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SYSTEM EQUATIONS


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SYSTEM EQUATIONS

A property common to all basic

laws of physics is that certain

fundamental quantities can bedefined by numerical values. The

physical laws define relationships

between these fundamental

quantities and are usuallyrepresented by equations.


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PARTIALAND ORDINARY

DIFFERENTIAL EQUATIONS


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PARTIAL DIFFERENTIAL EQUATION

Is an equality involving one or

more dependent and two or moreindependent variables, together with

partial derivatives of the dependent

with respect to the independent

variables


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!t

Tk

x

T

HH

HH

Diffusion Equation

EXAMPLE:

txTT ,!

t

x

Dependent Variable

Independent Variable


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ORDINARY DIFFERENTIAL EQUATION

Is an equality involving one or

more dependent variables, oneindependent variable, and one or

more derivatives of the dependent

variables with respect to the

independent variable


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!dt

dqRv

Ohms Law

EXAMPLE:

tvv

tqq

!

!

t

Dependent Variable

Independent Variable


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TIME VARIABILITY AND TIME

INVARIANCE

Time is the only independent variable, unless

otherwise specified. This variable is normally

designated t , except that in difference equations

the discrete variable k is often used, as an

abbreviation for the time instant t is used

instead ofy ( t k ) , etc.


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Term of a differential or difference

equation consists of products and/or quotients of

explicit functions of the independent variable, the

dependent variables, and, for differential

equations, derivatives of the dependent variables.

Equationrefers to either a differential equation

or a difference equation.


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Definition 3.5A time-variableequationis an

equation in which one or more terms depend

explicitly on the independent variable time.

Definition 3.6A time-invariant equationis an

equation in which none of the terms depends

explicitly on the independent variable time.


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EXAMPLE 3.12. The difference equation ky( k +

2) + y ( k ) = U( k), where U and y are dependent

variables, is time-variable because the term ky( k

+ 2) depends explicitly on the coefficient k, which

represents the time tk

EXAMPLE 3.13. Any differential equation of the

form:


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where the coefficients

are constants, is time-inoariant. The equation

depends implicitly on t, via the dependent

variables U and y and their derivatives.


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LINEAR AND NONLINEAR

DIFFERENTIAL AND DIFFERENCE

EQUATIONS

Definition3.7: A linear term is one which

is first degree in the dependent variables

and their derivatives.

Definition 3.8:A linear equation is an

equation consisting of a sum of linear terms.

All others are nonlinear equations.


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If any term of a differential equation contains

higher powers, products, or transcendental

functions of the dependent variables, it is

nonlinear. Such terms include ( d ~ / d t )u~(d,y/dt ) , and sin U, respectively.

For example, (5/cos t)( d 2 y / d t 2 ) is a term of

first degree in the dependent variabley ,

and2uy3(dy/dt) is a term of fifth degree in the

dependent variables Uandy .


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EXAMPLE 3.14. The ordinary differentialequations (dy/dt)*+ y = 0 and d2y/dt2+ cosy =0 are nonlinear because ( dy/dt)2is seconddegree in the first equation, and cosy in the

second equation is not first degree, which is trueof all transcendental functions.

EXAMPLE 3.15. The difference equationy ( k +2) +u(k +l ) y ( k +1) +y ( k ) =u ( k ) , inwhich Uandy are dependent variables, is anonlinear difference equation because U( k + l)y(k + 1) is second degree in Uand y. This type ofnonlinear equation is sometimes called bilinearin Uand y.


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EXAMPLE 3.16.Any difference equation

in which the coefficients a , ( k ) and b , ( k )

depend only upon the independent variable k, isa linear difference equation.


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EXAMPLE 3.17.Any ordinary differential

equation

where the coefficients a, ( t ) and b, ( t ) dependonly upon the independent variable t , is a linear

differential equation


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THE DIFFERENTIAL OPERATOR DAND

THE CHARACTERISTIC EQUATION


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