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Consider the first order differential equation

\[\frac{dy}{dy} = f(t,y) = -16 t^3 y^2\]

with initial condition $y(0)=1$

Using second order Adams-Bashforth method, write a Fortran programming to generate an approximate solution to the problem.

Solution

Program adams

Implicit None

Real, allocatable :: $y),t)$

Real:: yo, tend, h, k1,k2

Integer:: NI, i

Real,external ::f

!Asking to enter the initial vaule of yo, final time and numbers of step

Print*, 'Enter yo,Tend,NI'

read*, yo,Tend,NI

h= Tend/NI

Print*, 'This gives stepsize h=',h

allocate (t(0:NI), y(0:NI))

!Initial Conditions

t(0)=0

y(0) = 1

!After using runge kutta method, found out k1 =0 and k2= -16h^3,

k1=0

k2= -16*h**3

!we know that y(n+1) =y(n) + h/2(k1+K2) at n=0

y(1) = y(0) + h/2 *( k1+ k2)

! Loop through the number of steps to calculate the following at each step

do i=2, NI

t(i)= i*h

write(10,*) i, t(i),t(i-1)

!Second order Adam bashforth for all n

y(i)= y(i-1) + (h/2)*(3*f(t(i-1), y(i - 1))- f(t(i-2), y(i-2)))

end do

end program adams

!declaring function

Real Function f(t,y)

Real:: t

Real:: y

f = - 16*t**3*y**2

Return

End Function f

Now i want to plot this partial derivative ∂f/∂y = |-32 (4t^4 +1)^-1 t^3| against t using fortran to generate a file or something for the matlab to plot the graph.

Can someone help me

\[\frac{dy}{dy} = f(t,y) = -16 t^3 y^2\]

with initial condition $y(0)=1$

Using second order Adams-Bashforth method, write a Fortran programming to generate an approximate solution to the problem.

Solution

Program adams

Implicit None

Real, allocatable :: $y),t)$

Real:: yo, tend, h, k1,k2

Integer:: NI, i

Real,external ::f

!Asking to enter the initial vaule of yo, final time and numbers of step

Print*, 'Enter yo,Tend,NI'

read*, yo,Tend,NI

h= Tend/NI

Print*, 'This gives stepsize h=',h

allocate (t(0:NI), y(0:NI))

!Initial Conditions

t(0)=0

y(0) = 1

!After using runge kutta method, found out k1 =0 and k2= -16h^3,

k1=0

k2= -16*h**3

!we know that y(n+1) =y(n) + h/2(k1+K2) at n=0

y(1) = y(0) + h/2 *( k1+ k2)

! Loop through the number of steps to calculate the following at each step

do i=2, NI

t(i)= i*h

write(10,*) i, t(i),t(i-1)

!Second order Adam bashforth for all n

y(i)= y(i-1) + (h/2)*(3*f(t(i-1), y(i - 1))- f(t(i-2), y(i-2)))

end do

end program adams

!declaring function

Real Function f(t,y)

Real:: t

Real:: y

f = - 16*t**3*y**2

Return

End Function f

Now i want to plot this partial derivative ∂f/∂y = |-32 (4t^4 +1)^-1 t^3| against t using fortran to generate a file or something for the matlab to plot the graph.

Can someone help me

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