2.2. NUMERICAL APPROACHES 9
Substituting (2.13) into (2.16) and discarding the higher-order terms, we obtain the approxi-
mating system
y.x
0
/ D L; i D 0
y.x
iC1
/ 2y.x
i
/ Cy.x
i1
/
h
2
D f
x
i
; y.x
i
/;
y.x
iC1
/ y.x
i1
/
2h
; 1 i N 1
y.x
N
/ D U; i D N (2.17a)
which gives N C 1 nonlinear algebraic equations in the N C 1 unknowns. ere are a number
of methods for solving nonlinear systems in literature such as Newton’s method, Trapezoidal
scheme with Newton’s method, and the Jacobian method [42, 43]. e main advantage of this
method is very easy to implement since the system is solved with a tri-diagonal matrix and the
disadvantage of the finite difference method is less accurate due to computation of the equations
by omitting the higher-order terms in the system. In addition, this does not work very well on
large intervals [44].
2.2.3 FINITE ELEMENT METHODS
Finite element methods are among the most extensively studied methods for modeling problems
with complex geometries in science and engineering for structural, mechanical, heat transfer, and
fluid dynamics problems [
45–48]. e idea behind the finite element method is to somehow
make a combination of known functions, satisfying the boundary conditions, which represent
the true solution in the given interval. Obviously, there is a wide choice in selecting these basis
functions such as the collocation, the Galerkin, the Rayleigh-Ritz, and the least squares [49–
53]. In all these methods, the conditions for determining the unknown coefficient vectors will
yield a system of algebraic equations in the boundary value problem.
e solution of the boundary value problem y.x/ can be obtained using a piecewise poly-
nomial approximation w.x/ by a linear combination of N functions as [54]
y.x/ w.x/ D
N
X
j D0
c
j
j
.x/; (2.18)
where the
j
.x/ are basis functions, each of which satisfies the given boundary conditions, and
the c
j
are unknown coefficients. e finite element methods differ only in how these c
j
are
determined as below.
Collocation Method
In this method, the coefficients c
j
are determined by satisfying the problem exactly at N C 1
points, fx
i
ji D 0; 1; : : : ; N g, the collocation points in the interval. Now, consider again the linear
case of a two-point boundary value problem given in (2.1) and this can be rewritten as
w
00
.x
i
/ k
1
.x
i
/ w
0
.x
i
/ k
0
.x
i
/ w.x
i
/ D ˛.x
i
/; i D 0; 1; : : : ; N: (2.19)
10 2. NUMERICAL AND ANALYTICAL METHODS ON BOUNDARY VALUE PROBLEMS
Substituting (2.18) into (2.19), we have
N
X
j D0
c
j
Œ
00
j
.x
i
/ k
1
.x
i
/
0
j
.x
i
/ k
0
.x
i
/
j
.x
i
/ D ˛.x
i
/; i; j D 0; 1; : : : ; N: (2.20)
is can be expressed in matrix notation as [49]
Œˆ
ij
.N C1/.N C1/
Œc
.N C1/1
D Œd
.N C1/1
; (2.21)
where
Œˆ
ij
D Œ
00
j
.x
i
/ k
1
.x
i
/
0
j
.x
i
/ k
0
.x
i
/
j
.x
i
/;
Œc D Œc
0
c
1
: : : c
N
t
;
Œd D Œ˛.x
0
/ ˛.x
1
/ ˛.x
N
/
t
:
Once the values of x
i
are selected then the system can be solved.
Galerkin Method
Here, the c
j
are obtained by defining a residual function that makes the approximation solution
orthogonal to all the basic functions f
i
.x/ji D 0; 1; : : : ; N g, on the interval. So, first define a
residual function r.x/ for the approximate solution w.x/ as [50]
r.x/ D w
00
.x/ k
1
.x/ w
0
.x/ k
0
.x/ w.x/ ˛.x/; l x u: (2.22)
If w.x/ are the exact solution then r.x/ should be exactly zero. en, in order to satisfy the
orthogonality property of the functions,
i
.x/, we have
u
Z
l
Œw
00
.x/ k
1
.x/ w
0
.x/ k
0
.x/ w.x/ ˛.x/
i
.x/ dx D 0: (2.23)
Now, substituting (2.18) into (2.23) to obtain for i D 0; 1; : : : ; N as
N
X
j D0
c
j
u
Z
l
Œ
00
j
.x/ k
1
.x/
0
j
.x/ k
0
.x/
j
.x/
i
.x/ dx D
u
Z
l
˛.x/
i
.x/ dx (2.24)
gives N C 1 linear equations with N C 1 unknown coefficients, c
j
, which can be solved. How-
ever, both the collocation and Galerkin methods can be used together in some boundary value
problems [51].
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.