables. The partial derivatives may be in respect of individual independent variables or may be ‘mixed derivatives’. As in the case of an ordinary differential equation, the order of a PDE is the same as the order of the highest derivate in the equation. Thus, if z is a variable that depends upon two independent variables x and y, the PDE will involve derivatives such as ¶ ¶
¶ ¶
¶ ¶
¶ ¶ ¶
u
x
u
y u
x
u
x y , , ,
, etc. (the last one is a mixed derivative). A partial differential equation is linear
if the powers of the dependent variable or its derivatives are 1 or 0. If any of such quantities has
a power other than 1 or 0, the PDE is called non-linear. Thus, ¶ ¶
= ¶ ¶
u
x K u
y
2 is a linear PDE of
second order (since the power of the highest derivative is 1), while the equation ¶ ¶
= ¶ ¶
+u x
K u y
u 2
is a second-order non-linear PDE. The unknowns or dependent variables generally include physical quantities such as temperature
in a medium, concentration, velocity of a fluid, electrical or gravitational potential, displacement of an elastic wire or a membrane, etc. The independent variables generally are time and position. Problems involving one dependent variable and two or more independent variables are more common: for example variation of temperature with time and position in a medium. However, there may be physical problems that involve more than one dependent variable. A common example is variation of concentration and temperature in a non-isothermal catalyst pellet, or the variation of velocity components in a fluid with time and position (the Navier−Stokes equation). A few physical problems of practical importance have been formulated in terms of partial differential equations in Chapter 1.
