Experiment 7 Differential Equation First Order ( Euler Method )

      Differential Equation First Order ( Euler Method )

CONCEPT:-

(a)(Differential Equation - First Order) In many problems, the direct functional relation between the dependent variable y and the independent variable x is not known. However, the rate of change in y with respect to x is known and is given by a function f(x, y). The idea is to deduce the function y(x) from the rate equation                             

                                                             dy/dx = f(x, y(x)) 

(b)(Initial Value Problem) Any DE represent a family of curves/surfaces, however most problems pertains to picking out a particular curve which satisfies the given conditions. In nutshell, we seek a solution y(x) which satisfies the initial conditions 

                                     

                                                                    y(x0) = y0 

CODE/METHOD 

(a) (Euler Method - Forward) The derivative is approximated as y' = {y(x + h) − y(x)}/h and the ODE is represented by 

                               {y(x + h) − y(x)}/h = f(x, y(x)) ⇒ y(x + h) = y(x) + hf(x, y(x)) 

 The domain (a, b) is populated with N equispaced nodal points to get (a, x1, ..., xN , b) representative points where xi = a + ih and h = (b − a)/(N + 1). The corresponding yi is evaluated using the above recursive relation to yield (y0, y1, ..., yN , yN+1)

APPLICATION :-

(a) (Radioactive Decay) The time t and the population N bears the relation dN dt = −λN. Note f(t, N) = −N/τ where τ = 1/λ. 

(b)(RC Circuit) The time t and the voltage V bears the relation dV /dt = −V /τ where τ = RC. Note f(t, V ) = −V /τ . 

(c)(Stokes’ Law) The time t and the speed v bears the relation mdv/dt = −6πηav where all variables have their usual meaning. Note f(t, v) = −v/τ where τ = m/6πηa. Plot the time evolution of the respective solutions and highlight the main features of the solution. Make and display the tabulated output with time

Note Initial Values will be Given In the Question.

SCILAB CODE FOR THE GIVEN PROBLEM 

//***Euler MEthod****

/****************************

clc, clear, clf

a=input("enter the value of left boundary condition for x of RC circuit= ")

b=input("enter the value of right boundary condition for xof RC circuit= ")

k=input("enter the value of left boundary condition for x of Radioactive

decay= ")

l=input("enter the value of right boundary condition for x of Radioactive

decay- ")

e=input("enter the value of left boundary condition for x of viscosity= ")

f=input("enter the value of right boundary condition for x of visxosity= ")

N=50

//RC Circuit

A= 5 //initial value of y in volts

Az=A/2 //half life

R= 1000 //value of R in ohms

C=0.000001 //value of C in Farads

u=1/(R*C) //u is a constant

hf1= log(0.5)/(-u)

h=(b-a)/(N+1) // calculating the value of step size for RC circuit

x(1)=a;y(1)=A; //initialising x and y

for i=1:N+1

y(i+1)= y(i)+h*(-u*y(i)) //forward difference formula

x(i+1)= x(i) + h

end

ex= A*exp(-u*x)

disp([x,y])

scf

plot(x,y,'*')

plot(hf1,Az,"*k")

plot(x,ex, 'r')

xlabel("time(sec))")

ylabel("voltage(v)")

set(gca(), 'grid',[1 1]*color('red'))

title ("RC Circuit")

note - With Limits 0 to 0.01 we will get the graph like this 

                                     

//Radioactive Decay

lam= 0.3 //value of lambda

Nd= 10,000 //valyue of N- initial no of atoms

Ndz= Nd/2 //half life

dt= -lam//dt is a constant

hf2= log(0.5)/dt

ss= (l-k)/(N+1) //calculating the step size for Radioactive decay

xd(1)=k; yd(1)= Nd; //initialising x and y

for i=1:N+1

yd(i+1)=yd(i)+ss*(dt*yd(i)) //forward difference formula

xd(i+1)= xd(i) + ss

end

ex2= Nd*exp(dt*xd)

disp([xd,yd])

scf

plot(xd,yd,'*')

plot(xd,ex2, 'r')

plot(hf2,Ndz,"*k")

xlabel("time(sec)")

ylabel("Amount of substance(in thousands)")

set(gca(), 'grid',[1 1]*color('purple'))

title("radioactive decay")

//viscosity

Vo=1 //initial velocity in m s^-1

m=0.25 //mass of object in kg

sig= 0.09 //value of neta in kg m^-1 s^-1

Rv=0.007 // value of R

vt= -(6*%pi*sig*Rv)/m //vt is constant

sv= (f-e)/(N+1) //calculating the step size for viscosity

Vz= Vo/2 //half life

hf3=log(0.5)/ vt

xv(1)= e; yv(1)=Vo; //initialising x and y

for i=1:N+1

yv(i+1)=yv(i)+sv*(vt*yv(i)) //forward difference formula

xv(i+1)= xv(i) + sv

end

ex3= Vo*exp(vt*xv)

disp([xv,yv])

scf

plot(xv,yv,'*')

plot(xv,ex3, 'r')

plot(hf3,Vz,"*k")

xlabel("time(sec)")

ylabel("velocity(m/s)")

set(gca(), 'grid',[1 1]*color('green'))

title("viscosity")

Make Adjustment according to Your Given Problem .

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//university of Delhi //Department of Physics