MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 4 pptx
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 4 pptx
... equations in
physics and engineering can be solved by the method of separation
of
variables.
This method helps us to reduce a second-order partial differential equation
into
a
set of ordinary ... following integral definitions:
(6 .48 )
and
1
2
(1
-
t2)n-6
cosztdt,
(n
>
).
(6 .49 )
Jn(IL.1
=
6 .4
RECURSION RELATIONS
OF
THE BESSEL FUNCTIONS
Using the s...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 16 pptx
...
by determining the
u
and the
solutions in Equation (19.15)
as
sin
bx
and sin
ko(x
-
L),
respectively. We can now obtain a
closed
expression for
G(z,
d)
as
sin
kox
sin
ICg(x’ ...
dk'g(k')eikfZ
VG
-"
and
Their inverse Fourier transforms are
and
Using these in Equation (19.116) we get
which gives
us
Substituting this in Equation (19.118) we...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 1 potx
...
14. 7
Applications
of
Diflerintegrals in Science
and
Engineering
42 4
14. 7.2
Fractional Fokker-Planck Equations
42 7
Problems
42 9
14. 7.1 Continuous Time Random Walk (CTRW)
42 4
15 INFINITE ...
44 0
15.5 Useful Inequalities About Series
44 2
15.6 Series
of
Functions
44 2
15.6.1 Uniform Convergence 44 3
15.6.2 Weierstrass M-Test
44 3
15.6.3 Abel Test
44 4...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 2 pdf
... part
of the learning process.
In a vast area like mathematical methods in science and engineering, there
is
always room for new approaches,
new
applications, and new topics. In fact,
the number ... write
x
=
fl
in the generating function
Equation
(2.65)
we find
(2.85)
Expanding the left-hand side by using the binomial formula and comparing
equal powers oft, w...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 3 docx
...
as
(4. 53)
and defining
a
new parameter
(4. 54)
50
-
=
t,
2
and using the generating function
[Eq.
(4. 25)],
we obtain
(4. 55)
We now use the orthogonality relation
[Eq.
(4. 46)] ...
(z)
,
(n
2
m).
(4. 41)
J-00
Using the Rodriguez formula, we write Equation
(4. 41)
as
After n-fold integration by parts and since
n
>
m,
we obtain
(4. 43)
Since...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 5 ppsx
... exists
a
minimum value,
mmin,
thus determining
X
as
X
=
mmin
(9. 143 )
To
find
mmin
we equate the two expressions
[Eqs.
(9. 141 )
and
(9. 143 )]
for
X
to obtain
(9. 144 )
mmin
=
-1. ...
Substituting this into Equation (9.83) and integrating gives
us
ko(z)
=
cacota(z
+p)
+
.
d
sin+
+
p)
’
(9.89)
where
p
and
d
are integration constants.
With thes...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 6 pps
...
1
0
-
sin
$
cosCp sin4cos$
cosdsin+ -sin4 cos~cos$
(10.93)
Reversing the order we get
cos
$
(10. 94)
sin
$
sin
Cp
-
sin
$
cos
Cp
R2R1
=
0
cos
Cp
sin
4
[
sin
$
-
cos ... choosing three mutually orthogonal straight lines.
A
point is
defined by giving its coordinates,
(q,z2,q),
or by using the position vector
163
188
COORDINATES AND TENSORS
Hence
aa...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 7 pdf
...
be expressed in terms of the poten-
tials
-A'
and
4
as
In the Lorentz gauge
2
and
4
satisfy
and
(10.373)
(10.3 74)
(10.375)
(10.376)
(10.377)
respectively. Defining
a
four-potential ... transformation. It
gives
i3,
as
a
linear
Ah'
combination
of
Gar.
Using
we obtain
(10. 340 )
(10. 341 )
(10. 342 )
222
COORDINATES AND TENSORS
and
10.2...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 8 pps
...
(Q 74) Km(Q,
4)
=
4P
fimmf
and the completeness relation
(1 1. 240 )
(
1
1. 24 1)
c c
Y,L(Q’ ,4 )Km(B ,4)
=
G(cosQ-cosQ’)6 (4- 4’),
I=O
m=-I
(11. 242 )
where
dQ
=
sin
Qdsdq5.
In the ...
256
CONTINUOUS GROUPS AND REPRESENTATIONS
and
(11. 244 )
aa
Glrn(e ,4)
=
o(-
-,Q ,4) Krn(Q ,4) ,
a6
'
a4
where
O($,
&,
8 ,4)
is some differential operator...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 9 pps
...
(11 .44 1)
(11 .44 2)
(11 .44 3)
(11 .44 4)
12
COMPLEX
VARIABLES
and
FUNCTIONS
Even though the complex numbers do not exist directly in nature, they are
very useful in physics and engineering applications: ... dimensions.
2.
The method of analytic continuation
is
very useful in finding solutions
of differential equations and evaluating some definite integrals....
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