Page 136 - IJOCTA-15-3

Srinivasarao Thota et al. / IJOCTA, Vol.15, No.3, pp.503-516 (2025)
Table 1. Summary of convergence

Analysis type Result
n
Local convergence Ninth-order convergence (e ) confirmed via Taylor expansion and
9
error analysis
Semi-local convergence The method is robust under smoothness conditions and has a large
basin of attraction
Practical implications Faster convergence than the Newton–Raphson; better stability for
distant initial guesses

The approximate solution is x = 6.308777130.
t 1 = 1.4044916466908547473726183272184, This example further highlights the simplicity of
s 1 = 1.4044916482153412286835185849637, the proposed algorithm. It can be easily applied
to real-world problems involving nonlinear equa-
x 2 = 1.40449164821534122520772018543,
tions.
|f(x 2 )| = 0.000000000000000002053915, Expanding its utility to engineering applica-
tions, Example 3 examines the open-channel flow

x 2 − x 1
= 0.00101435436464643787660. problem, originally discussed as Example 3 in

x 2 38
Kalantari and Hans Lee. It remains a challenge
Iteration 3
to relate water flow to elements like drainage
ditches, gutters, sewers, and canals, all of which
t 2 = 1.404491648215341226035086817786868,
affect flow dynamics within open channels. The
s 3 = 1.404491648215341226035086817786868, flow rate in such cases refers to the volume of wa-
x 4 = 1.404491648215341226035086817786868, ter that passes through a certain region over a
given time. A particularly problematic scenario
|f(x 3 )| = 0.0000000000000000000000000000,
arises when the channel is poorly maintained.

x 3 − x 2 The water flow in an open channel with uniform
= 0.000000000000000000589086.
flow conditions is determined by Manning’s equa-
x 3
It is evident that f(x 3 ) = 0, hence confirming tion:
the approximate root as: √ 2/3
m Wh
Q = Wh , (15)
n W + 2h
x = 1.404491648215341226035086817786868
where m is the slope of the channel, n is the
as stated in Equation (14). Manning’s roughness coefficient, W is the channel
This example demonstrates that the proposed width, and h is the depth of water. The equation
algorithm rapidly converges to the approximate above can be simplified to determine the water
root. depth (h) in the channel for a given quantity of
Example 2. Further reinforces the applica- water, as follows:
bility of the proposed method by solving a tran-
√ 2/3
scendental equation with an initial approximation m Wh
f(h) = Wh − Q. (16)
x 0 = 5. The stopping conditions are |f(x)| ≤ ε or n W + 2h

≤ δ, where ε = 10 , δ = 10 . 38
x n+1 −x n −200 −600
In Kalantari and Hans Lee, the authors con-
x n+1
3
sidered the parameters: Q = 14.15 m /s, W =
5
x + x − 10000 = 0 4.572 m, n = 0.017, and m = 0.0015, with the ini-
tial guess h 0 = 8.5 m. Substituting these values
Following the proposed algorithm as demon- into Equation (16) yields:
strated in Example 1, we obtained the following
2/3
values during the iteration: h
f(h) = 28.69285373h − 14.15.
4.572 + 2h
t 2 = 6.308777130,
(17)
s 2 = 6.308777130, Applying the proposed algorithm, the follow-
x 3 = 6.308777130, ing results were obtained:
Iteration 1: 1.446410377, f(h) = 0.25324599,
f(x 3 ) = 0,
Iteration 2: 1.465091222, f(h) = 0.2 × 107,

x 3 − x 2 and
= 0.

x 3
Iteration 3: 1.465091220, f(h) = 0.
508