Mathematical Methods for Physicists: A concise introduction - Site Map

SIMPLE LINEAR INTEGRAL EQUATIONS
Then using the relation
Z t
a
Z t
a
f …y†dydt ˆ
we can rewrite the last equation as
x…t† ˆ
‡
Z t
a
Z t
a
Z t
a
…t y† f …y†dy;
A…y†‡…t y† B…y†A 0 …y†
x…y†dy
…t y†g…y†dy ‡ A…a†x 0 ‡ x0
0
…t a†‡x0 ; …11:38†
which can be put into the form of a Volterra equation of the second kind
with
x…t† ˆf …t†‡
Z t
a
K…t; y†x…y†dy;
K…t; y† ˆ…y t†‰B…y†A 0 …y†Š A…y†;
…11:39†
…11:39a†
f …t† ˆ
Z t
0
…t y†g…y†dy ‡‰A…a†x 0 ‡ x 0 0Š…t a†‡x 0 :
…11:39b†
Use of integral equations
We have learned how linear integral equations of the more common types may be
solved. We now show some uses of integral equations in physics; that is, we are
going to state some physical problems in integral equation form. In 1823, Abel
made one of the earliest applications of integral equations to a physical problem.
Let us take a brief look at this old problem in mechanics.
Abel's integral equation
Consider a particle of mass m falling along a smooth curve in a vertical plane, the
yz plane, under the in¯uence of gravity, which acts in the negative z direction.
Conservation of energy gives
1
2 m… _z2 ‡ _y 2 †‡mgz ˆ E;
where _z ˆ dz=dt; and _y ˆ dy=dt: If the shape of the curve is given by y ˆ F…z†, we
can write _y ˆ…dF=dz† _z. Substituting this into the energy conservation equation
and solving for _z, we obtain
p
p
2E=m2gz E=mgz
_z ˆ q
ˆ
; …11:40†
1 ‡…dF=dz† 2 u…z†
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