APPLICATIONS OF DIFFERENTIAL EQUATIONS-ZBJ

APPLICATIONS OF DIFFERENTIAL
EQUATIONS
PRESENTED TO:DR.SADIA ARSHAD
PRESENTED BY:ASHHAD ABBAS GILANI(026)
SHAHAB ARSHAD(058)
RIAZ HUSSAIN(060)
MUHAMMAD YOUSUF(082)
ZUHAIR BIN JAWAID(094)

INVENTIONOF DIFFERENTIAL
EQUATION:
• In mathematics, the history of differential equations traces the development of
"differential equations" from calculus, which itself was independently invented
by English physicist Isaac Newton and German mathematician Gottfried Leibniz.
• The history of the subject of differential equations, in concise form, from a
synopsis of the recent article “The History of Differential Equations,1670-1950”
“Differential equations began with Leibniz, the Bernoulli brothers, and others
from the 1680s, not long after Newton’s ‘fluxional equations’ in the 1670s.”

DIFFERENTIAL EQUATION:
• A Differential Equation is an equation containing the derivative of one
or more dependent variables with respect to one or more
independent variables.
• For Example,

TYPES OF DIFFERENTIAL EQUATION:
ODE (ORDINARY DIFFERENTIAL EQUATION):
An equation contains only ordinary derivates of one or more
dependent variables of a single independent variable.
For Example,
dy/dx + 5y = ex, (dx/dt) + (dy/dt) = 2x + y
PDE (PARTIAL DIFFERENTIAL EQUATION):
An equation contains partial derivates of one or more dependent
variables of two or more independent variables.
For Example,

 FIRST ORDERODE:
• A first order differential equation is an
equation involving the unknown function y, its
derivative y' and the variable x. We will only
talk about explicit differential equations.
• General Form,
• For Example,
32  x
dx
dy

INITIALANDBOUNDARY VALUE PROBLEMS:
• Boundary value problems are similar to initial value problems. A boundary value problem
has conditions specified at the extremes ("boundaries") of the independent variable in the
equation whereas an initial value problem has all of the conditions specified at the same
value of the independent variable (and that value is at the lower boundary of the domain,
thus the term "initial" value).
• For example, if the independent variable is time over the domain [0,1], a boundary value
problem would specify values for at both and , whereas an initial value problem would
specify a value of and at time .
• Finding the temperature at all points of an iron bar with one end kept at absolute zero and
the other end at the freezing point of water would be a boundary value problem.
• If the problem is dependent on both space and time, one could specify the value of the
problem at a given point for all time the data or at a given time for all space.
• Concretely, an example of a boundary value (in one spatial dimension) is the problem

APPLICATIONS OF ODE:
 MODELLING WITH FIRST-ORDER EQUATIONS
 Newton’s Law of Cooling
 Electrical Circuits
 MODELLING FREE MECHANICAL OSCILLATIONS
 No Damping
 Light Damping
 Heavy Damping
 MODELLING FORCED MECHANICAL OSCILLATIONS
 COMPUTER EXERCISE OR ACTIVITY

Examples of PDE:
PDEs are used to model many systems in many
different fields of science and engineering.
Important Examples:
Laplace Equation
Heat Equation
Wave Equation

LAPLACE EQUATION:
• Laplace Equation is used to describe the
steady state distribution of heat in a body.
• Also used to describe the steady state
distribution of electrical charge in a body.
0
),,(),,(),,(
2
2
2
2
2
2









z
zyxu
y
zyxu
x
zyxu

HEATEQUATION:
• The function u(x,y,z,t) is used to represent the
temperature at time t in a physical body at a
point with coordinates (x,y,z)
•  is the thermal diffusivity. It is sufficient to
consider the case  = 1.



















2
2
2
2
2
2
),,,(
z
u
y
u
x
u
t
tzyxu


WAVE EQUATION:
• The function u(x,y,z,t) is used to represent the
displacement at time t of a particle whose
position at rest is (x,y,z) .
• The constant c represents the propagation
speed of the wave.



















2
2
2
2
2
2
2
2
2
),,,(
z
u
y
u
x
u
c
t
tzyxu

APPLICATIONS OF DIFFERENTIAL EQUATIONS-ZBJ

NEWTON’S SECOND LAW
• THE RATE OF CHANGE IN MOMENTUM
ENCOUNTERED BY A MOVING OBJECT IS
EQUAL TO THE NET FORCE APPLIED TO IT. IN
MATHEMATICAL TERMS,

Kirchhoff's law , sum of voltage drop across R and L = E

WHEN COMPARE TO
I.F IS
MULITIPLING I.F BOTH SIDES
When ‘t=0’ then ‘i=0’ we get c = - E/R

RADIOACTIVE HALF-LIFE
• A stochastic (random) process
• The RATE of decay is dependent upon the
number of molecules/atoms that are there
• Negative because the number is decreasing
• K is the constant of proportionality
kN
dt
dN


Law: The rate of change of the temperature of an
object is proportional to the difference between its
own temperature and the temperature of its
surroundings.
Therefore,
dθ / dt = E A (θ – θr ) ; E- A constant that depends upon
the object , A – surface area, θ – A certain
temperature, θr – Room/ ambient temperature or the
temperature of the surroundings.
Newton’s law of cooling

APPLICATIONS OF DIFFERENTIAL EQUATIONS-ZBJ

More Related Content

What's hot (20)

Viewers also liked (19)

Similar to APPLICATIONS OF DIFFERENTIAL EQUATIONS-ZBJ (20)

More from Zuhair Bin Jawaid (12)

Recently uploaded (20)

APPLICATIONS OF DIFFERENTIAL EQUATIONS-ZBJ