Applications of differential equations

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1. APPLICATIONS OF DIFFERENTIAL EQUATIONS PRESENTED BY PRESENTED TO Md . Sohag Em@il : [email protected] Daffodil international University

2. 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.”

3. 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,

4. 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 + 5 y = e x , ( dx / dt ) + ( dy / dt ) = 2 x + y PDE (PARTIAL DIFFERENTIAL EQUATION): An equation contains partial derivates of one or more dependent variables of two or more independent variables. For Example,

5. 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

6. GAME APPS DEveLOPMENT Game theorytic models ,building block concept and many applications are solve with differential Equation. graphical interference of analyzing data and creating browser based on partial differential equation solving with finite element method.

7. ROBOTIC INDUSTRIALIZATION Auto motion and robotic technologies for customized component, module and building Prefabrication are based on differential equation.

8. Motivating Examples Differential equations have wide applications in various engineering and science disciplines . In general , modeling variations of a physical quantity, such as temperature, pressure , displacement, velocity, stress, strain, or concentration of a pollutant, with the change of time t or location, such as the coordinates (x , y , z) , or both would require differential equations. Similarly, studying the variation of a physical quantity on other physical quantities would lead to differential equations . For example , the change of strain on stress for some viscoelastic materials follows a differential equation.

9. 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

10. 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.

11. 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 .

12. 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.

13. Physical applications ds / dt If a body be moving along with the time t respectly ,

14. 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

15. ELECTRIC R-L SERIES CIRCUIT Kirchhoff's law , sum of voltage drop across R and L = E Let a series circuit contain only a resistor and an inductor. By Kirchhoff’s second law the sum of the voltage drop across the inductor and the voltage drop across the resistor ( iR ) is the same as the impressed voltage (E(t)) on the circuit. Current at time t, i (t), is the solution of the differential equation.

16. 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

17. 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

18. Population Growth and Decay We have seen in section that the differential equation where N(t) denotes population at time t and k is a constant of proportionality, serves as a model for population growth and decay of insects, animals and human population at certain places and duration. Solution of this equation is : N(t)= Ce kt , where C is the constant of integration: Integrating both sides we get lnN (t)= kt+ln C or N(t)= Ce kt C can be determined if N(t) is given at certain time.

19. 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, d θ / dt = E A ( θ a – θ r ) ; E= constant that depends upon the object , A is the surface area, θ a certain temperature, θ r = Room/ ambient temperature or the temperature of the surroundings. Newton’s law of cooling

20. Applications on Newton’s Law of Cooling :

21. Example : A global company such as Intel is willing to produce a new cooling system for their processors that can cool the processors from a temperature of 50℃ to 27℃ in just half an hour when the temperature outside is 20℃ but they don’t know what kind of materials they should use or what the surface area and the geometry of the shape are. So what should they do ? Simply they have to use the general formula of Newton’s law of cooling T ( t ) = T e + ( T − T e ) e –t k And by substituting the numbers they get 27 = 20 + (50 − 20) e -0.5k Solving for k we get k =2.9 so they need a material with k=2.9 (k is a constant that is related to the heat capacity , thermodynamics of the material and also the shape and the geometry of the material)

22. EVERY ONE !!!!