Section 2: Occurrence and Description of Different Types of Equation

This section introduces the various different types of equations that will be met in this module. It describes how they arise, and the nature of their solution.

Section 2.1: Introduction to Equations

Equations and Reality

There are several different kinds or categories of equation, all of which require a different approach to their solution. Indeed, the fundamental nature of the solution differs significantly for the various categories.

There is also a connection between the physical system and the equations which describe it, in that certain types of equation describe certain situations or phenomena. An understanding of the relationship between the `real' situation and the types of equations is thus essential in developing mathematical models in the form of equations.

Classification of Equations and Systems

What kinds of equation are there?
How do these relate to engineering situations?
What is the nature of the solution?

What is an equation?

An equation is anything with an equals sign in it, i.e.:
(Something) = (Something else)

There are three major categories of equation.

Algebraic Equations

These are most easily identifiable by what they do not contain, i.e. No derivatives or integrals but any of the normal arithmetic operators or higher algebraic functions. The following are thus all algebraic equations: x - 3 = 5
sin 3 x = y cos x
a x² + b x + c = 0

The solution is a numerical value for a single equation, or a set of numbers, as many as there are unknowns and equations, for a set of algebraic equations.

One significant characteristic of an equation is whether it is linear or nonlinear in a particular variable. The equation is linear only if the variable appears to a power of one, and does not appear as the argument of a higher function.

Thus only the first equation above is linear in x. However the second equation is linear in y, and the third is linear in a, b, and c.

There is more about algebraic equations in section 3.1.

Ordinary Differential Equations: o.d.es

These are readily identifiable as they contain derivatives:

NB: these must all be `Straight d' derivatives if the equations are o.d.e.s.

There are two different kinds of variable in this type of equation: the dependent and independent variables. There may only be one independent variable, and this will appear on the bottom line of the derivatives. The variables which appear on the top line are the dependent variables and there should be the same number of these as there are equations.

The solution is the dependent variable or variables as a function or set of functions in terms of the independent variable.

There is more about o.d.e.s in section 4.1.

Partial Differential Equations: p.d.e.s

If there is more than one independent variable, then derivatives must normally be written as partial derivatives, giving rise to p.d.e.s with `curly ds', e.g.:

P.d.e.s are hard to solve! The solution is the dependent variables as functions of all the independent variables.

Section 2.2: Examples

Equations in Engineering Problems

The following example show how the different categories of equation arise in various situations.

Example 1: Simple Mixing Process

See Figure below

Consider the behaviour of the process over a period of time when the two flowrates do not change, and any disturbances to the process resulting from previous changes in the flows have settled out.

total outflow rate = total inflow rate

F₁ + F₂ - F = 0

Notes

The condition of no change, etc, means that the process may be said to be at steady state.
The properties of the process variables, i.e. the flows, are associated with one value at one point in the process, e.g. an inlet stream. There are a finite number of such points and in particular, not a continuum. This is called a lumped system.
The equation is an algebraic equation, i.e. no integrals or derivatives.
The solution is a number.

Example 2: Tank Filling

See Figure below

Liquid flows at a rate F kg/s into a tank. The mass of material in the tank initially, at say time t=0 is M₀ kg. There is no outflow from the tank.

rate of accumulation = net rate of inflow - net rate of outflow

With initial condition: M(t) = M₀ at t=0

Notes

This is a dynamic situation because M changes with time.
This is a lumped system because the property, holdup, is associated with a single point and is not distributed in space. That is, we can talk about the mass of material in the tank but not the mass of material at point x in the tank.
The equation is an o.d.e. whose independent variable is time.
The solution is functional variation of M(t).

Example 3: Heat Transfer in a Slab

See Figure below

Heat flows at a constant steady rate Q watts across a uniform slab. Each side of the slab is held at constant temperature. The temperature of the slab, of cross sectional area A and thermal conductivity k, varies across its thickness x and is denoted by T(x).

Notes

This is called a distributed system because a property, the temperature of the slab, is distributed in space.
There is one spatial dimension.
There is no change in time because the heat flow is constant, thus this is once again a steady state situation.
The equation describing the system is an ordinary differential equation.
The independent variable in the o.d.e. is distance.
The solution is the functional variation of the distributed property with distance, i.e. T(x).

Example 4: Time Varying Convection by Plug Flow in a Pipe

This is described by a p.d.e. whose independent variables are time and distance along the pipe. Consider the situation where the temperature T is changing with distance x down a pipe of total length L where the velocity of flow is u.

This is described by the convective flow terms of the Navier-Stokes equation:

Although this is a complicated looking equation it turns out to have a simple solution. The temperature of the fluid leaving the pipe at time t is exactly the same as the temperature entering the pipe at a previous time (t - L/u) since L/u is the residence time for the fluid in the pipe.

More Complicated Situations

Steady state systems in two or more dimensions are described by p.d.e.s whose independent variables are all the spatial dimensions involved.
The unsteady state of distributed systems involves equations containing p.d.e.s with independent variables of both space and time.
The solutions of p.d.e.s are functions or functionals.
Distributed problems may contain integrals, leading to integral equations.

Summary and Further Points

Steady state lumped systems are described by algebraic equations
Steady state distributed systems by o.d.e.s in distance
Unsteady state lumped systems by o.d.e.s in time
Anything more complicated by p.d.e.s.
All these are different kinds of equations, requiring different solution methods.
Algebraic equations also describe dynamic systems with no capacity and thus instant response.