Scales and scalings

This short sections introduces simple concepts of dimensional analysis and scalings. It also illustrates the existence of self-similar solutions to partial differential equations.

• Dimensional analysis and scalings
• Self-similar solutions of partial differential equations

Dimensional analysis and scalings

Dimensional analysis

Dimensional analysis is a simple technique that allows you to determine the dimension of quantities that appear in mathematical descriptions. Each term in a formula is either dimensionless, or can be assigned a dimension according to the following criteria.

• T typically represents quantities which have the dimension of a time. Such quantities may be expressed in different units, such as seconds, minutes, days, years, etc.
• M represents quantities which have the dimension of a mass. The corresponding units may be grams, ounces, kilograms, etc.
• L represents quantities which have the dimension of a length. Here again, such quantities my have different units, such as meters, feet, yards, kilometers, etc.

It is thus important to distinguish between dimensions and units. Dimensional analysis can be used to check that a mathematical model is consistent. In particular, quantities that are equal to one another should have the same dimension, and quantities that are the arguments of a mathematical function (e.g. the argument of an exponential, a log, a sine, or a cosine) should be dimensionless.

Scalings

The number of parameters in a model can often be significantly reduced by rescaling its variables. The resulting dimensionless model sometimes has no parameters at all, which immensely simplifies its analysis. A series of examples will be discussed in class.

Self-similar solutions of partial differential equations

We will discuss how dimensional analysis may be applied to the search of exact solutions to nonlinear partial differential equations.

Just-in-time mathematics

• Partial differential equations: introduction