An Effective Approximation Algorithm for SecondOrder Singular Functional Differential Equations
Abstract
:1. Introduction
2. The Bessel Matrix Technique
Algorithm 1: The computation of sderivative of the vector ${\mathbf{\Xi}}_{N}\left(x\right)$. 

Initial Conditions in the Matrix Form
3. Computational Simulations
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Bessel ($\mathit{N}=3$)  ANNs ($\mathit{\ell}=1$) [17]  

x  $\mathit{\ell}=1$  $\mathit{\ell}=5$  $\mathit{\ell}=20$  Min  Mean  S.D 
$0.1\ell $  ${9.2006}_{19}$  ${5.4584}_{16}$  ${1.1589}_{14}$  ${6.20}_{09}$  ${1.42}_{05}$  ${2.03}_{05}$ 
$0.2\ell $  ${3.3134}_{18}$  ${1.9357}_{15}$  ${3.3996}_{14}$  ${1.60}_{07}$  ${7.36}_{05}$  ${4.17}_{04}$ 
$0.3\ell $  ${6.6298}_{18}$  ${3.7979}_{15}$  ${4.8678}_{14}$  ${5.21}_{07}$  ${1.44}_{04}$  ${8.90}_{04}$ 
$0.4\ell $  ${1.0319}_{17}$  ${5.7610}_{15}$  ${3.7092}_{14}$  ${2.86}_{07}$  ${2.16}_{04}$  ${1.37}_{03}$ 
$0.5\ell $  ${1.3831}_{17}$  ${7.4534}_{15}$  ${1.9303}_{14}$  ${1.34}_{07}$  ${2.80}_{04}$  ${1.80}_{03}$ 
$0.6\ell $  ${1.6615}_{17}$  ${8.5035}_{15}$  ${1.3905}_{13}$  ${1.72}_{07}$  ${3.24}_{04}$  ${2.11}_{03}$ 
$0.7\ell $  ${1.8121}_{17}$  ${8.5398}_{15}$  ${3.4069}_{13}$  ${1.96}_{07}$  ${3.48}_{04}$  ${2.30}_{03}$ 
$0.8\ell $  ${1.7799}_{17}$  ${7.1907}_{15}$  ${6.4277}_{13}$  ${6.20}_{07}$  ${3.58}_{04}$  ${2.40}_{03}$ 
$0.9\ell $  ${1.5099}_{17}$  ${4.0846}_{15}$  ${1.0638}_{12}$  ${9.39}_{07}$  ${3.67}_{04}$  ${2.47}_{03}$ 
$1.0\ell $  ${9.4700}_{18}$  ${1.1500}_{15}$  ${1.6224}_{12}$  ${5.82}_{07}$  ${3.79}_{04}$  ${2.56}_{03}$ 
Bessel  ANNs ($\mathit{\ell}=1$) [17]  

x  $\mathit{\ell}=1,\mathit{N}=10$  $\mathit{\ell}=2,\mathit{N}=20$  $\mathit{\ell}=5,\mathit{N}=30$  Min  Mean  S.D 
$0.1\ell $  ${1.4633}_{6}$  ${1.9321}_{9}$  ${1.7737}_{7}$  ${3.30}_{8}$  ${4.51}_{5}$  ${8.65}_{5}$ 
$0.2\ell $  ${4.7314}_{6}$  ${9.8570}_{9}$  ${2.8830}_{7}$  ${1.09}_{6}$  ${1.81}_{4}$  ${3.67}_{4}$ 
$0.3\ell $  ${8.2497}_{6}$  ${2.3617}_{8}$  ${9.0792}_{8}$  ${3.79}_{7}$  ${4.22}_{4}$  ${8.68}_{4}$ 
$0.4\ell $  ${1.0729}_{5}$  ${4.0088}_{8}$  ${1.1621}_{7}$  ${2.04}_{6}$  ${7.52}_{4}$  ${1.59}_{3}$ 
$0.5\ell $  ${1.1221}_{5}$  ${5.5145}_{8}$  ${3.1451}_{8}$  ${1.05}_{5}$  ${1.16}_{3}$  ${2.50}_{3}$ 
$0.6\ell $  ${9.1480}_{6}$  ${6.5255}_{8}$  ${3.1422}_{7}$  ${1.51}_{5}$  ${1.58}_{3}$  ${3.50}_{3}$ 
$0.7\ell $  ${4.2955}_{6}$  ${6.8403}_{8}$  ${6.4486}_{7}$  ${2.80}_{7}$  ${1.96}_{3}$  ${4.47}_{3}$ 
$0.8\ell $  ${3.2249}_{6}$  ${6.4385}_{8}$  ${6.9379}_{7}$  ${2.02}_{5}$  ${2.26}_{3}$  ${5.30}_{3}$ 
$0.9\ell $  ${1.3031}_{5}$  ${5.4593}_{8}$  ${3.7913}_{7}$  ${3.87}_{6}$  ${2.43}_{3}$  ${5.85}_{3}$ 
$1.0\ell $  ${2.4539}_{5}$  ${4.1499}_{8}$  ${1.7989}_{7}$  ${5.39}_{6}$  ${2.45}_{3}$  ${6.05}_{3}$ 
Bessel  ANNs ($\mathit{\ell}=1$) [17]  

x  $\mathit{\ell}=1,\mathit{N}=10$  $\mathit{\ell}=\mathit{\pi},\mathit{N}=20$  $\mathit{\ell}=2\mathit{\pi},\mathit{N}=30$  Min  Mean  S.D 
$0.1\ell $  ${1.3172}_{8}$  ${4.9934}_{12}$  ${5.9707}_{11}$  ${3.51}_{8}$  ${1.03}_{6}$  ${1.29}_{6}$ 
$0.2\ell $  ${3.4686}_{8}$  ${1.3423}_{11}$  ${7.7946}_{11}$  ${7.83}_{9}$  ${2.37}_{6}$  ${2.34}_{6}$ 
$0.3\ell $  ${4.1486}_{8}$  ${1.7895}_{11}$  ${8.3353}_{11}$  ${8.63}_{8}$  ${3.90}_{6}$  ${3.94}_{6}$ 
$0.4\ell $  ${1.8096}_{8}$  ${2.0670}_{11}$  ${7.7780}_{11}$  ${6.81}_{9}$  ${5.52}_{6}$  ${5.42}_{6}$ 
$0.5\ell $  ${4.1676}_{8}$  ${2.2442}_{11}$  ${6.2475}_{11}$  ${8.48}_{8}$  ${6.83}_{6}$  ${6.74}_{6}$ 
$0.6\ell $  ${1.3460}_{7}$  ${2.3120}_{11}$  ${3.7118}_{11}$  ${6.03}_{8}$  ${7.52}_{6}$  ${7.82}_{6}$ 
$0.7\ell $  ${2.4942}_{7}$  ${2.3020}_{11}$  ${1.3218}_{12}$  ${1.37}_{7}$  ${7.74}_{6}$  ${8.37}_{6}$ 
$0.8\ell $  ${3.6947}_{7}$  ${2.2240}_{11}$  ${4.5159}_{11}$  ${1.28}_{7}$  ${7.61}_{6}$  ${8.51}_{6}$ 
$0.9\ell $  ${4.7614}_{7}$  ${2.0646}_{11}$  ${1.0211}_{10}$  ${8.94}_{9}$  ${7.53}_{6}$  ${8.38}_{6}$ 
$1.0\ell $  ${5.5254}_{7}$  ${1.8200}_{11}$  ${1.6874}_{10}$  ${2.68}_{8}$  ${7.70}_{6}$  ${8.27}_{6}$ 
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Izadi, M.; Srivastava, H.M.; Adel, W. An Effective Approximation Algorithm for SecondOrder Singular Functional Differential Equations. Axioms 2022, 11, 133. https://doi.org/10.3390/axioms11030133
Izadi M, Srivastava HM, Adel W. An Effective Approximation Algorithm for SecondOrder Singular Functional Differential Equations. Axioms. 2022; 11(3):133. https://doi.org/10.3390/axioms11030133
Chicago/Turabian StyleIzadi, Mohammad, Hari M. Srivastava, and Waleed Adel. 2022. "An Effective Approximation Algorithm for SecondOrder Singular Functional Differential Equations" Axioms 11, no. 3: 133. https://doi.org/10.3390/axioms11030133