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Numerical solution of doubly singular nonlinear systems using neural networks-based integrated intelligent computing

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Abstract

In this paper, a bio-inspired computational intelligence technique is presented for solving nonlinear doubly singular system using artificial neural networks (ANNs), genetic algorithms (GAs), sequential quadratic programming (SQP) and their hybrid GA–SQP. The power of ANN models is utilized to develop a fitness function for a doubly singular nonlinear system based on approximation theory in the mean square sense. Global search for the parameters of networks is performed with the competency of GAs and later on fine-tuning is conducted through efficient local search by SQP algorithm. The design methodology is evaluated on number of variants for two point doubly singular systems. Comparative studies with standard results validate the correctness of proposed schemes. The consistent correctness of the proposed technique is proven through statistics using different performance indices.

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Author information

Authors and Affiliations

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

    Muhammad Asif Zahoor Raja

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

    Jabran Mehmood

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

    Zulqurnain Sabir

  4. School of Mathematical Sciences, Universiti Sains Malaysia, 11800, Gelugor, Penang, Malaysia

    A. Kazemi Nasab

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

    Muhammad Anwaar Manzar

Authors
  1. Muhammad Asif Zahoor Raja
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  2. Jabran Mehmood
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  3. Zulqurnain Sabir
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  4. A. Kazemi Nasab
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  5. Muhammad Anwaar Manzar
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Corresponding author

Correspondence to Muhammad Asif Zahoor Raja.

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Appendix

Appendix

Derived solutions of the proposed methodology are listed here by using 14 decimal places of accuracy in the values of the weights of ANN models. The equation listed with same number as in the manuscript for easy reference.

$$\begin{aligned} y_{1} (x) & = \frac{5.06090285540006}{{1 + {\text{e}}^{ - ( - 0.0702478033774433x - 0.583753149084774)} }} + \frac{5.79491328148513}{{1 + {\text{e}}^{ - (0.286789695866269x + 2.07332599418136)} }} \\ & \quad + \frac{ - 0.485814402406440}{{1 + {\text{e}}^{ - ( - 0.711413479269947x - 1.07592810200203)} }} + \frac{ - 6.28871115473857}{{1 + {\text{e}}^{ - ( - 1.87100189303373x + 2.52309316357794)} }} \\ & \quad + \frac{ - 0.521022827437516}{{1 + {\text{e}}^{ - ( - 0.216373045273920x - 1.01342365705573)} }} + \frac{6.62234542330856}{{1 + {\text{e}}^{ - (2.78418517351797x - 4.99709612589177)} }} \\ & \quad + \frac{ - 5.49358058709068}{{1 + {\text{e}}^{ - ( - 0.273599551623518x - 1.11311923766351)} }} + \frac{4.72182331263108}{{1 + {\text{e}}^{ - (2.49854913845262x + 2.81658358340508)} }} \\ & \quad + \frac{3.42004110701236}{{1 + {\text{e}}^{ - ( - 1.19839985113893x + 0.228071378383934)} }} + \frac{ - 4.92130765418310}{{1 + {\text{e}}^{ - ( - 4.90249044271862x + 10.2185031974595)} }} \\ \end{aligned}$$
(45)
$$\begin{aligned} y_{2} (x) & = \frac{ - 2.16167819757990}{{1 + {\text{e}}^{ - ( - 1.22566299279515x - 0.567044158218105)} }} + \frac{5.35374994888277}{{1 + {\text{e}}^{ - (2.64718320786792x + 3.45639674203448)} }} \\ & \quad + \frac{3.27520102970372}{{1 + {\text{e}}^{ - ( - 0.0256570513055492x - 0.108830638896546)} }} + \frac{ - 3.09511267926457}{{1 + {\text{e}}^{ - (1.49418360470716x - 0.0916529279004633)} }} \\ & \quad + \frac{0.581937101311438}{{1 + {\text{e}}^{ - (1.90407327264843x + 1.42957286496310)} }} + \frac{1.17399279719794}{{1 + {\text{e}}^{ - (1.17745379843572x + 2.75304442640134)} }} \\ & \quad + \frac{6.57499699312264}{{1 + {\text{e}}^{ - (2.47525684319937x - 4.12048750231603)} }} + \frac{ - 5.79011456869788}{{1 + {\text{e}}^{ - ( - 3.99911492076474x + 8.01608749271995)} }} \\ & \quad + \frac{3.94396483584132}{{1 + {\text{e}}^{ - (1.83252188055622x - 2.17698744981887)} }} + \frac{3.44491398396493}{{1 + {\text{e}}^{ - ( - 0.151752869143717x - 2.64327645722025)} }} \\ \end{aligned}$$
(46)
$$\begin{aligned} y_{3} (x) & = \frac{ - 4.89255640340418}{{1 + {\text{e}}^{ - (1.09631183106634x + 2.54718959776047)} }} + \frac{ - 1.65607670949300}{{1 + {\text{e}}^{ - ( - 0.317075103221240x - 0.0753235627418367)} }} \\ & \quad + \frac{6.70303733222117}{{1 + {\text{e}}^{ - (3.25931491085795x - 6.32765371454161)} }} + \frac{0.631601445451111}{{1 + {\text{e}}^{ - (0.877810670872484x + 0.834349299727913)} }} \\ & \quad + \frac{2.79226553988273}{{1 + {\text{e}}^{ - ( - 1.55197211384629x + 1.15125555795079)} }} + \frac{ - 0.361932186913774}{{1 + {\text{e}}^{ - (0.798735533729565x + 3.26900589535334)} }} \\ & \quad + \frac{1.39254298785122}{{1 + {\text{e}}^{ - (0.201897840407861x + 0.154600651687391)} }} + \frac{11.2307689596373}{{1 + {\text{e}}^{ - (1.72905279108444x - 2.64810269614498)} }} \\ & \quad + \frac{2.97127358135085}{{1 + {\text{e}}^{ - (3.02677052642493x + 2.53101175809415)} }} + \frac{ - 0.285857402161480}{{1 + {\text{e}}^{ - (0.104728001449634x - 0.102733082814781)} }} \\ \end{aligned}$$
(47)
$$\begin{aligned} y(x) & = \frac{ - 2.54719091290125}{{1 + {\text{e}}^{ - ( - 1.95421289444200x + 3.12584730370828)} }} + \frac{4.00876994736113}{{1 + {\text{e}}^{ - (0.907697852594049x - 1.49562714049952)} }} \\ & \quad + \frac{0.0339707387060374}{{1 + {\text{e}}^{ - (2.82835239375004x + 0.809060471573114)} }} + \frac{3.32763985181839}{{1 + {\text{e}}^{ - (0.710085051995562x - 0.850815525467989)} }} \\ & \quad + \frac{1.00761357046961}{{1 + {\text{e}}^{ - ( - 1.38860606211098x - 0.589674970805058)} }} + \frac{ - 0.705188625728285}{{1 + {\text{e}}^{ - ( - 0.642941851398015x - 0.305949374778106)} }} \\ & \quad + \frac{7.01504909472222}{{1 + {\text{e}}^{ - (2.76862180499161x - 6.24888097992903)} }} + \frac{ - 5.45348866742368}{{1 + {\text{e}}^{ - ( - 1.98024194421427x + 7.06046812197553)} }} \\ & \quad + \frac{3.17977599883819}{{1 + {\text{e}}^{ - ( - 1.94110207672956x + 6.13828775080669)} }} + \frac{5.17699764318622}{{1 + {\text{e}}^{ - ( - 0.0865958521204367x + 1.10487099998281)} }} \\ \end{aligned}$$
(48)
$$\begin{aligned} y(x) & = \frac{ - 0.491885577669344}{{1 + {\text{e}}^{ - ( - 1.04360826793733x - 0.779183616251543)} }} + \frac{1.57194204454147}{{1 + {\text{e}}^{ - ( - 0.937739180556172x + 0.0190788365203624)} }} \\ & \quad + \frac{2.89682160556479}{{1 + {\text{e}}^{ - (1.66535922258982x - 2.93956779256959)} }} + \frac{ - 1.12679278519487}{{1 + {\text{e}}^{ - ( - 1.28679406133602x + 1.48521429792892)} }} \\ & \quad + \frac{8.09616718066885}{{1 + {\text{e}}^{ - (2.55362190979163x - 6.04565842550422)} }} + \frac{1.74825098806409}{{1 + {\text{e}}^{ - (0.437188354092352x - 1.28447199950952)} }} \\ & \quad + \frac{ - 1.34135459930309}{{1 + {\text{e}}^{ - ( - 1.36281293309862x + 1.98252265429720)} }} + \frac{5.55517846129658}{{1 + {\text{e}}^{ - (1.09092680201182x - 1.43539317190078)} }} \\ & \quad + \frac{ - 2.38943961873907}{{1 + {\text{e}}^{ - (1.43558647066204x - 1.54607444041500)} }} + \frac{1.29663126574339}{{1 + {\text{e}}^{ - ( - 0.278804474173689x + 3.75511817517950)} }} \\ \end{aligned}$$
(49)
$$\begin{aligned} y(x) & = \frac{0.423335701446074}{{1 + {\text{e}}^{ - (0.0748838110620104x - 0.640531930970829)} }} + \frac{3.79397716028822}{{1 + {\text{e}}^{ - ( - 1.08984400898693x + 3.05775654559903)} }} \\ & \quad + \frac{4.00030985948602}{{1 + {\text{e}}^{ - (0.824081733331562x + 2.08988750718457)} }} + \frac{ - 0.707611019644300}{{1 + {\text{e}}^{ - (1.02028389138862x - 1.06880714695559)} }} \\ & \quad + \frac{ - 5.80530819297031}{{1 + {\text{e}}^{ - ( - 3.06568135646585x + 7.15076807390266)} }} + \frac{5.51015424348853}{{1 + {\text{e}}^{ - (1.84262927139645x - 3.56433465666046)} }} \\ & \quad + \frac{ - 1.05730524457473}{{1 + {\text{e}}^{ - (1.57597538792983x + 2.28924283191326)} }} + \frac{2.56311572064720}{{1 + {\text{e}}^{ - (1.29657883298244x - 1.48930497350755)} }} \\ & \quad + \frac{4.81615064967417}{{1 + {\text{e}}^{ - (1.04846523537790x - 2.39048832853946)} }} + \frac{ - 0.578737364598431}{{1 + {\text{e}}^{ - (0.420035532981895x + 0.934815594102052)} }} \\ \end{aligned}$$
(50)
$$\begin{aligned} y(x) & = \frac{ - 0.393812573610744}{{1 + {\text{e}}^{ - (0.305009475702023x - 0.322335473054778)} }} + \frac{ - 0.368976546094913}{{1 + {\text{e}}^{ - (0.0695910880403198x - 0.619273066257462)} }} \\ & \quad + \frac{0.115659305765670}{{1 + {\text{e}}^{ - (0.321824227356529x + 0.886571836580560)} }} + \frac{ - 1.20863099243674}{{1 + {\text{e}}^{ - (0.309989201918084x + 2.26568457321997)} }} \\ & \quad + \frac{ - 0.248857095269561}{{1 + {\text{e}}^{ - ( - 0.411269795492479x - 0.566907515343194)} }} + \frac{0.516160537708405}{{1 + {\text{e}}^{ - (1.10020219908361x + 1.77777116552109)} }} \\ & \quad + \frac{1.58017457770858}{{1 + {\text{e}}^{ - ( - 0.250015164825119x - 0.896129783051629)} }} + \frac{0.990454631804859}{{1 + {\text{e}}^{ - ( - 0.243373544838292x - 0.0987306114157956)} }} \\ & \quad + \frac{0.232934428664574}{{1 + {\text{e}}^{ - (0.0251535317301115x + 0.573023573768020)} }} + \frac{ - 1.68287900841249}{{1 + {\text{e}}^{ - (0.921155849011818x + 2.15345379527144)} }} \\ \end{aligned}$$
(51)
$$\begin{aligned} y(x) & = \frac{ - 1.34531009927343}{{1 + {\text{e}}^{ - ( - 0.403069293933660x + 2.77850041715483)} }} + \frac{1.70374265870458}{{1 + {\text{e}}^{ - ( - 0.835683594061887x - 2.0798238862202483)} }} \\ & \quad + \frac{ - 0.750422431499470}{{1 + {\text{e}}^{ - ( - 0.858465963250953x + 0.715204584403003)} }} + \frac{ - 0.926032227758271}{{1 + {\text{e}}^{ - (0.459780226967856x - 0.440075452834427)} }} \\ & \quad + \frac{1.40523632749511}{{1 + {\text{e}}^{ - ( - 0.354312197292955x + 0.897982655098785)} }} + \frac{ - 0.149584246718296}{{1 + {\text{e}}^{ - ( - 0.617267704349377x + 0.0683357154528507)} }} \\ & \quad + \frac{1.47789768683490}{{1 + {\text{e}}^{ - ( - 0.768663842288795x + 0.0641394736934732)} }} + \frac{0.0225074523269556}{{1 + {\text{e}}^{ - (0.0894999590846092x + 1.00575559906559)} }} \\ & \quad + \frac{ - 1.72242762172707}{{1 + {\text{e}}^{ - ( - 0.836480213660271x - 0.653345699522128)} }} + \frac{ - 0.829959414853381}{{1 + {\text{e}}^{ - (0.767320605753489x + 0.691438358983869)} }} \\ \end{aligned}$$
(52)
$$\begin{aligned} y(x) & = \frac{ - 1.24674786348219}{{1 + {\text{e}}^{ - (0.199695265853276x - 0.518095421301578)} }} + \frac{ - 0.205615048645665}{{1 + {\text{e}}^{ - (0.151732228103460x + 0.390410762284336)} }} \\ & \quad + \frac{ - 0.0828850517219031}{{1 + {\text{e}}^{ - (0.272250663138720x + 1.07630788154172)} }} + \frac{0.633141849230506}{{1 + {\text{e}}^{ - ( - 0.0275241634444990x + 0.359315488981845)} }} \\ & \quad + \frac{0.969180706528858}{{1 + {\text{e}}^{ - (0.0584420842252987x + 0.242230530069149)} }} + \frac{ - 0.418472862231310}{{1 + {\text{e}}^{ - (0.394874323897873x - 0.967919422711968)} }} \\ & \quad + \frac{ - 0.374075783852076}{{1 + {\text{e}}^{ - (0.802544594333259x + 1.91988811145830)} }} + \frac{ - 1.01703806790783}{{1 + {\text{e}}^{ - (0.547432637506353x + 1.19599907244523)} }} \\ & \quad + \frac{ - 0.490769143582236}{{1 + {\text{e}}^{ - (1.50439316568563x + 3.91271635644075)} }} + \frac{0.162867932113670}{{1 + {\text{e}}^{ - ( - 0.279524330031746x - 0.779552034606925)} }} \\ \end{aligned}$$
(53)

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Raja, M.A.Z., Mehmood, J., Sabir, Z. et al. Numerical solution of doubly singular nonlinear systems using neural networks-based integrated intelligent computing. Neural Comput & Applic 31, 793–812 (2019). https://doi.org/10.1007/s00521-017-3110-9

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