2.3. ANALYTICAL APPROACHES 11
Rayleigh-Ritz Method
e basic idea behind this method is, to first determine the unknown coefficients c
j
to replace
the boundary value problem by a variational formulation and then choose these c
j
appropriately
to minimize this formulation related to the problem [52].
Consider a simple case of the problem by taking k
1
D 0; L D 0; U D 0 in (2.10) and re-
arranging as
w
00
.x/ C k
0
.x/ w.x/ D ˛.x/; l < x < u with y.l/ D 0; y.u/ D 0: (2.25)
Now, defining the functional as [24]
I.w/ D
u
Z
l
Œk
1
.x/
w
0
.x/
2
C k
0
.x/
.
w.x/
/
2
2˛.x/w.x/ dx (2.26)
for any w.x/ 2 C
2
Œl; u subspace. en, the solution w.x/ is first approximated by the basic
function which is defined in (2.18) and substituted into (2.26) as
I
0
@
N
X
j D0
c
j
j
.x/
1
A
D
u
Z
l
(
k
1
.x/
0
@
N
X
j D0
c
j
0
j
.x/
1
A
2
C k
0
.x/
0
@
N
X
j D0
c
j
j
.x/
1
A
2
2˛.x/
N
X
j D0
c
j
j
.x/
)
dx:
(2.27)
Next the coefficients c
j
are minimized as
@
@c
i
I
0
@
N
X
j D0
c
j
j
.x/
1
A
D 0; i D 0; 1; : : : ; N (2.28)
and these N C 1 equations can be simplified by using Spline or Piecewise cubic polynomials [55,
56].
2.3 ANALYTICAL APPROACHES
e analytical methods can compete with the numerical methods in most real physical pro-
cesses on boundary value problems [57]. Generally, in analytical approaches we replace the given
boundary value problems by some analytically equivalent mathematical models and attempt to
solve it numerically. ere are a vast number of analytical methods for solving boundary value
problems in practical sciences [28, 58, 59]. We briefly describe some of them in what follows.
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