APPLICATIONS OF DIFFERENTIAL EQUATIONS

PRESENTED TO:DR.SADIA ARSHADPRESENTED BY:ASHHAD ABBAS GILANI(026)

SHAHAB ARSHAD(058) RIAZ HUSSAIN(060)

MUHAMMAD YOUSUF(082) ZUHAIR BIN JAWAID(094)

INVENTION OF 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 ORDER ODE:

• 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 xdx

dy

INITIAL AND BOUNDARY 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 EquationHeat EquationWave 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

HEAT EQUATION:

• 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

22

2

2

),,,(

z

u

y

u

x

uc

t

tzyxu

Physical applications

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,

F = m a

ELECTRIC R-L SERIES CIRCUIT

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

THIS CIRCUIT CAN SOLVE BY INTERGRATION FACTOR

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

kNdt

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