Transcript Document

Green functions is a type of function
that solves inhomogeneous
differential equations with initial
values or boundary conditions.
The concept of Green's functions
was developed in 1830 by George
Green. George Green was a early
nineteenth century self-taught
English mathematician
Drawing of a RC circuit where R represents a
resistor, C represents a capacitor, and ε
represents a emf which is our driving force (ex.
battery)
where I is the current, R is the resistance, C is
the capacitor, ε is the emf (battery), and q is the
charge.
Using the Green's Function, the solution to a
inhomogeneous system with a general force
function can be rewritten by superimposing the
effects of unit impulses.
where ε is the driving force and G(t',t) is the
Green's Function and it contains everything
about system
Green's function must be of the following form by
causality.
Must solve the differential equation
Which can be reduced to
This gives equations whose solution can be
superimposed to give g(t)
This gives the g(t,t') to be
Which then tells us what G(t,t') is using the
following equation
Hence,
Ultimately, we are now able to solve for charge
as a function of time using
The initial conditions for this problem is that q=0 at
t=0 because a capacitor is initially uncharged. The
emf for our problem is
Combining the following equations
Yields
From here, it is a matter of solving the integral to
get the charge as a function of time.