Nondimensionalization is the partial or full removal of physical dimensions from an

_{c}'' is the characteristic unit used to scale it.
As an illustrative example, consider a first order differential equation with _{c}'', hence it is easiest to choose to set this to unity first: $$\backslash frac\; =\; 1\; \backslash Rightarrow\; t\_c\; =\; \backslash frac.$$ Subsequently, $$\backslash frac\; =\; \backslash frac\; =\; 1\; \backslash Rightarrow\; x\_c\; =\; \backslash frac.$$
# The final dimensionless equation in this case becomes completely independent of any parameters with units: $$\backslash frac\; +\; \backslash chi\; =\; F(\backslash tau).$$

_{c} and ''t''_{c} with the same units as ''x'' and ''t'' respectively, such that these conditions hold:
$$\backslash tau\; =\; \backslash frac\; \backslash Rightarrow\; t\; =\; \backslash tau\; t\_c$$
$$\backslash chi\; =\; \backslash frac\; \backslash Rightarrow\; x\; =\; \backslash chi\; x\_c.$$
These equations are used to replace ''x'' and ''t'' when nondimensionalizing. If differential operators are needed to describe the original system, their scaled counterparts become dimensionless differential operators.

_{''c''} and ''t''_{''c''} so that the coefficients become normalized. Since there are two free parameters, at most only two coefficients can be made to equal unity.

_{''c''}:
#If $t\_c\; =\; \backslash frac$ the first order term is normalized.
#If $t\_c\; =\; \backslash sqrt$ the zeroth order term is normalized.
Both substitutions are valid. However, for pedagogical reasons, the latter substitution is used for second order systems. Choosing this substitution allows ''x''_{''c''} to be determined by normalizing the coefficient of the forcing function:
$$1\; =\; \backslash frac\; =\; \backslash frac\; \backslash Rightarrow\; x\_c\; =\; \backslash frac.$$
The differential equation becomes
$$\backslash frac\; +\; \backslash frac\; \backslash frac\; +\; \backslash chi\; =\; F(\backslash tau).$$
The coefficient of the first order term is unitless. Define
$$2\; \backslash zeta\; \backslash \; \backslash stackrel\backslash \; \backslash frac.$$
The factor 2 is present so that the solutions can be parameterized in terms of ''ζ''. In the context of mechanical or electrical systems, ''ζ'' is known as the

^{−2}* $m$ = mass of the block g* $B$ = damping constant of dashpot −1">g⋅s^{−1}* $k$ = force constant of spring −2">g⋅s^{−2}
Suppose the applied force is a sinusoid , the differential equation that describes the motion of the block is
$$m\; \backslash frac\; +\; B\; \backslash frac\; +\; kx\; =\; F\_0\; \backslash cos(\backslash omega\; t)$$
Nondimensionalizing this equation the same way as described under second order system yields several characteristics of the system.
The intrinsic unit ''x_{c}'' corresponds to the distance the block moves per unit force
$$x\_c\; =\; \backslash frac.$$
The characteristic variable ''t_{c}'' is equal to the period of the oscillations
$$t\_c\; =\; \backslash sqrt$$
and the dimensionless variable 2''ζ'' corresponds to the linewidth of the system. ''ζ'' itself is the

Analysis of differential equation models in biology: a case study for clover meristem populations

(Application of nondimensionalization to a problem in biology).

Course notes for Mathematical Modelling and Industrial Mathematics

''Jonathan Evans, Department of Mathematical Sciences,

Scaling of Differential Equations

''Hans Petter Langtangen, Geir K. Pedersen, Center for Biomedical Computing, Simula Research Laboratory and Department of Informatics, University of Oslo.'' Dimensional analysis

equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...

involving physical quantities by a suitable substitution of variables
Substitution may refer to:
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*Chord substitution, in music, swapping one chord for a related one within a chord progression
*Substitution (poetry), a variation in poetic scansion
* "Substitution" (song), a 2009 song by Silversun Pic ...

. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (suc ...

. In some physical systems, the term scaling is used interchangeably with ''nondimensionalization'', in order to suggest that certain quantities are better measured relative to some appropriate unit. These units refer to quantities intrinsic
In science and engineering, an intrinsic property is a property of a specified subject that exists itself or within the subject. An extrinsic property is not essential or inherent to the subject that is being characterized. For example, m ...

to the system, rather than units such as SI units. Nondimensionalization is not the same as converting extensive quantities in an equation to intensive quantities, since the latter procedure results in variables that still carry units.
Nondimensionalization can also recover characteristic properties of a system. For example, if a system has an intrinsic resonance frequency, length, or time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a sy ...

, nondimensionalization can recover these values. The technique is especially useful for systems that can be described by differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

s. One important use is in the analysis of control system
A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial c ...

s.
One of the simplest characteristic units is the doubling time The doubling time is the time it takes for a population to double in size/value. It is applied to population growth, inflation, resource extraction, consumption of goods, compound interest, the volume of malignant tumours, and many other things th ...

of a system experiencing exponential growth
Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...

, or conversely the half-life
Half-life (symbol ) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable at ...

of a system experiencing exponential decay
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate c ...

; a more natural pair of characteristic units is mean age/mean lifetime
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...

, which correspond to base ''e'' rather than base 2.
Many illustrative examples of nondimensionalization originate from simplifying differential equations. This is because a large body of physical problems can be formulated in terms of differential equations. Consider the following:
* List of dynamical systems and differential equations topics
* List of partial differential equation topics
A ''list'' is any set of items in a row. List or lists may also refer to:
People
* List (surname)
Organizations
* List College, an undergraduate division of the Jewish Theological Seminary of America
* SC Germania List, German rugby union ...

* Differential equations of mathematical physics
Differential may refer to:
Mathematics
* Differential (mathematics) comprises multiple related meanings of the word, both in calculus and differential geometry, such as an infinitesimal change in the value of a function
* Differential algebra
* ...

Although nondimensionalization is well adapted for these problems, it is not restricted to them. An example of a non-differential-equation application is dimensional analysis; another example is normalization
Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to:
* Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...

in statistics.
Measuring device
A measuring instrument is a device to measure a physical quantity. In the physical sciences, quality assurance, and engineering, measurement is the activity of obtaining and comparing physical quantities of real-world objects and events. Esta ...

s are practical examples of nondimensionalization occurring in everyday life. Measuring devices are calibrated relative to some known unit. Subsequent measurements are made relative to this standard. Then, the absolute value of the measurement is recovered by scaling with respect to the standard.
Rationale

Suppose apendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...

is swinging with a particular period ''T''. For such a system, it is advantageous to perform calculations relating to the swinging relative to ''T''. In some sense, this is normalizing the measurement with respect to the period.
Measurements made relative to an intrinsic property of a system will apply to other systems which also have the same intrinsic property. It also allows one to compare a common property of different implementations of the same system. Nondimensionalization determines in a systematic manner the characteristic units of a system to use, without relying heavily on prior knowledge of the system's intrinsic properties
(one should not confuse characteristic units of a ''system'' with natural units of ''nature''). In fact, nondimensionalization can suggest the parameters which should be used for analyzing a system. However, it is necessary to start with an equation that describes the system appropriately.
Nondimensionalization steps

To nondimensionalize a system of equations, one must do the following: #Identify all the independent and dependent variables; #Replace each of them with a quantity scaled relative to a characteristic unit of measure to be determined; #Divide through by the coefficient of the highest order polynomial or derivative term; #Choose judiciously the definition of the characteristic unit for each variable so that the coefficients of as many terms as possible become 1; #Rewrite the system of equations in terms of their new dimensionless quantities. The last three steps are usually specific to the problem where nondimensionalization is applied. However, almost all systems require the first two steps to be performed.Conventions

There are no restrictions on the variable names used to replace "''x''" and "''t''". However, they are generally chosen so that it is convenient and intuitive to use for the problem at hand. For example, if "''x''" represented mass, the letter "''m''" might be an appropriate symbol to represent the dimensionless mass quantity. In this article, the following conventions have been used: * ''t'' – represents the independent variable – usually a time quantity. Its nondimensionalized counterpart is $\backslash tau$. * ''x'' – represents the dependent variable – can be mass, voltage, or any measurable quantity. Its nondimensionalized counterpart is $\backslash chi$. A subscripted ''c'' added to a quantity's variable name is used to denote the characteristic unit used to scale that quantity. For example, if ''x'' is a quantity, then ''xconstant coefficients
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = ...

:
$$a\backslash frac\; +\; bx\; =\; Af(t).$$
# In this equation the independent variable here is ''t'', and the dependent variable is ''x''.
# Set $x\; =\; \backslash chi\; x\_c,\; \backslash \; t\; =\; \backslash tau\; t\_c$. This results in the equation $$a\; \backslash frac\; \backslash frac\; +\; b\; x\_c\; \backslash chi\; =\; A\; f(\backslash tau\; t\_c)\; \backslash \; \backslash stackrel\backslash \; A\; F(\backslash tau).$$
# The coefficient of the highest ordered term is in front of the first derivative term. Dividing by this gives $$\backslash frac\; +\; \backslash frac\; \backslash chi\; =\; \backslash frac\; F(\backslash tau).$$
# The coefficient in front of $\backslash chi$ only contains one characteristic variable ''tSubstitutions

Suppose for simplicity that a certain system is characterized by two variables - a dependent variable ''x'' and an independent variable ''t'', where ''x'' is afunction
Function or functionality may refer to:
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* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orien ...

of ''t''. Both ''x'' and ''t'' represent quantities with units. To scale these two variables, assume there are two intrinsic units of measurement ''x''Differential operators

Consider the relationship $$\backslash ,\backslash !\; t\; =\; \backslash tau\; t\_c\; \backslash Rightarrow\; dt\; =\; t\_c\; d\backslash tau\; \backslash Rightarrow\; \backslash frac\; =\; \backslash frac.$$ The dimensionless differential operators with respect to the independent variable becomes $$\backslash frac\; =\; \backslash frac\; \backslash frac\; =\; \backslash frac\; \backslash frac\; \backslash Rightarrow\; \backslash frac\; =\; \backslash left(\; \backslash frac\; \backslash right)^n\; =\; \backslash left(\; \backslash frac\; \backslash frac\; \backslash right)^n\; =\; \backslash frac\; \backslash frac.$$Forcing function

If a system has a forcing function $\backslash ,\backslash !\; f(t)$ then $$\backslash ,\backslash !\; f(t)\; =\; f(\backslash tau\; t\_c)\; =\; f(t(\backslash tau))\; =\; F(\backslash tau).$$ Hence, the new forcing function $\backslash ,\backslash !\; F$ is made to be dependent on the dimensionless quantity $\backslash ,\backslash !\; \backslash tau$.Linear differential equations with constant coefficients

First order system

Consider the differential equation for a first order system: $$a\backslash frac\; +\; bx\; =\; Af(t).$$ The derivation of the characteristic units for this system gives $$t\_c\; =\; \backslash frac,\; \backslash \; x\_c\; =\; \backslash frac.$$Second order system

A second order system has the form $$a\; \backslash frac\; +\; b\backslash frac\; +\; cx\; =\; A\; f(t).$$Substitution step

Replace the variables ''x'' and ''t'' with their scaled quantities. The equation becomes $$a\; \backslash frac\; \backslash frac\; +\; b\; \backslash frac\; \backslash frac\; +\; c\; x\_c\; \backslash chi\; =\; A\; f(\backslash tau\; t\_c)\; =\; A\; F(\backslash tau)\; .$$ This new equation is not dimensionless, although all the variables with units are isolated in the coefficients. Dividing by the coefficient of the highest ordered term, the equation becomes $$\backslash frac\; +\; t\_c\; \backslash frac\; \backslash frac\; +\; t\_c^2\; \backslash frac\; \backslash chi\; =\; \backslash frac\; F(\backslash tau).$$ Now it is necessary to determine the quantities of ''x''Determination of characteristic units

Consider the variable ''t''damping ratio
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples in ...

, and is an important parameter required in the analysis of control system
A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial c ...

s. 2''ζ'' is also known as the linewidth of the system. The result of the definition is the universal oscillator equation.
$$\backslash frac\; +\; 2\; \backslash zeta\; \backslash frac\; +\; \backslash chi\; =\; F(\backslash tau)\; .$$
Higher order systems

The general n-th order linear differential equation with constant coefficients has the form: $$a\_n\; \backslash frac\; +\; a\_\; \backslash frac\; +\; \backslash ldots\; +\; a\_1\; \backslash frac\; +\; a\_0\; x(t)\; =\; \backslash sum\_^n\; a\_k\; \backslash frac\; =\; Af(t).$$ The function ''f''(''t'') is known as the forcing function. If the differential equation only contains real (not complex) coefficients, then the properties of such a system behaves as a mixture of first and second order systems only. This is because the roots of itscharacteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The cha ...

are either real, or complex conjugate pairs. Therefore, understanding how nondimensionalization applies to first and second ordered systems allows the properties of higher order systems to be determined through superposition.
The number of free parameters in a nondimensionalized form of a system increases with its order. For this reason, nondimensionalization is rarely used for higher order differential equations. The need for this procedure has also been reduced with the advent of symbolic computation
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressio ...

.
Examples of recovering characteristic units

A variety of systems can be approximated as either first or second order systems. These include mechanical, electrical, fluidic, caloric, and torsional systems. This is because the fundamental physical quantities involved within each of these examples are related through first and second order derivatives.Mechanical oscillations

Suppose we have a mass attached to a spring and a damper, which in turn are attached to a wall, and a force acting on the mass along the same line. Define * $x$ = displacement from equilibrium * $t$ = time * $f$ = external force or "disturbance" applied to system −2">g⋅m⋅sdamping ratio
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples in ...

.
$$2\; \backslash zeta\; =\; \backslash frac$$
Electrical oscillations

= First-order series RC circuit

= For a series RC attached to avoltage source
A voltage source is a two- terminal device which can maintain a fixed voltage. An ideal voltage source can maintain the fixed voltage independent of the load resistance or the output current. However, a real-world voltage source cannot supply u ...

$$R\; \backslash frac\; +\; \backslash frac\; =\; V(t)\; \backslash Rightarrow\; \backslash frac\; +\; \backslash chi\; =\; F(\backslash tau)$$
with substitutions
$$Q\; =\; \backslash chi\; x\_c,\; \backslash \; t\; =\; \backslash tau\; t\_c,\; \backslash \; x\_c\; =\; C\; V\_0,\; \backslash \; t\_c\; =\; RC,\; \backslash \; F\; =\; V.$$
The first characteristic unit corresponds to the total charge
Charge or charged may refer to:
Arts, entertainment, and media Films
* '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* '' Charge!!'', an album by The Aq ...

in the circuit. The second characteristic unit corresponds to the time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a sy ...

for the system.
= Second-order series RLC circuit

= For a series configuration of ''R'',''C'',''L'' components where ''Q'' is the charge in the system $$L\; \backslash frac\; +\; R\; \backslash frac\; +\; \backslash frac\; =\; V\_0\; \backslash cos(\backslash omega\; t)\; \backslash Rightarrow\; \backslash frac\; +\; 2\; \backslash zeta\; \backslash frac\; +\; \backslash chi\; =\; \backslash cos(\backslash Omega\; \backslash tau)$$ with the substitutions $$Q\; =\; \backslash chi\; x\_c,\; \backslash \; t\; =\; \backslash tau\; t\_c,\; \backslash \; \backslash \; x\_c\; =\; C\; V\_0,\; \backslash \; t\_c\; =\; \backslash sqrt,\; \backslash \; 2\; \backslash zeta\; =\; R\; \backslash sqrt,\; \backslash \; \backslash Omega\; =\; t\_c\; \backslash omega.$$ The first variable corresponds to the maximum charge stored in the circuit. The resonance frequency is given by the reciprocal of the characteristic time. The last expression is the linewidth of the system. The Ω can be considered as a normalized forcing function frequency.Quantum mechanics

Quantum harmonic oscillator

TheSchrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...

for the one-dimensional time independent quantum harmonic oscillator
量子調和振動子 は、 古典調和振動子 の 量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最 ...

is
$$\backslash left(-\backslash frac\; \backslash frac\; +\; \backslash fracm\; \backslash omega^2\; x^2\backslash right)\; \backslash psi(x)\; =\; E\; \backslash psi(x).$$
The modulus square of the wavefunction represents probability density that, when integrated over , gives a dimensionless probability. Therefore, has units of inverse length. To nondimensionalize this, it must be rewritten as a function of a dimensionless variable. To do this, we substitute
$$\backslash tilde\; x\; \backslash equiv\; \backslash frac,$$
where is some characteristic length of this system. This gives us a dimensionless wave function $\backslash tilde\; \backslash psi$ defined via
$$\backslash psi(x)\; =\; \backslash psi(\backslash tilde\; x\; x\_)\; =\; \backslash psi(x(x\_))\; =\; \backslash tilde\; \backslash psi(\backslash tilde\; x).$$
The differential equation then becomes
$$\backslash left(-\backslash frac\; \backslash frac\; \backslash frac\; +\; \backslash frac\; m\; \backslash omega^2\; x\_^2\; \backslash tilde\; x^2\; \backslash right)\; \backslash tilde\; \backslash psi(\backslash tilde\; x)\; =\; E\; \backslash ,\; \backslash tilde\; \backslash psi(\backslash tilde\; x)\; \backslash Rightarrow\; \backslash left(-\backslash frac\; +\; \backslash frac\; \backslash tilde\; x^2\; \backslash right)\; \backslash tilde\; \backslash psi(\backslash tilde\; x)\; =\; \backslash frac\; \backslash tilde\; \backslash psi(\backslash tilde\; x).$$
To make the term in front of $\backslash tilde\; x^2$ dimensionless, set
$$\backslash frac\; =\; 1\; \backslash Rightarrow\; x\_\; =\; \backslash sqrt\; .$$
The fully nondimensionalized equation is
$$\backslash left(-\backslash frac\; +\; \backslash tilde\; x^2\; \backslash right)\; \backslash tilde\; \backslash psi(\backslash tilde\; x)\; =\; \backslash tilde\; E\; \backslash tilde\; \backslash psi(\backslash tilde\; x),$$
where we have defined
$$E\; \backslash equiv\; \backslash frac\; \backslash tilde\; E.$$
The factor in front of $\backslash tilde\; E$ is in fact (coincidentally) the ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...

energy of the harmonic oscillator. Usually, the energy term is not made dimensionless as we are interested in determining the energies of the quantum states. Rearranging the first equation, the familiar equation for the harmonic oscillator becomes
$$\backslash frac\; \backslash left(\; -\backslash frac\; +\; \backslash tilde\; x^2\; \backslash right)\; \backslash tilde\; \backslash psi(\backslash tilde\; x)\; =\; E\; \backslash tilde\; \backslash psi(\backslash tilde\; x).$$
Statistical analogs

In statistics, the analogous process is usually dividing a difference (a distance) by a scale factor (a measure ofstatistical dispersion
In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquart ...

), which yields a dimensionless number, which is called ''normalization
Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to:
* Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...

.'' Most often, this is dividing errors or residuals by the standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, w ...

or sample standard deviation, respectively, yielding standard scores and studentized residual
In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. It is a form of a Student's ''t''-statistic, with the estimate of error varying between points.
This i ...

s.
See also

*Buckingham π theorem
In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physical ...

* Dimensionless number
* Natural units
* System equivalence
* RLC circuit
An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components ...

* RL circuit
A resistor–inductor circuit (RL circuit), or RL filter or RL network, is an electric circuit composed of resistors and inductors driven by a voltage or current source. A first-order RL circuit is composed of one resistor and one inductor, either ...

* RC circuit
A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC ...

* Logistic equation
{{colend
External links

Analysis of differential equation models in biology: a case study for clover meristem populations

(Application of nondimensionalization to a problem in biology).

Course notes for Mathematical Modelling and Industrial Mathematics

''Jonathan Evans, Department of Mathematical Sciences,

University of Bath
(Virgil, Georgics II)
, mottoeng = Learn the culture proper to each after its kind
, established = 1886 (Merchant Venturers Technical College) 1960 (Bristol College of Science and Technology) 1966 (Bath University of Technology) 1971 (univ ...

.'' (see Chapter 3).Scaling of Differential Equations

''Hans Petter Langtangen, Geir K. Pedersen, Center for Biomedical Computing, Simula Research Laboratory and Department of Informatics, University of Oslo.'' Dimensional analysis