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Numerical treatment of nonlinear singular Flierl–Petviashivili systems using neural networks models

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Abstract

In this study, new intelligent computing methodologies have been developed for highly nonlinear singular Flierl–Petviashivili (FP) problem having boundary condition at infinity by exploiting three different neural network models integrated with active-set algorithm (ASA). A modification in the modeling is introduced to cater the singularity, avoid divergence in results for unbounded inputs and capable of dealing with strong nonlinearity. Three models have been constructed in an unsupervised manner for solving the FP equation using log-sigmoid, radial basis and tan-sigmoid transfer functions in the hidden layers of the network. The training of adaptive adjustable variables of each model is carried out with a constrained optimization technique based on ASA. The proposed models have been evaluated on three variants of the two FP equations. The designed models have been examined with respect to precision, stability and complexity through statistics.

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Authors and Affiliations

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

    Muhammad Asif Zahoor Raja

  2. Department of Computer Science and Engineering, HITEC University Texila, Rawalpindi, Pakistan

    Junaid Ali Khan

  3. Department of Computer and Information Science, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad, Pakistan

    Aneela Zameer

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

    Najeeb Alam Khan

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

    Muhammad Anwaar Manzar

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  1. Muhammad Asif Zahoor Raja
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  2. Junaid Ali Khan
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  3. Aneela Zameer
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  4. Najeeb Alam Khan
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  5. Muhammad Anwaar Manzar
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Corresponding author

Correspondence to Aneela Zameer.

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Appendix

Appendix

Complete expressions for the solution of the proposed scheme with fourteen places of decimal are provided here for the reproduction and verification of the results. Same equation numbers are referred here for better identification with the main manuscript text.

$$\hat{u}_{\text{LS}} \left( x \right) = \left( {\frac{{1 + x^{2} }}{{1 + x^{3} }} + \frac{{x^{2} }}{{1 + x^{3} }}\left( \begin{aligned} \frac{4.72632510851474}{{1 + {\text{e}}^{{ - \left( {1.65710173577515x + 2.27779096744896} \right)}} }} + \frac{ - 4.45031709267048}{{1 + {\text{e}}^{{ - \left( { - 3.56958568913710x + 6.62613183690404} \right)}} }} \hfill \\ + \frac{ - 2.23650019923638}{{1 + {\text{e}}^{{ - \left( { - 0.71859866765235x - 2.59945765214019} \right)}} }} + \frac{0.73190624969960}{{1 + {\text{e}}^{{ - \left( { - 1.91940354568530x - 3.61059364483905} \right)}} }} \hfill \\ + \frac{5.11507523597824}{{1 + {\text{e}}^{{ - \left( {0.89408386567960x + 2.28476124378728} \right)}} }} + \frac{3.83538037784671}{{1 + {\text{e}}^{{ - \left( { - 1.30345287906150x - 0.64487671895220} \right)}} }} \hfill \\ + \frac{1.84790897996908}{{1 + {\text{e}}^{{ - \left( {2.78382423750435x - 3.85784033875989} \right)}} }} + \frac{ - 0.52523154216449}{{1 + {\text{e}}^{{ - \left( { - 0.71294819225919x - 1.93880716849144} \right)}} }} \hfill \\ + \frac{6.53699521328446}{{1 + {\text{e}}^{{ - \left( {0.96411186775932x - 1.66126218702275} \right)}} }} + \frac{ - 7.18576280760247}{{1 + {\text{e}}^{{ - \left( { - 4.60722995331281x + 8.87260939644230} \right)}} }} \hfill \\ \end{aligned} \right)} \right)$$
(28)
$$\hat{u}_{\text{RB}} \left( r \right) = \left( {\frac{{1 + x^{2} }}{{1 + x^{3} }} + \frac{{x^{2} }}{{1 + x^{3} }}\left( \begin{aligned} - 0.72323894205343{\text{e}}^{{ - \left( {0.553303195007x + 1.77010552750341} \right)^{2} }} \hfill \\ - 2.91568268441046{\text{e}}^{{ - \left( {1.28822893617979x - 1.96477480403763} \right)^{2} }} \hfill \\ - 1.93226171990883{\text{e}}^{{ - \left( { - 0.70978529328024x - 0.41643049423644} \right)^{2} }} \hfill \\ - 0.02959622516246{\text{e}}^{{ - \left( {0.31583918575973x + 1.40299752833714} \right)^{2} }} \hfill \\ + 3.9872944467341{\text{e}}^{{ - \left( {0.62389115059975x - 0.6828302080878} \right)^{2} }} \hfill \\ + 3.84326621512007{\text{e}}^{{ - \left( { - 0.88728479960531x + 1.91402739675086} \right)^{2} }} \hfill \\ - 1.67692510938239{\text{e}}^{{ - \left( { - 1.48410653387875x + 1.98449500968432} \right)^{2} }} \hfill \\ - 2.28175466645687{\text{e}}^{{ - \left( {1.09978049675893x - 0.71425448496625} \right)^{2} }} \hfill \\ + 5.27367648116228{\text{e}}^{{ - \left( { - 1.73564376929631x + 3.8967359801976} \right)^{2} }} \hfill \\ + 2.25764575903348{\text{e}}^{{ - \left( { - 1.49851072245445x + 2.12478829653603} \right)^{2} }} \hfill \\ \end{aligned} \right)} \right)$$
(29)
$$u_{\text{TS}} \left( x \right) = \left( {\frac{{1 + x^{2} }}{{1 + x^{3} }} + \frac{{x^{2} }}{{1 + x^{3} }}\left( \begin{aligned} - 7.311383357855470 + \frac{1.29421376172621 \times 2}{{1 + {\text{e}}^{{ - 2\left( {1.51432531177837x - 1.94096678634061} \right)}} }} \hfill \\ + \frac{0.24564615594904 \times 2}{{1 + {\text{e}}^{{ - 2\left( {1.31335392628889x - 0.87050667836733} \right)}} }} + \frac{ - 1.78354087566988 \times 2}{{1 + {\text{e}}^{{ - 2\left( { - 0.2217120222215x - 0.6812838775957} \right)}} }} \hfill \\ + \frac{0.43104856261592 \times 2}{{1 + {\text{e}}^{{ - 2\left( {1.37969766064208x + 1.55574402348314} \right)}} }} + \frac{ - 1.46857857194636 \times 2}{{1 + {\text{e}}^{{ - 2\left( { - 0.62748038623427x - 1.69163507309337} \right)}} }} \hfill \\ + \frac{1.50311683748809 \times 2}{{1 + {\text{e}}^{{ - 2\left( {0.41140560603128x - 1.5791752769795} \right)}} }} + \frac{0.09619087525675 \times 2}{{1 + {\text{e}}^{{ - 2\left( {1.54023349256904x + 0.73179062527214} \right)}} }} \hfill \\ + \frac{ - 0.68343571033137 \times 2}{{1 + {\text{e}}^{{ - 2\left( { - 1.11310616902629x - 2.80171020977017} \right)}} }} + \frac{ - 1.82634479444537 \times 2}{{1 + {\text{e}}^{{ - 2\left( { - 3.7325288255196x - 5.72045675105031} \right)}} }} \hfill \\ + \frac{ - 3.09473729629636 \times 2}{{1 + {\text{e}}^{{ - 2\left( { - 2.36919201936288x + 4.01770385003562} \right)}} }} \hfill \\ \end{aligned} \right)} \right)$$
(30)
$$u_{\text{LS}} \left( x \right) = \left( {\frac{{1 + x^{2} }}{{1 + x^{3} }} + \frac{{x^{2} }}{{1 + x^{3} }}\left( \begin{aligned} \frac{15}{{1 + {\text{e}}^{{ - \left( {0.62787566019007x + 2.52838216054528} \right)}} }} + \frac{ - 10.7048232606873}{{1 + {\text{e}}^{{ - \left( { - 9.95405229455268x - 11.3180441306054} \right)}} }} \hfill \\ + \frac{ - 10.9204550541312}{{1 + {\text{e}}^{{ - \left( { - 2.18278617021429x - 7.17487716951065} \right)}} }} + \frac{ - 11.4985988300121}{{1 + {\text{e}}^{{ - \left( { - 8.41693250597733x - 10.205931560291} \right)}} }} \hfill \\ + \frac{ - 10.211941012151}{{1 + {\text{e}}^{{ - \left( { - 3.61662960636564x - 7.62440963072436} \right)}} }} + \frac{ - 10.855539537289}{{1 + {\text{e}}^{{ - \left( { - 9.4802032106731x - 10.3577493488161} \right)}} }} \hfill \\ + \frac{ - 12.7110988320754}{{1 + {\text{e}}^{{ - \left( { - 2.99868359771211x + 6.45601387989685} \right)}} }} + \frac{ - 1.98064169969149}{{1 + {\text{e}}^{{ - \left( { - 2.05876442801075x + 2.66747342710029} \right)}} }} \hfill \\ + \frac{ - 10.0881400939881}{{1 + {\text{e}}^{{ - \left( { - 10.6585383968093x - 10.6782642975707} \right)}} }} + \frac{ - 11.1082232011744}{{1 + {\text{e}}^{{ - \left( { - 6.33009565379611x - 9.63981989859923} \right)}} }} \hfill \\ \end{aligned} \right)} \right)$$
(38)
$$\hat{u}_{\text{RB}} \left( x \right) = \left( {\frac{{1 + x^{2} }}{{1 + x^{3} }} + \frac{{x^{2} }}{{1 + x^{3} }}\left( \begin{aligned} 1.68324391564392{\text{e}}^{{ - \left( { - 0.10712388714561x - 2.21511672890905} \right)^{2} }} \hfill \\ - 2.05499453801537{\text{e}}^{{ - \left( { - 0.52476179013829x - 3.37036767681847} \right)^{2} }} \hfill \\ + 1.55463824003447{\text{e}}^{{ - \left( {0.97423598176392x - 3.58266002843627} \right)^{2} }} \hfill \\ + 11.516335371765{\text{e}}^{{ - \left( { - 1.0850655003947x - 0.42130431141011} \right)^{2} }} \hfill \\ - 0.11796545133915{\text{e}}^{{ - \left( {1.2701966304861x + 4.66842062255857} \right)^{2} }} \hfill \\ - 0.41062548803041{\text{e}}^{{ - \left( {4.17183593775754x - 0.96015828675471} \right)^{2} }} \hfill \\ + 0.94786138308886{\text{e}}^{{ - \left( {2.84397615832785x + 0.74556129044324} \right)^{2} }} \hfill \\ + 0.21293802731812{\text{e}}^{{ - \left( { - 3.08518023831259x - 1.69573892787157} \right)^{2} }} \hfill \\ + 0.40985976734602{\text{e}}^{{ - \left( { - 1.63951673862173x + 2.04616296380706} \right)^{2} }} \hfill \\ + 2.13933788243694{\text{e}}^{{ - \left( { - 0.42594411510417x - 0.9319605433395} \right)^{2} }} \hfill \\ \end{aligned} \right)} \right)$$
(39)
$$\hat{u}_{\text{RB}} \left( x \right) = \left( {\frac{{1 + x^{2} }}{{1 + x^{3} }} + \frac{{x^{2} }}{{1 + x^{3} }}\left( \begin{aligned} - 7.311383357855470 + \frac{ - 1.99121552809285 \times 2}{{1 + {\text{e}}^{{ - 2\left( { - 0.85229344702745x + 1.36773934178084} \right)}} }} \hfill \\ + \frac{1.07500099939665 \times 2}{{1 + {\text{e}}^{{ - 2\left( { - 0.34902276066273x + 1.34857227942967} \right)}} }} + \frac{ - 0.59901142221974 \times 2}{{1 + {\text{e}}^{{ - 2\left( { - 1.03555424382006x - 2.12498182205873} \right)}} }} \hfill \\ + \frac{11.78812131451280 \times 2}{{1 + {\text{e}}^{{ - 2\left( {1.43933625849589x - 3.53689075669933} \right)}} }} + \frac{ - 0.46852496693387 \times 2}{{1 + {\text{e}}^{{ - 2\left( { - 1.26141785737207x - 3.63752443720412} \right)}} }} \hfill \\ + \frac{ - 8.10362905397083 \times 2}{{1 + {\text{e}}^{{ - 2\left( { - 0.40992951879207x - 1.50554728413777} \right)}} }} + \frac{1.54986401688038 \times 2}{{1 + {\text{e}}^{{ - 2\left( {5.94103063881747x + 4.32534775108113} \right)}} }} \hfill \\ + \frac{1.43825323424858 \times 2}{{1 + {\text{e}}^{{ - 2\left( { - 0.84887990769193x + 5.84607362112051} \right)}} }} + \frac{0.0302804784842 \times 2}{{1 + {\text{e}}^{{ - 2\left( {0.85893427468416x + 2.56877250502613} \right)}} }} \hfill \\ + \frac{0.50141177678916 \times 2}{{1 + {\text{e}}^{{ - 2\left( {4.44680325577113x + 3.20940133395529} \right)}} }} \hfill \\ \end{aligned} \right)} \right)$$
(40)

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Raja, M.A.Z., Khan, J.A., Zameer, A. et al. Numerical treatment of nonlinear singular Flierl–Petviashivili systems using neural networks models. Neural Comput & Applic 31, 2371–2394 (2019). https://doi.org/10.1007/s00521-017-3193-3

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