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Intelligent computing approach to analyze the dynamics of wire coating with Oldroyd 8-constant fluid

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Abstract

In the study, intelligent computing technique is developed for solving the nonlinear system for wire coating analysis with the bath of Oldroyd 8-constant fluid having pressure gradient using feedforward artificial neural networks (ANNs), evolutionary computing, active-set algorithm (ASA) and their hybrid. Original partial differential equations of wire coating process are converted to nonlinear ordinary differential equation (NL-ODEs) in dimensionless form using similarity transformation. Strength of ANNs is exploited to develop mathematical model of the transformed equations by defining an unsupervised error. Training of design variables of the network is carried out globally using evolutionary computing techniques based on genetic algorithms (GAs) hybrid with ASA for rapid local convergence. Design scheme is applied to analyze the dynamics of the problem for number of variants based on dilatant constant, the pseudoplastic constant, the pressure gradient, shear stress under the effect of viscosity parameter and varying the coating thickness of the polymer. Results of the proposed method are compared with standard numerical solver for NL-ODEs based of Adams method to establish its correctness. Reliability of the method is further validated through the results of statistics based on different performance measures for accuracy and computational complexity.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan

    Annum Munir

  2. Department of Mathematics, Preston University Kohat, Islamabad Campus, Kohat, Pakistan

    Annum Munir

  3. Hamdard Institute of Engineering and Technology, Hamdard University, Islamabad, Pakistan

    Muhammad Anwaar Manzar

  4. Department of Mathematical Sciences, University of Karachi, Karachi, 75270, Pakistan

    Najeeb Alam Khan

  5. Department of Electrical Engineering, COMSATS Institute of Information Technology, Attock, Pakistan

    Muhummad Asif Zahoor Raja

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  1. Annum Munir
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  2. Muhammad Anwaar Manzar
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  3. Najeeb Alam Khan
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  4. Muhummad Asif Zahoor Raja
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Correspondence to Muhammad Anwaar Manzar.

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Appendices

Appendix 1

In this appendix, the solution of nonlinear Riccati differential equation (NRDE) with known exact solution is provided for general understanding of the method.

The quadratic NRDE with associated condition is given as:

$$\begin{aligned} w^{\prime } (r) & = - w^{2} + 1,\quad 0 \le r \le 1, \\ w(0) & = 0, \\ \end{aligned}$$

with exact solution as:

$$w(r) = \frac{{{\text{e}}^{2r} - 1}}{{{\text{e}}^{2r} + 1}}.$$

The fitness function as defined in Eq. (11) for 10 input grid points, i.e., r = 0.1, 0.2, …, 1 NRDE is written as:

$$\varepsilon = \frac{1}{10}\,\sum\limits_{m = \,1}^{10} {\left( {w_{m}^{\prime } + w_{m}^{2} - 1} \right)^{2} + w_{0}^{2} }.$$

Optimization of the fitness function \(\varepsilon\) is performed using the settings of the parameter as given in Table 1 for GA and ASA using the built-in routines of optimization toolbox. By following the similar procedure as given in pseudocode in step-wise form, we obtained the following final weight tabulated in Table 8.

Table 8 One set of weight values for nonlinear Riccati differential equation
Full size table

Using the weights of Table 8 in Eq. (8), one may derive the expression for the solutions for NRDE as given in “Appendix 3”. The solution obtained by the weights of Table 8 is compared with the exact results, and it is found matching of results of the order 6–7 decimal places of the accuracy. The procedure of the proposed provided in the appendix can simply extending for solving the highly nonlinear and stiff models arising in different applications of practical interest.

Appendix 2

The solution parameters \(k_{XX}\) and \(\varLambda_{xx}\) of optimal homotopy asymptotic method are defined as:

$$\begin{aligned} \varLambda_{11} & = 1 - 0.25\varOmega , \\ \varLambda_{12} & = 0.25\varOmega , \\ \varLambda_{13} & = - \frac{0.25}{\ln \delta }\left( {4 - \varOmega + \varOmega \delta^{2} } \right), \\ \varLambda_{14} & = C_{1} \left( {\varLambda_{12} \varLambda_{13}^{4} \alpha \beta - 0.5\varLambda_{13}^{3} \alpha + 0.5\varLambda_{13}^{3} \beta - 0.25\varLambda_{13}^{4} \varOmega \beta^{2} } \right), \\ \varLambda_{15} & = - \varLambda_{12} - \varLambda_{12} C_{1} + \cdots - 0.25\varLambda_{13}^{4} \varOmega \beta^{2} C_{1} , \\ \varLambda_{16} & = \varLambda_{12} + \varLambda_{12} C_{1} - \cdots - 6\varLambda_{12}^{2} \varLambda_{13}^{3} \varOmega \beta^{2} C_{1} , \\ \varLambda_{17} & = C_{1} \left( {2\varLambda_{12}^{3} \alpha - 0.5\varLambda_{12}^{3} \varOmega \beta + 6\varLambda_{12}^{4} \varLambda_{13} \alpha \beta - 2\varLambda_{12}^{4} \varLambda_{13} \varOmega \beta^{2} } \right), \\ \varLambda_{18} & = 9C_{1} \left( {16\varLambda_{12}^{5} \alpha \beta - 4\varLambda_{13}^{3} \varOmega \beta^{2} } \right), \\ \varLambda_{19} & = \frac{1}{\ln \delta }\left( {\varLambda_{12} - 0.25\varOmega + \cdots - \varLambda_{13}^{2} \varOmega \beta C_{1} } \right), \\ k_{10} & = \frac{1}{9}\varLambda_{13}^{4} \varLambda_{14} \alpha \beta C_{1} , \\ k_{11} & = C_{1} \left( {1.5\varLambda_{13}^{2} \varLambda_{14} \alpha - \varLambda_{13}^{2} \varLambda_{14} \beta + \varLambda_{13}^{4} \varLambda_{14} \varOmega \beta^{2} } \right), \\ k_{12} & = \varLambda_{14} + \varLambda_{14} C_{1} + \cdots + \varLambda_{13}^{3} \varLambda_{19} \varOmega \beta^{2} C_{1} , \\ k_{13} & = - \varLambda_{14} - \varLambda_{16} - \cdots - 2\varLambda_{13}^{3} \varLambda_{14} \varLambda_{17} \varOmega \beta^{2} C_{1} , \\ k_{14} & = \varLambda_{16} + \varLambda_{16} C_{1} + \cdots - 12\varLambda_{12}^{2} \varLambda_{13} \varLambda_{19} \varOmega \beta^{2} C_{1} , \\ k_{15} & = \varLambda_{17} + \varLambda_{17} C_{1} + \cdots - 2\varLambda_{12}^{3} \varLambda_{19} \varOmega \beta^{2} C_{1} , \\ k_{16} & = \varLambda_{18} + \varLambda_{18} C_{1} + \cdots - 4\varLambda_{12}^{2} \varLambda_{19} \varOmega \beta^{2} C_{1} , \\ k_{17} & = 9\varLambda_{12}^{2} \varLambda_{18} \alpha C_{1} - 1.5\varLambda_{12}^{2} \varLambda_{18} \beta C_{1} - \cdots - 4.5\varLambda_{12}^{2} \varLambda_{13} \varLambda_{18} \varOmega \beta^{2} C_{1} , \\ k_{18} & = 0.04C_{1} \left( { - 48\varOmega \beta + 336\varLambda_{12} } \right)\varLambda_{12}^{3} \varLambda_{18} \beta , \\ k_{19} & = \varLambda_{14} + \varLambda_{16} + \cdots + \varPsi_{11} + \varPsi_{12} + \varPsi_{13} + \varPsi_{14} , \\ \end{aligned}$$

where the parameter \(\alpha\), \(\beta\), \(\varPsi_{11}\), \(\varPsi_{12}\), \(\varPsi_{13}\), and \(\varPsi_{14}\) are provided in [3].

Appendix 3

Proposed approximate solutions are given here with 14 decimal places of the accuracy to avoid the problem of rounding of error.

Equations (38)–(41) represent the solution for case 1–4 of problem 1, respectively, as:

$$\hat{w}_{c - 1} \left( r \right) = \left( {\begin{array}{*{20}l} {\frac{0.647145641366172}{{1 + {\text{e}}^{{ - \left( {4.15748066641636r - 1.36847255989593} \right)}} }} - \frac{1.70694044391665}{{1 + {\text{e}}^{{ - \left( {0.102271757151059r + 0.366206348573016} \right)}} }}} \hfill \\ { - \frac{ - 1.95154392315358}{{1 + {\text{e}}^{{ - \left( {3.68595208480142r + 2.61870854325221} \right)}} }} + \frac{ - 2.23442195623440}{{1 + {\text{e}}^{{ - \left( {0.612334318551994r + 1.10608374073241} \right)}} }}} \hfill \\ { + \frac{ - 3.28047853696769}{{1 + {\text{e}}^{{ - \left( {0.136839681547638r + 0.308243294115322} \right)}} }} + \frac{ - 0.427893887946368}{{1 + {\text{e}}^{{ - \left( {4.30583797951597r - 1.66507129404126} \right)}} }}} \hfill \\ { + \frac{7.60672386815794}{{1 + {\text{e}}^{{ - \left( { - 0.840568898011056r + 4.95538635020765} \right)}} }} + \frac{ - 2.47465994723031}{{1 + {\text{e}}^{{ - \left( { - 1.19935732975813r - 0.424920571264802} \right)}} }}} \hfill \\ { + \frac{ - 1.49775100399665}{{1 + {\text{e}}^{{ - \left( {1.20716216474360r + 0.953166961735882} \right)}} }} + \frac{ - 1.17681357539180}{{1 + {\text{e}}^{{ - \left( {0.865768739684841r - 2.77178679297078} \right)}} }}} \hfill \\ \end{array} } \right)$$
(38)
$$\hat{w}_{c - 2} \left( r \right) = \left( \begin{aligned} \frac{0.187671397412595}{{1 + {\text{e}}^{{ - \left( {1.38541084731558r - 0.499167632148370} \right)}} }} - \frac{1.37565011128720}{{1 + {\text{e}}^{{ - \left( { - 0.0440544170467705r - 0.0184410606500222} \right)}} }} \hfill \\ - \frac{2.07703009215308}{{1 + {\text{e}}^{{ - \left( { - 0.592860434748386r + 3.21739705884779} \right)}} }} + \frac{1.51902564519803}{{1 + {\text{e}}^{{ - \left( {0.767565345613089r - 3.57709006393788} \right)}} }} \hfill \\ + \frac{ - 3.54895636251377}{{1 + {\text{e}}^{{ - \left( {0.533248661450748r - 3.23805588196301} \right)}} }} + \frac{ - 3.34671109114714}{{1 + {\text{e}}^{{ - \left( {0.639746625876463r - 2.83889161270586} \right)}} }} \hfill \\ + \frac{ - 1.74286998320735}{{1 + {\text{e}}^{{ - \left( { - 0.0556024317587073r - 0.0556024317587073} \right)}} }} + \frac{0.268767800072797}{{1 + {\text{e}}^{{ - \left( { - 4.70835173496243r - 0.979600693022802} \right)}} }} \hfill \\ + \frac{ - 0.682981155099476}{{1 + {\text{e}}^{{ - \left( {0.463408846853167r - 0.0684508827320506} \right)}} }} + \frac{ - 0.215017652709061}{{1 + {\text{e}}^{{ - \left( { - 1.09231229234423r + 4.12948301656718} \right)}} }} \hfill \\ \end{aligned} \right)$$
(39)
$$\hat{w}_{c - 3} \left( r \right) = \left( \begin{aligned} \frac{3.35422878884439}{{1 + {\text{e}}^{{ - \left( {0.486426254957742r + 3.18127708523477} \right)}} }} - \frac{ - 1.41973294822556}{{1 + {\text{e}}^{{ - \left( {1.93936830738782r + 2.01387720423554} \right)}} }} \hfill \\ - \frac{ - 2.98186626956667}{{1 + {\text{e}}^{{ - \left( {0.401957369178460r - 1.10237669603314} \right)}} }} + \frac{0.111884728287358}{{1 + {\text{e}}^{{ - \left( {0.563119419739604r + 0.260450940092143} \right)}} }} \hfill \\ + \frac{ - 1.01991511957443}{{1 + {\text{e}}^{{ - \left( {4.93444256800362r + 2.50854726644657} \right)}} }} + \frac{ - 4.40947997441380}{{1 + {\text{e}}^{{ - \left( {0.523928615824856r - 3.85654227383059} \right)}} }} \hfill \\ + \frac{1.36271675341221}{{1 + {\text{e}}^{{ - \left( { - 3.69254745684856r - 2.96190657584362} \right)}} }} + \frac{1.64175441526743}{{1 + {\text{e}}^{{ - \left( { - 0.386694611126202r + 1.95157757777880} \right)}} }} \hfill \\ + \frac{ - 0.130265051773553}{{1 + {\text{e}}^{{ - \left( { - 0.976728118952166r - 0.476044976021596} \right)}} }} + \frac{ - 0.153502354867538}{{1 + {\text{e}}^{{ - \left( {0.0333138889585737r + 1.67192434738641} \right)}} }} \hfill \\ \end{aligned} \right)$$
(40)
$$\hat{w}_{c - 4} \left( r \right) = \left( \begin{aligned} \frac{ - 0.905963487546473}{{1 + {\text{e}}^{{ - \left( {0.355065729913624r - 0.143489021401752} \right)}} }} - \frac{1.81247178806508}{{1 + {\text{e}}^{{ - \left( { - 0.670166597292294r + 2.33091993952271} \right)}} }} \hfill \\ - \frac{ - 2.59358707979362}{{1 + {\text{e}}^{{ - \left( {0.0661183176842188r - 0.426057547152923} \right)}} }} + \frac{1.77408245135325}{{1 + {\text{e}}^{{ - \left( { - 2.64889167930521r - 1.34190087841978} \right)}} }} \hfill \\ + \frac{0.158910620934395}{{1 + {\text{e}}^{{ - \left( {3.05579204611093r - 0.334743766049303} \right)}} }} + \frac{1.02687467800744}{{1 + {\text{e}}^{{ - \left( {0.711034333738193r - 2.89161984846872} \right)}} }} \hfill \\ + \frac{2.81292985547109}{{1 + {\text{e}}^{{ - \left( { - 0.696398523918427r + 4.06480470058236} \right)}} }} + \frac{ - 1.34725173332563}{{1 + {\text{e}}^{{ - \left( {0.123004324988019r + 0.719234761644911} \right)}} }} \hfill \\ + \frac{ - 1.80738491761624}{{1 + {\text{e}}^{{ - \left( {0.316595392528044r + 1.89778577535915} \right)}} }} + \frac{1.16981968035838}{{1 + {\text{e}}^{{ - \left( { - 0.176721291279196 + 0.261855681205052} \right)}} }} \hfill \\ \end{aligned} \right)$$
(41)

Equations (42)–(45) represent the solution for case 1–4 of problem 2, respectively, as:

$$\hat{w}_{c - 1} \left( r \right) = \left( \begin{aligned} \frac{ - 0.916765800501836}{{1 + {\text{e}}^{{ - \left( { - 0.963389580421896r - 1.37496049201293} \right)}} }} - \frac{ - 1.74330324948378}{{1 + {\text{e}}^{{ - \left( {3.99533098686048r + 3.38547567884518} \right)}} }} \hfill \\ - \frac{0.0182432564497315}{{1 + {\text{e}}^{{ - \left( { - 1.71379007441375r + 1.11416991590706} \right)}} }} + \frac{ - 2.06364432745519}{{1 + {\text{e}}^{{ - \left( { - 0.512954573104975r - 0.478612064486545} \right)}} }} \hfill \\ + \frac{2.72126528392210}{{1 + {\text{e}}^{{ - \left( { - 0.475966844761314r + 0.945097493753776} \right)}} }} + \frac{5.67124347726436}{{1 + {\text{e}}^{{ - \left( {0.866179425508759r + 7.57482901579570} \right)}} }} \hfill \\ + \frac{ - 8.94021752832487}{{1 + {\text{e}}^{{ - \left( {0.456601074795190r - 3.17606891228861} \right)}} }} + \frac{ - 1.54900901646979}{{1 + {\text{e}}^{{ - \left( {1.83924885536412r + 2.88921217062805} \right)}} }} \hfill \\ + \frac{ - 1.91474877853051}{{1 + {\text{e}}^{{ - \left( {7.65648624600581r + 8.59457108244669} \right)}} }} + \frac{0.420784097480516}{{1 + {\text{e}}^{{ - \left( { - 0.492830674545496r - 1.94618800909803} \right)}} }} \hfill \\ \end{aligned} \right)$$
(42)
$$\hat{w}_{c - 2} \left( r \right) = \left( \begin{aligned} \frac{0.992335956462138}{{1 + {\text{e}}^{{ - \left( {3.09054868933327r + 1.78403914961692} \right)}} }} - \frac{ - 0.417156967915766}{{1 + {\text{e}}^{{ - \left( { - 1.49919210359934r + 0.171812352445208} \right)}} }} \hfill \\ - \frac{2.37786623757731}{{1 + {\text{e}}^{{ - \left( { - 0.0511044892057031r - 0.744130540402797} \right)}} }} + \frac{ - 4.06846893640396}{{1 + {\text{e}}^{{ - \left( {0.566407833578774r - 2.11083666514436} \right)}} }} \hfill \\ + \frac{5.14410648391462}{{1 + {\text{e}}^{{ - \left( { - 0.886704871589830r + 5.16797559962547} \right)}} }} + \frac{ - 1.95468519729372}{{1 + {\text{e}}^{{ - \left( { - 0.610059874393097r + 0.675721597667982} \right)}} }} \hfill \\ + \frac{0.333990081774987}{{1 + {\text{e}}^{{ - \left( {1.39546151792236r + 2.60847378505353} \right)}} }} + \frac{ - 3.54288865902479}{{1 + {\text{e}}^{{ - \left( {0.0909692082756472r + 0.954931075270838} \right)}} }} \hfill \\ + \frac{ - 3.02495681952819}{{1 + {\text{e}}^{{ - \left( {0.163726180031139r + 0.510256703397404} \right)}} }} + \frac{1.50356797420596}{{1 + {\text{e}}^{{ - \left( { - 0.0737958243873805r - 1.26838690316516} \right)}} }} \hfill \\ \end{aligned} \right)$$
(43)
$$\hat{\varvec{w}}_{{\varvec{c} - 3}} \left( \varvec{r} \right) = \varvec{ }\left( \begin{aligned} \frac{ - 3.19407568942891}{{1 + \varvec{e}^{{ - \left( { - 0.322730912803841{\text{r}} + 6.30346502725742} \right)}} }} - \frac{0.0278226503353777}{{1 + \varvec{e}^{{ - \left( { - 3.14691955594440{\text{r}} + 7.50690234764292} \right)}} }} \hfill \\ - \frac{1.63890667673029}{{1 + \varvec{e}^{{ - \left( { - 1.20905530334225{\text{r}} + 2.73234929358647} \right)}} }} + \frac{ - 2.17837605054832}{{1 + \varvec{e}^{{ - \left( {0.496681110974480{\text{r}} + 4.13825463386980} \right)}} }} \hfill \\ + \frac{3.94061712738066}{{1 + \varvec{e}^{{ - \left( { - 1.99761550601316{\text{r}} + 8.02577551919528} \right)}} }} + \frac{0.283054734207603}{{1 + \varvec{e}^{{ - \left( { - 3.06376534319879{\text{r}} + 9.36530514069546} \right)}} }} \hfill \\ + \frac{0.359746463672984}{{1 + \varvec{e}^{{ - \left( { - 1.18134158202104{\text{r}} - 2.77744639516325} \right)}} }} + \frac{3.59429255199355}{{1 + \varvec{e}^{{ - \left( {0.332589525912578{\text{r}} + 0.387030907616429} \right)}} }} \hfill \\ + \frac{2.31655365155107}{{1 + \varvec{e}^{{ - \left( {3.20863465375263{\text{r}} + 1.44728132303346} \right)}} }} + \frac{ - 4.08588295178087}{{1 + \varvec{e}^{{ - \left( { - 0.0101234174102530{\text{r}} + 3.47527224199378} \right)}} }} \hfill \\ \end{aligned} \right)$$
(44)
$$\hat{w}_{c - 4} \left( r \right) = \left( \begin{aligned} \frac{ - 0.0352701468557131}{{1 + {\text{e}}^{{ - \left( { - 7.68577088358541r + 19.5124036823225} \right)}} }} - \frac{ - 0.0364724585730827}{{1 + {\text{e}}^{{ - \left( { - 6.02063109322370r + 14.1803253613733} \right)}} }} \hfill \\ - \frac{0.541990023233788}{{1 + {\text{e}}^{{ - \left( {2.50399748327207r - 3.68933482542127} \right)}} }} + \frac{ - 7.55577430741644}{{1 + {\text{e}}^{{ - \left( {1.39221031097525r - 6.24780343329729} \right)}} }} \hfill \\ + \frac{10.4075387141080}{{1 + {\text{e}}^{{ - \left( { - 0.129453779454824r + 1.94661801985007} \right)}} }} + \frac{ - 9.21035751796697}{{1 + {\text{e}}^{{ - \left( {1.37562922254829r - 2.98045121759701} \right)}} }} \hfill \\ + \frac{ - 11.6013710605871}{{1 + {\text{e}}^{{ - \left( { - 6.33851402305966r - 1.31910947982297} \right)}} }} + \frac{ - 4.97377745014209}{{1 + {\text{e}}^{{ - \left( { - 1.93528471563286r + 4.33625674501382} \right)}} }} \hfill \\ + \frac{0.266324879207348}{{1 + {\text{e}}^{{ - \left( { - 0.410453571222274r - 2.42972581555362} \right)}} }} + \frac{ - 4.87898842511696}{{1 + {\text{e}}^{{ - \left( { - 1.01003196340724r + 0.523421404726410} \right)}} }} \hfill \\ \end{aligned} \right)$$
(45)

Equations. (29)–(32) represent the solution for case 1–4 of problem 3, respectively, as:

$$\hat{w}_{c - 1} \left( r \right) = \left( \begin{aligned} \frac{ - 0.557884722614297}{{1 + {\text{e}}^{{ - \left( {5.39177576350898r - 3.96001310692469} \right)}} }} - \frac{4.05407149524991}{{1 + {\text{e}}^{{ - \left( {0.898802130591326r - 1.71640395713524} \right)}} }} \hfill \\ - \frac{1.52647617394133}{{1 + {\text{e}}^{{ - \left( { - 1.50186562756171r + 2.07164913302750} \right)}} }} + \frac{1.10249805269285}{{1 + {\text{e}}^{{ - \left( {1.34031499788130r + 1.14162199862648} \right)}} }} \hfill \\ + \frac{3.00820715531389}{{1 + {\text{e}}^{{ - \left( { - 1.04690259078557r + 2.50234663176738} \right)}} }} + \frac{ - 2.03243551229239}{{1 + {\text{e}}^{{ - \left( {5.10502778349739r - 1.40557692630696} \right)}} }} \hfill \\ + \frac{0.901675570666091}{{1 + {\text{e}}^{{ - \left( { - 2.70106160520870r + 2.35794479686030} \right)}} }} + \frac{ - 3.33833425208156}{{1 + {\text{e}}^{{ - \left( {9.14888397370770r - 2.27473017651894} \right)}} }} \hfill \\ + \frac{8.06474620020193}{{1 + {\text{e}}^{{ - \left( { - 4.12658649057811r - 3.86231585275885} \right)}} }} + \frac{0.719051336852431}{{1 + {\text{e}}^{{ - \left( {5.86123809863325r + 1.05850015002654} \right)}} }} \hfill \\ \end{aligned} \right)$$
(46)
$$\hat{w}_{c - 2} \left( r \right) = \left( \begin{aligned} \frac{ - 3.99917733396647}{{1 + {\text{e}}^{{ - \left( {0.891332697184988r - 5.90451761582513} \right)}} }} - \frac{ - 3.63048568061670}{{1 + {\text{e}}^{{ - \left( {7.90340838756394r + 0.968241198302790} \right)}} }} \hfill \\ - \frac{ - 2.31974126742829}{{1 + {\text{e}}^{{ - \left( {0.616708012312368r - 2.73282250710302} \right)}} }} + \frac{2.62417803088949}{{1 + {\text{e}}^{{ - \left( { - 1.52058501459065r - 1.49138470011232} \right)}} }} \hfill \\ + \frac{1.43101988902660}{{1 + {\text{e}}^{{ - \left( { - 0.340632687941473r - 0.276390450125065} \right)}} }} + \frac{2.09537715399110}{{1 + {\text{e}}^{{ - \left( { - 0.386365719160677r - 0.341253162796568} \right)}} }} \hfill \\ + \frac{2.69349856107792}{{1 + {\text{e}}^{{ - \left( {0.168790032941893r + 0.911527814377524} \right)}} }} + \frac{2.29282819004317}{{1 + {\text{e}}^{{ - \left( { - 3.79881982215620r - 1.41599087016578} \right)}} }} \hfill \\ + \frac{1.54672003167370}{{1 + {\text{e}}^{{ - \left( {0.813664616645804r + 1.25208989253170} \right)}} }} + \frac{1.06272327855314}{{1 + {\text{e}}^{{ - \left( { - 0.475417792457132r - 0.969174610316326} \right)}} }} \hfill \\ \end{aligned} \right)$$
(47)
$$\hat{w}_{c - 3} \left( r \right) = \left( \begin{aligned} \frac{0.876851623473768}{{1 + {\text{e}}^{{ - \left( {2.49349648377317r - 1.04413757040828} \right)}} }} - \frac{2.98841106964488}{{1 + {\text{e}}^{{ - \left( {2.76757517014916r - 7.56729418144732} \right)}} }} \hfill \\ - \frac{ - 3.08704875988994}{{1 + {\text{e}}^{{ - \left( {2.74246487988205r - 7.54823803463353} \right)}} }} + \frac{1.95002243864153}{{1 + {\text{e}}^{{ - \left( { - 1.17343162417066r + 5.19485772133905} \right)}} }} \hfill \\ + \frac{ - 2.91727788537681}{{1 + {\text{e}}^{{ - \left( { - 0.146638709571110r + 1.06692803800891} \right)}} }} + \frac{ - 0.213985534940599}{{1 + {\text{e}}^{{ - \left( { - 0.123559731081878r - 1.19126954449528} \right)}} }} \hfill \\ + \frac{4.60685282549067}{{1 + {\text{e}}^{{ - \left( {6.33918703814984r - 0.620132812860196} \right)}} }} + \frac{ - 4.16309707518349}{{1 + {\text{e}}^{{ - \left( { - 0.301272972108148r + 3.05524036710699} \right)}} }} \hfill \\ + \frac{ - 9.73735558321735}{{1 + {\text{e}}^{{ - \left( {1.12731778159919r - 5.14273412780001} \right)}} }} + \frac{1.57621773601807}{{1 + {\text{e}}^{{ - \left( { - 3.65581063096570r - 3.84842030920905} \right)}} }} \hfill \\ \end{aligned} \right)$$
(48)
$$\hat{w}_{c - 4} \left( r \right) = \left( \begin{aligned} \frac{ - 7.15726364209788}{{1 + {\text{e}}^{{ - \left( {0.885195118826721r - 4.78209598133979} \right)}} }} - \frac{ - 6.17255622348657}{{1 + {\text{e}}^{{ - \left( {1.49564692571066r - 5.54337224493422} \right)}} }} \hfill \\ - \frac{ - 0.413072960150419}{{1 + {\text{e}}^{{ - \left( { - 10.2266486130385r + 8.09232728970431} \right)}} }} + \frac{1.82564122081174}{{1 + {\text{e}}^{{ - \left( {0.0633336044282149r - 1.14400399368649} \right)}} }} \hfill \\ + \frac{1.03508823950399}{{1 + {\text{e}}^{{ - \left( {3.62602092550570r - 1.28772588614558} \right)}} }} + \frac{ - 3.72551673972169}{{1 + {\text{e}}^{{ - \left( {0.516181919854073r - 2.86969311405169} \right)}} }} \hfill \\ + \frac{0.303941969629559}{{1 + {\text{e}}^{{ - \left( {4.00536572420897r - 9.85028475535404} \right)}} }} + \frac{ - 0.360905913713567}{{1 + {\text{e}}^{{ - \left( {3.58805786961814r - 9.16851747736320} \right)}} }} \hfill \\ + \frac{2.48005813826967}{{1 + {\text{e}}^{{ - \left( {1.10236542530956r - 2.66486193626854} \right)}} }} + \frac{ - 0.780692363450426}{{1 + {\text{e}}^{{ - \left( { - 3.09968654105565r + 2.16718640176508} \right)}} }} \hfill \\ \end{aligned} \right)$$
(49)

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Munir, A., Manzar, M.A., Khan, N.A. et al. Intelligent computing approach to analyze the dynamics of wire coating with Oldroyd 8-constant fluid. Neural Comput & Applic 31, 751–775 (2019). https://doi.org/10.1007/s00521-017-3107-4

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