A novel design of Gudermannian function as a neural network for the singular nonlinear delayed, prediction and pantograph differential models

Zulqurnain Sabir; Hafiz Abdul Wahab; Juan L.G. Guirao; Zulqurnain Sabir; Hafiz Abdul Wahab; Juan L.G. Guirao

doi:10.3934/mbe.2022030

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 1: 663-687. doi: 10.3934/mbe.2022030

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Research article Special Issues

A novel design of Gudermannian function as a neural network for the singular nonlinear delayed, prediction and pantograph differential models

1.
Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan
2.
Department of Applied Mathematics and Statistics, Technical University of Cartagena, Hospital de Marina 30203-Cartagena, Spain
3.
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

Academic Editor: Haiyan Wang

Received: 30 July 2021 Accepted: 28 October 2021 Published: 19 November 2021

The present work is to solve the nonlinear singular models using the framework of the stochastic computing approaches. The purpose of these investigations is not only focused to solve the singular models, but the solution of these models will be presented to the extended form of the delayed, prediction and pantograph differential models. The Gudermannian function is designed using the neural networks optimized through the global scheme "genetic algorithms (GA)", local method "sequential quadratic programming (SQP)" and the hybridization of GA-SQP. The comparison of the singular equations will be presented with the exact solutions along with the extended form of delayed, prediction and pantograph based on these singular models. Moreover, the neuron analysis will be provided to authenticate the efficiency and complexity of the designed approach. For the correctness and effectiveness of the proposed approach, the plots of absolute error will be drawn for the singular delayed, prediction and pantograph differential models. For the reliability and stability of the proposed method, the statistical performances "Theil inequality coefficient", "variance account for" and "mean absolute deviation'' are observed for multiple executions to solve singular delayed, prediction and pantograph differential models.

Keywords:

Citation: Zulqurnain Sabir, Hafiz Abdul Wahab, Juan L.G. Guirao. A novel design of Gudermannian function as a neural network for the singular nonlinear delayed, prediction and pantograph differential models[J]. Mathematical Biosciences and Engineering, 2022, 19(1): 663-687. doi: 10.3934/mbe.2022030

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Abstract

1. Introduction

The models based on Lane-Emden (LE) have a variety of valuable applications in physiological, physical models, mathematics science and engineering studies. The celebrated form of the LE has a great significance due to the singular point and this model is explored by famous scientists Lane and Emden ^[1,2] working on the thermal gas performance and the thermodynamics state ^[3]. The literature/generic form of the LE model is given as ^[4,5,6,7]:

$\left\{ \begin{array}{l} \frac{{{d^2}\aleph }}{{d{\tau ^2}}} + \frac{\upsilon }{\tau }\frac{{d\aleph }}{{d\tau }} + g(\tau , \aleph ) = f(\tau ), \, \, \, \, \, \upsilon \geqslant 0, \, \, 0 < \tau \leqslant 1, \hfill \\ \aleph (0) = {u_1}, \, \, \frac{{d\aleph (0)}}{{d\tau }} = {u_2}, \, \, \hfill \end{array} \right.$

(1)

where $\upsilon$ is the shape factor, $g(\tau, \aleph)$ is the continuous real valued based function, ${u_1}$ and ${u_2}$ are the initial conditions (ICs).

The LE based singular nonlinear systems define an area of physical sciences ^[8], gaseous density stars ^[9], stellar models ^[10], morphogenesis ^[11], oscillating fields ^[12], mathematical systems ^[13], dusty fluidics ^[14] and an isotropic source ^[15]. To present the solutions of the singular systems is not easy and considered tough due to the involvement of singularity at the origin. Only a few schemes in literature are available to solve the singular based LE types of models ^[16,17].

A variety of physical phenomena are investigated using the sense of differential systems, among them the delay differential (DD) form is the most prominent due to the vast applications in technology as well as engineering. The DD systems were introduced few centuries ago and has number of applications in the variety of scientific areas including a communication system, engineering models, population dynamics, transport systems and economic circumstances ^{[18,19,20,21]}. Few associated investigations of DD systems are the geometric functions reliability through DD systems to exploit the factors of delay-dependent studied by Beretta et al. ^[22]. The biological based mathematical systems based on the DD models were studied by the Forde ^[23]. The implementations based on the Galerkin wavelet scheme together with the Taylor series investigations to obtain the numerical outcomes of the DD system defined by the Frazier ^[24]. The coupled variation iteration approach to calculate the analytical outcomes of the DD system introduced by Rangkuti et al. ^[25]. The Runge-Kutta approach to calculate the numerical procedures of the DD systems is implemented by the Chapra ^[26]. Few more schemes have been implemented to solve the DD models are reported in the literature ^[27,28,29]. The PD system is considered using the ideas of the DD system. The prediction differential system is applied in weather forecasting, transport, stock markets, technology, engineering, astrophysics and biological networks ^[30]. The literature form of the DD and PD systems are provided in Eqs (2) and (3) as ^[31]:

$\left\{ \begin{array}{l} \frac{{{d^2}\aleph }}{{d{\tau ^2}}} = h\left( {\tau , \, \aleph (\tau ), \, \aleph (\tau - {r_1})} \right), \;\;\;\; {r_1} > 0, \, \, l \leqslant \tau \leqslant m, \hfill \\ \aleph (\tau ) = \phi (\tau ), \;\;\;\; \varepsilon \leqslant \tau \leqslant l, \;\;\;\;\;\;\; 0 \leqslant {r_1} \leqslant \left| {l - \varepsilon } \right|, \hfill \\ \frac{{d\aleph (l)}}{{d\tau }} = w, \hfill \end{array} \right.$

(2)

where $\phi (\tau)$ shows the ICs and $\aleph (\tau - {r_1})$ represents the delayed form, while ${r_1}$ is delayed term, l and m are constants. The term w is the value derivative of $\aleph$ at l and $\varepsilon$ represents a small positive constant. As $\varepsilon \leqslant \tau \leqslant l$ , then the necessary condition for the delay term bound should be 0 ≤ r1 ≤ | l − ε | such that system in Eq (2), remained DD equation.

When any of the value is added in $\tau$ , it becomes prediction, i.e., $\aleph (\tau + {r_1})$ , where $r$ is used as a prediction term and PD system is given as:

$\left\{ \begin{array}{l} \frac{{{d^2}\aleph }}{{d{\tau ^2}}} = h\left( {\tau , \, \aleph (\tau ), \, \aleph (\tau + {r_1})} \right), \;\;\;\;\; {r_1} > 0, \, \, l \leqslant \tau \leqslant m, \hfill \\ \aleph (\tau ) = \phi (\tau ), \;\;\;\;\;\varepsilon \leqslant \tau \leqslant l,\;\;\;\;\;\;\;\;0 \leqslant {r_1} \leqslant \left| {\varepsilon - l} \right|, \hfill \\ \frac{{d\aleph (l)}}{{d\tau }} = w, \hfill \end{array} \right.$

(3)

The pantograph differential systems (PDSs) have a variety of submission due to its enormous significance in the engineering fields, biological models and science areas. Some well-known applications are light absorption in the solid, dynamical population networks, communication networks, control systems, infectious viruses, propagation systems, electronic frameworks, transports and quantum mechanism ^{[32,33,34,35,36]}. There are various techniques that have been implemented to treat the PDSs, e.g., one-dimensional transformation approach, Taylor polynomial method and Direchlet series scheme ^{[37,38,39,40,41]}. The generic form of the PDSs is given as:

$\left\{ \begin{array}{l} \beta \frac{{{d^2}}}{{d{\tau ^2}}}\aleph (\beta \tau ) + \frac{\upsilon }{\tau }\frac{d}{{d\tau }}\aleph (\beta \tau ) + g(\tau , \aleph ) = f(\tau ), & \, \upsilon \geqslant 0, \, \, 0 < \tau \leqslant 1, \hfill \\ \aleph (0) = {u_1}, \, \, \frac{{d\aleph (0)}}{{d\tau }} = {u_2}, \hfill \end{array} \right.$

(4)

For solving the LE based singular systems, every technique has specific sensitivity, correctness, potential and efficiency, over and above, flaws, demerits and weaknesses. The wide-ranging computing potential schemes are used for the singular LE systems, DD models, PD systems and PDSs. To solve these singular systems, Gudermannian function is designed as a neural network optimized with the global scheme "genetic algorithms (GA)", local method "sequential quadratic programming (SQP)" and the hybridization of GA-SQP. The stochastic numerical schemes have been explored to solve various applications like HIV infection nonlinear system based infected latently cells ^{[42,43,44,45]}, higher order singular nonlinear systems ^[46,47,48], mosquito dispersal nonlinear system ^[49], heat conduction system based human head ^[50], doubly singular systems ^[51,52] and SIR nonlinear dengue fever system ^[53].These well-known submissions authenticated the implication of the stochastic computing solvers in terms of stability, exactitude and convergence. Hence, the design of Gudermannian function that work as a neural network is never been applied to solve the LE systems, DD models, PD systems and PDSs by using the optimization through the GA-SQP procedures. Some novel topographies of the proposed Gudermannian neural network along with GA-SQP are summarized as:

● A novel Gudermannian function is designed as a neural network is presented for solving the singular LE systems, DD models, PD systems and PDSs by using the optimization through the GA-SQP procedures.

● The numerical solutions of the stochastic procedures for solving four different examples of the singular systems are found precise and accurate.

● The analysis based on small and large neurons are effectively provided to authenticate the efficiency and complexity of the designed approach.

● The matching of the best and mean outcomes obtained by the stochastic procedures authenticate the consistency, accuracy and perfection of the singular LE systems, DD models, PD systems and PDSs.

● The reliability of the outcomes obtained by the proposed stochastic procedures through single/multiple executions via performance operators based on mean, Theil inequality coefficient (TIC), median (Med), variance account for (VAF), semi-interquartile range (SIR), maximum (Max) and mean absolute deviation (MAD) improve the capability of the scheme.

The remaining parts of the paper are given as: Section 2 labels the designed structure along with the statistical performance, Section 3 indicates the details of numerical results together with clarifications of the results. The neuron analysis will be presented in Section 4. The concluding remarks are provided in the last Section.

2. Methodology

In this section, the Gudermannian function is presented as a neural network for the singular LE systems, DD models, PD systems and PDSs by using the optimization through the GA-SQP procedures. The differential operators, objective function and optimization-based procedures using the proposed scheme are also presented.

2.1. Designed procedures through Gudermannian function as a neural network

The artificial neural networks are considered important to form the consistent and steadfast solutions for frequent submission arising in the various fields. In this modelling, $\hat \aleph (\tau)$ indicates the obtained performances via the Gudermannian function as a neural network and its derivatives are described as:

$\hat \aleph (\tau ) = \sum\limits_{m\, = \, 1}^k {{s_m}y({w_m}\tau + {z_m})}$

${\hat \aleph ^{(n)}}(\tau ) = \sum\limits_{m\, = \, 1}^k {{s_m}{y^{(n)}}({w_m}\tau + {z_m})}$

(5)

where, k shows the neurons and n represents the derivative. The objective function is y, whereas, s, $w$ , $z$ are the unidentified weights that are $\mathit{\boldsymbol{W}} = [\mathit{\boldsymbol{s}}, \mathit{\boldsymbol{w}}, \mathit{\boldsymbol{z}}]$ , for $\mathit{\boldsymbol{s}} = [{s_1}, {s_2}, {s_3}, ..., {s_k}]$ , $\mathit{\boldsymbol{w}} = [{w_1}, {w_2}, {w_3}, ..., {w_k}]$ and $\mathit{\boldsymbol{z}} = [{z_1}, {z_2}, {z_3}, ..., {z_k}]$ . The mathematical form of the Gudermannian function is given as:

$y(\tau ) = 2{\tan ^{ - 1}}{e^\tau } - 0.5\pi$

(6)

The approximate form of the continuous mapping based differential operations is shown as:

$\hat \aleph (\tau ) = \sum\limits_{m = 1}^k {{s_m}\left( {2{{\tan }^{ - 1}}{e^{({w_m}\tau + {z_m})}} - 0.5\pi } \right)} ,$

$\hat \aleph '(\tau ) = \sum\limits_{m = 1}^k {2{s_m}{w_m}\left( {\frac{{{e^{({w_m}\tau + {z_m})}}}}{{1 + {{\left( {{e^{({w_m}\tau + {z_m})}}} \right)}^2}}}} \right)} ,$

(7)

$\hat \aleph ''(\tau ) = \sum\limits_{m = 1}^k {2{s_m}w_m^2\left( {\frac{{{e^{({w_m}\tau + {z_m})}}}}{{1 + {{\left( {{e^{({w_m}\tau + {z_m})}}} \right)}^2}}} - \frac{{2{e^{{{({w_m}\tau + {z_m})}^3}}}}}{{{{\left( {1 + {{\left( {{e^{({w_m}\tau + {z_m})}}} \right)}^2}} \right)}^2}}}} \right)} ,$

To solve the singular LE systems, DD models, PD systems and PDSs, the objective function is formulated in terms of mean squared error, which is written as:

${E_{Fit}} = {E_{Fit - 1}} + {E_{Fit - 2}},$

(8)

where ${E_{Fit}}$ is an unsupervised learning-based error function related to the LE systems, DD models, PD systems and PDSs. ${E_{Fit - 1}}$ and ${E_{Fit - 2}}$ are the error based objective functions based on the differential model and ICs of the system (1)–(4). The fitness function ${E_{Fit - 1}}$ is constructed using the system (1), while for the system (2)–(4) can also be constructed.

${E_{Fit - 1}} = \frac{1}{N}{\sum\limits_{i = 1}^N {\left( {\frac{{{d^2}\hat \aleph ({\tau _i})}}{{d{\tau ^2}}} + \frac{\upsilon }{{{\tau _i}}}\frac{{d\hat \aleph ({\tau _i})}}{{d\tau }} + g\left( {{\tau _i}, \hat \aleph ({\tau _i})} \right) - f({\tau _i})\, } \right)} ^2},$

(9)

${E_{Fit - 2}} = \frac{1}{2}\left( {{{({{\hat \aleph }_0} - {u_1})}^2} + \, {{\left( {\frac{{d{{\hat \aleph }_0}}}{{d\tau }} - {u_2}} \right)}^2}\, } \right),$

(10)

where $Nh = 1, \, \, g(\tau, \aleph) = g({\tau _i}, \aleph ({\tau _i}))$ and ${\tau _i}\, = ih$ .

2.2. Network optimization GA-SQP

In this section, the Gudermannian function as a neural network is designed and numerical solutions for solving the singular LE systems, DD models, PD systems and PDSs through the optimization of GA-SQP are presented.

The evolutionary intelligent computing approach "GA" is designed on the basis of natural growth. GA is introduced by Holland in the previous century and then it is used as a leading factor in optimization using the constrained/unconstrained systems ^[54]. GA shows the optimal performances of mutation, crossover, heuristic and selection and broadly implemented in various fields of robotics, Bioinformatics, optics, astrophysics, digital communication, financial based mathematics, signal processing, nuclear based power systems, chemical industry and economics. Recently, it is implemented in the optimization of wind power systems ^[55], pipe networks ^[56], intrusion detection system ^[57], energy management models ^[58], circularity error unified evaluation ^[59], heterogeneous celebration ^[60], drying process of carrot ^[61], 2D industrial packing problems ^[62] and aquatic weed systems ^[63]. These reputed submissions inspired the authors to optimize through GA the singular LE systems, DD models, PD systems and PDSs for finding the best proposed outputs.

The hybridization of global search with any local search method performs the rapid convergence using the hybridize with the local search scheme. The best GA values are assigned as an initial input. The local search SQP is applied to normalize the parameters. SQP has been applied in various directions, e.g., bilinear model predictive control of a HVAC system ^[64], optimization of the engineering models ^[65], optimal control of building HVAC & R systems ^[66], ESP-implemented wells ^[67], optimal coordination of automated vehicles at intersections ^[68], dynamic combined economic emission dispatch models ^[69], cost minimization of a hybrid photovoltaic, diesel generator, and battery energy storage system ^[70].

Figure 1 indicates the graphical illustrations based Gudermannian function as a neural network and optimization procedures of GA-SQP, while the details of the optimization steps based on GA-SQP are given in Table 1 for singular LE systems, DD models, PD systems and PDSs. The settings of the parameter are adopted based on the experiments, experience, performance advantages and knowledge using different applications in the current work.

Figure 1. Graphical illustrations of the Gudermannian function as a neural network for the singular LE systems, DD models, PD systems and PDSs using the optimization-based GA-SQP.

DownLoad: Full-Size Img PowerPoint

Table 1. Optimization procedures of the designed scheme for the nonlinear singular LE systems, DD models, PD systems and PDSs.

[GA procedures]

Inputs: The chromosomes are defined with equal network entries as:

$\mathit{\boldsymbol{W}} = [\mathit{\boldsymbol{s}}, \mathit{\boldsymbol{w}}, \mathit{\boldsymbol{z}}]$
Population: The chromosomes vectors are indicated as:

$\mathit{\boldsymbol{s}} = [{s_1}, {s_2}, {s_3}, ..., {s_k}]$ ,

$\mathit{\boldsymbol{w}} = [{w_1}, {w_2}, {w_3}, ..., {w_k}]$ and

$\mathit{\boldsymbol{z}} = [{z_1}, {z_2}, {z_3}, ..., {z_k}]$ .
   Output: The global best weight vectors are W_GA-Best
   Initialization: Form a W, which is a weight vector having real elements to choose a chromosome. For the initial population, the design of 'W' is presented. To regulate the generations and assertions for the gaoptimset.
   Fitness assessment: Proficient the fitness (E_Fit) in population for W using Eqs (8)–(10).
   Stopping standards: Terminate if any forms is obtained
       • [E_Fit = 10^–20], [StallLimit = 150], [TolFun = TolCon = 10^–18], [PopulationSize = 285], [Generations = 90].
       • Other values: default
  Go to [storage], if stopping criteria meets
   Ranking: Rank the W to achieve E_Fit
   Reproduction:
       • [Selection~@uniform], [Mutations~adaptfeasible] & [Crossover~heuristic].
   Store: Save W_GA-Best, E_Fit, Generations, time and function counts.
End of [GA] process

Process of SQP
   Inputs: Input: W_GA-Best
  Output: Best GA-SQP are indicated as W_GA-SQP.
   Initialize: Use W_GA-Best, Bounded constraints, assignments, generations and other decelerations.
   Terminate: The procedure stops, if any on the below procedure obtains [E_Fit = 10^–17], [Iterations = 650], [TolCon = TolX = TolFun = 10^–22], [MaxEvalsFun = 286000], While [Stop]
  Fitness assessment: Evaluate the E_Fit, W, using Eqs (8)–(10). Modifications: Using the SQP, Invoke [fmincon]
   Accumulate
  Adjust count of function, W_GA-Best, time, iterations and E_Fit for the present values of SQP.
End of SQP
Data Generations
  The process of GASQP is replicate 50 times for a larger dataset for the singular LE systems, DD models, PD systems and PDSs using the optimization-based GA-SQP through the statistical clarifications.

| Show Table

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2.3. Performance procedures

In this section, the performances through different statistical gages T.I.C, VAF, S.I.R and MAD is provided to validate the constancy and reliability of the proposed Gudermannian function as a neural network using the optimizations of GA-SQP for solving the nonlinear singular LE systems, DD models, PD systems and PDSs. The mathematical formulations of these measures are provided as:

${\text{T}}{\text{.I}}{\text{.C = }}\frac{{\sqrt {\frac{1}{n}{{\sum\limits_{i = 1}^k {\left( {{\aleph _i} - {{\hat \aleph }_i}} \right)} }^2}} }}{{\left( {\sqrt {\frac{1}{n}\sum\limits_{i = 1}^k {\aleph _i^2} } + \sqrt {\frac{1}{n}\sum\limits_{i = 1}^k {\hat \aleph _i^2} } } \right)}},$

(11)

$\left\{ \begin{array}{l} VAF = \left( {1 - \frac{{{var} \left( {{\aleph _i} - {{\hat \aleph }_i}} \right)}}{{{var} \left( {{\aleph _i}} \right)}}} \right) \times 100, \hfill \\ EVAF = \left| {VAF - 100} \right|, \hfill \end{array} \right.$

(12)

$\left\{ \begin{array}{l} {\text{S}}{\text{.I}}{\text{.R}} = - 0.5 \times \left( {{Q_1} - {Q_3}} \right), \hfill \\ {Q_1}{\text{ = }}{1^{st}}\, {\text{quartile}}\, \, {\rm{\& }}\, \, \, {Q_3}\, = {3^{rd}}{\text{ quartile, }}\, \hfill \end{array} \right.$

(13)

${\text{MAD = }}\sum\limits_{i = 1}^k {\left| {{\aleph _i} - {{\hat \aleph }_i}} \right|}$

(14)

3. Simulations of the results and discussions

In this section, the simulation of the results and comprehensive discussions is performed to solve the singular LE systems, DD models, PD systems and PDSs by exploiting the Gudermannian function as a neural network along with the optimal performances of GA-SQP. One example of each nonlinear singular model based on the singular LE systems, DD models, PD systems and PDSs along with the statistical procedures is numerically discussed through the designed scheme.

Problem Ⅰ: Consider a nonlinear singular differential model having an exponential function

$\left\{ \begin{array}{l} \frac{{{d^2}\aleph }}{{d{\tau ^2}}} + \frac{{0.5}}{\tau }\frac{{d\aleph }}{{d\tau }} + ({e^{2\aleph (\tau )}} - 0.5{e^{\aleph (\tau )}}) = 0, \, \, \, \hfill \\ \aleph (0) = \log (2), \, \, \, \aleph (1) = 0.\, \, \hfill \end{array} \right.$

(15)

The fitness function of the above function is given as:

${E_{Fit}} = \frac{1}{N}{\sum\limits_{i = 1}^N {\left( {{\tau _i}\frac{{{d^2}\hat \aleph ({\tau _i})}}{{d{\tau ^2}}} + 0.5\frac{{d\hat \aleph ({\tau _i})}}{{d\tau }} + {\tau _i}({e^{2\hat \aleph ({\tau _i})}} - 0.5{e^{\hat \aleph ({\tau _i})}})} \right)} ^2} + \frac{1}{2}\left( {{{\left( {{{\hat \aleph }_0} - \log (2)} \right)}^2} + {{\left( {{{\hat \aleph }_N}} \right)}^2}} \right).$

(16)

The exact form of the solution is $\log 2 - \log ({\tau ^2} + 1)$ .

Problem 2: Consider a nonlinear singular DD form of the equations with multiple-trigonometric functions (MTFs) in its forcing function

$\left\{ \begin{array}{l} \frac{{{d^2}\aleph (\tau - 1)}}{{d{\tau ^2}}} + \frac{3}{\tau }\frac{{d\aleph (\tau - 1)}}{{d\tau }} + {\aleph ^{ - 3}} = {\sec ^3}\tau - \cos (1 - \tau ) + \frac{3}{\tau }\sin (1 - \tau ), \hfill \\ \aleph (0) = 1, \, \, \, \, \frac{{d\aleph (0)}}{{d\tau }} = 0. \hfill \end{array} \right.$

(17)

The fitness function of the above function is given as:

$\begin{array}{l} {E_{Fit}} = \frac{1}{N}{\sum\limits_{i = 1}^N {\left( {{\tau _i}\frac{{{d^2}\hat \aleph ({\tau _i} - 1)}}{{d{\tau ^2}}} + 3\frac{{d\hat \aleph ({\tau _i} - 1)}}{{d\tau }} + {\tau _i}\hat \aleph _i^{ - 3} + {\tau _i}\cos \left( {1 - {\tau _i}} \right) - {\tau _i}{{\sec }^3}({\tau _i}) - 3\sin \left( {1 - {\tau _i}} \right)} \right)} ^2} \hfill \\ \, \, \, \, \, \, \, \, \, \, + \frac{1}{2}\left( {{{\left( {{{\hat \aleph }_0} - 1} \right)}^2} + {{\left( {\frac{{d{{\hat \aleph }_0}}}{{d\tau }}} \right)}^2}} \right). \hfill \end{array}$

(18)

The exact form of the solution is $\cos \tau$ .

Problem 3: Consider a nonlinear singular PD form of the equations with MTFs in its forcing function

$\left\{ \begin{array}{l} \frac{{{d^2}\aleph (\tau + 1)}}{{d{\tau ^2}}} + \frac{1}{\tau }\frac{{d\aleph (\tau + 1)}}{{d\tau }} + {\aleph ^3} = - \sin (\tau + 1) + {\sin ^3}\tau + \frac{1}{\tau }\cos (\tau + 1), \hfill \\ \aleph (0) = 0, \, \, \, \, \frac{{d\aleph (0)}}{{d\tau }} = 1. \hfill \end{array} \right.$

(19)

The fitness function of the above function is given as:

$\begin{array}{l} {E_{Fit}} = \frac{1}{N}{\sum\limits_{i = 1}^N {\left( {{\tau _i}\frac{{{d^2}\hat \aleph ({\tau _i} + 1)}}{{d{\tau ^2}}} + 3\frac{{d\hat \aleph ({\tau _i} + 1)}}{{d\tau }} + {\tau _i}\hat \aleph _i^3 + {\tau _i}\sin \left( {{\tau _i} + 1} \right) - {\tau _i}{{\sin }^3}({\tau _i}) - \cos \left( {{\tau _i} + 1} \right)} \right)} ^2} \hfill \\ \, \, \, \, \, \, \, \, \, \, + \frac{1}{2}\left( {{{\left( {{{\hat \aleph }_0}} \right)}^2} + {{\left( {\frac{{d{{\hat \aleph }_0}}}{{d\tau }} - 1} \right)}^2}} \right). \hfill \end{array}$

(20)

The exact form of the solution is $\sin \tau$ .

Problem 4: Consider a nonlinear singular PDS having MTFs in its forcing function

$\left\{ \begin{array}{l} 0.5\frac{{{d^2}}}{{d{\tau ^2}}}\aleph (0.5\tau ) + \frac{3}{\tau }\frac{d}{{d\tau }}\aleph (0.5\tau ) + \frac{1}{{{\aleph ^2}}} = {\sec ^2}(\tau ) - 0.5\cos \left( {0.5\tau } \right) - \frac{3}{\tau }\sin \left( {0.5\tau } \right), \hfill \\ \aleph (0) = 1, \, \, \, \, \frac{{d\aleph (0)}}{{d\tau }} = 0, \hfill \end{array} \right.$

(21)

The fitness function of the above function is given as:

$\begin{array}{l} {E_{Fit}} = \frac{1}{N}{\sum\limits_{i = 1}^N {\left( {0.5{\tau _i}\frac{{{d^2}\hat \aleph (0.5{\tau _i})}}{{d{\tau ^2}}} + 3\frac{{d\hat \aleph (0.5{\tau _i})}}{{d\tau }} + {\tau _i}\hat \aleph _i^{ - 2} + 0.5{\tau _i}\cos \left( {0.5{\tau _i}} \right) - {\tau _i}{{\sec }^2}({\tau _i}) + 3\sin \left( {\frac{1}{2}{\tau _i}} \right)} \right)} ^2} \hfill \\ \, \, \, \, \, \, \, \, \, \, + \frac{1}{2}\left( {{{\left( {{{\hat \aleph }_0} - 1} \right)}^2} + {{\left( {\frac{{d{{\hat \aleph }_0}}}{{d\tau }}} \right)}^2}} \right). \hfill \end{array}$

(22)

The exact form of the solution is $\cos \tau$ .

To solve the nonlinear singular LE systems, DD models, PD systems and PDSs, the optimization procedures through GA-SQP based on the Gudermannian function as a neural network for hundred independent trials are applied. The mathematical for of the best weight sets that authenticate the estimated outcomes for 30 variables. The convergence/learning curves along with the updated iterations through the merit functions, are drawn in Figures 2–5 for problems 1–4, respectively. It can be found that the performance of GA initially performed fast using the optimization procedures, but after a few generations the convergence ability decreased, enhanced further with the SQP hybridization process. Hence, the GA-SQP scheme provided reliable convergent results for all four problems. In addition, it can be authenticated that the stochastic scheme is reliable for solving the singular LE systems, DD models, PD systems and PDSs.

$\begin{array}{l} {{\hat \aleph }_1}(\tau ) = - 0.538(2{\tan ^{ - 1}}{e^{(0.4672\tau + 1.6674)}} - 0.5\pi ) + 0.8698(2{\tan ^{ - 1}}{e^{( - 0.4560\tau - 0.0525)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, - 1.3056(2{\tan ^{ - 1}}{e^{( - 1.509\tau - 0.3651)}} - 0.5\pi ) + 1.0878(2{\tan ^{ - 1}}{e^{( - 0.0517\tau - 0.1826)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, - 1.4714(2{\tan ^{ - 1}}{e^{(1.2901\tau - 0.1349)}} - 0.5\pi ) - 0.3913(2{\tan ^{ - 1}}{e^{(0.5111\tau - 0.56293)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, - 1.0740(2{\tan ^{ - 1}}{e^{( - 1.439\tau - 1.7516)}} - 0.5\pi ) - 0.9514(2{\tan ^{ - 1}}{e^{(0.5719\tau + 0.9982)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, + 0.6917(2{\tan ^{ - 1}}{e^{( - 0.113\tau + 0.1376)}} - 0.5\pi ) - 0.5887(2{\tan ^{ - 1}}{e^{( - 1.142\tau - 0.2003)}} - 0.5\pi ), \hfill \end{array}$

(23)

$\begin{array}{l} {{\hat \aleph }_2}(\tau ) = - 6.391(2{\tan ^{ - 1}}{e^{( - 5.200\tau + 15.5023)}} - 0.5\pi ) + 6.4850(2{\tan ^{ - 1}}{e^{( - 8.572\tau + 15.306)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, + 5.0955(2{\tan ^{ - 1}}{e^{( - 0.198\tau + 16.0699)}} - 0.5\pi ) + 6.5070(2{\tan ^{ - 1}}{e^{(0.2473\tau - 18.900)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, + 11.937(2{\tan ^{ - 1}}{e^{(3.6410\tau - 8.6838)}} - 0.5\pi ) - 0.0660(2{\tan ^{ - 1}}{e^{( - 2.363\tau + 0.1083)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, - 15.457(2{\tan ^{ - 1}}{e^{(0.6768\tau + 1.5181)}} - 0.5\pi ) - 18.200(2{\tan ^{ - 1}}{e^{(0.5861\tau + 1.6859)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, + 2.6041(2{\tan ^{ - 1}}{e^{( - 0.919\tau + 9.5774)}} - 0.5\pi ) + 19.024(2{\tan ^{ - 1}}{e^{(0.8335\tau + 1.3028)}} - 0.5\pi ), \hfill \end{array}$

(24)

$\begin{array}{l} {{\hat \aleph }_3}(\tau ) = 1.8340(2{\tan ^{ - 1}}{e^{( - 0.714\tau + 2.0500)}} - 0.5\pi ) + 0.3524(2{\tan ^{ - 1}}{e^{(0.9476\tau + 0.2285)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, + 2.4940(2{\tan ^{ - 1}}{e^{(1.0860\tau + 0.1740)}} - 0.5\pi ) - 1.0561(2{\tan ^{ - 1}}{e^{(0.6352\tau + 0.1740)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, + 1.2784(2{\tan ^{ - 1}}{e^{( - 1.188\tau + 0.3277)}} - 0.5\pi ) + 0.383(2{\tan ^{ - 1}}{e^{( - 1.0618\tau - 0.8162)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, + 1.8638(2{\tan ^{ - 1}}{e^{(0.7798\tau - 0.1478)}} - 0.5\pi ) - 0.890(2{\tan ^{ - 1}}{e^{(0.2622\tau + 0.9342)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, + 1.1649(2{\tan ^{ - 1}}{e^{( - 2.145\tau - 0.6517)}} - 0.5\pi ) + 0.357(2{\tan ^{ - 1}}{e^{(3.0119\tau + 0.5433)}} - 0.5\pi ), \hfill \end{array}$

(25)

$\begin{array}{l} {{\hat \aleph }_4}(\tau ) = - 12.68(2{\tan ^{ - 1}}{e^{(3.1706\tau + 2.0684)}} - 0.5\pi ) - 0.4048(2{\tan ^{ - 1}}{e^{(1.7796\tau - 6.5517)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, + 7.3132(2{\tan ^{ - 1}}{e^{(8.6833\tau + 9.5117)}} - 0.5\pi ) + 7.0854(2{\tan ^{ - 1}}{e^{(0.4758\tau + 7.1872)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, + 4.5364(2{\tan ^{ - 1}}{e^{( - 20.00\tau - 0.2227)}} - 0.5\pi ) + 11.966(2{\tan ^{ - 1}}{e^{( - 1.907\tau - 1.6907)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, + 8.3815(2{\tan ^{ - 1}}{e^{(17.309\tau + 1.0122)}} - 0.5\pi ) - 7.6265(2{\tan ^{ - 1}}{e^{( - 2.4212\tau + 0.8309)}} - 0.5\pi ) \hfill \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, + 0.9174(2{\tan ^{ - 1}}{e^{( - 5.101\tau - 5.8219)}} - 0.5\pi ) - 1.2451(2{\tan ^{ - 1}}{e^{(0.0011\tau + 17.6256)}} - 0.5\pi ), \hfill \end{array}$

(26)

Figure 2. Convergence performances through the optimization procedures for Problem 1. (a) Convergence performances of GA. (b) Convergence performances of GA-SQP.

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Figure 3. Convergence performances through the optimization procedures for Problem 2. (a) Convergence performances of GA. (b) Convergence performances of GA-SQP.

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Figure 4. Convergence performances through the optimization procedures for Problem 3. (a) Convergence performances of GA. (b) Convergence performances of GA-SQP.

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Figure 5. Convergence performances through the optimization procedures for Problem 4. (a) Convergence performances of GA. (b) Convergence performances of GA-SQP.

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The plots of the best results obtained through the best weight vectors, absolute error (AE), results comparison, performance indices based different operators and convergence measures are provided in the Figures 6–9. The best weights and comparison outcomes for 30 variables is drawn in Figure 3. The best weights show the best trials, which are drawn in Figure 6(a)–(d) and comparison of the best, exact and mean results is performed in –. It is observed that the best, exact and mean results for each singular system are overlapped. One can prove the witness of the proposed schemes through the demonstration of these results. The performances of the operators ${E_{Fit}}$ , EVAF, TIC and MAD are plotted in Figure 7(a), while the graphs of AE are drawn in Figure 7(b). It is seen that the best AE values lie around 10^–5–10^–6 for nonlinear singular problem (1), 10^–5–10^–7 for nonlinear singular DD problem (2), 10^–3–10^–5 for nonlinear singular PD problem (3) and 10^–4–10^–6 for nonlinear singular PDSs (4), respectively. These AE values indicate the correctness of the designed Gudermannian function as a neural network using the optimization procedures of the GA-SQP for solving the singular system. The ${E_{Fit}}$ performances are calculated around 10^–9–10^–10 for nonlinear singular problem (1), 10^–8–10^–9 for nonlinear singular DD problem (2), 10^–11–10^–12 for nonlinear singular PD problem (3) and 10^–8–10^–10 for nonlinear singular PDSs (4), respectively. The EVAF performances are found around 10^–11–10^–12 for nonlinear singular problem (1), 10^–9–10^–10 for nonlinear singular DD problem (2), 10^–7–10^–8 for nonlinear singular PD problem (3) and 10^–8–10^–9 for nonlinear singular PDSs (4), respectively. The TIC performances are calculated around 10^–9–10^–10 for nonlinear singular problem (1), 10^–8–10^–10 for nonlinear singular DD problem (2), 10^–7–10^–8 for nonlinear singular PD problem (3) and 10^–8–10^–9 for nonlinear singular PDSs (4), respectively. The MAD performances lie around 10^–5–10^–6 for nonlinear singular problem (1) and DD problem (2), 10^–3–10^–4 for nonlinear singular PD problem (3) and 10^–5–10^–6 for nonlinear singular PDSs (4), respectively.

Figure 6. Best set of weight vectors for 30 variables along with the mean, best and exact solutions for the singular models.

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Figure 7. AE and performances based different operators for the singular systems 1–4. (a) AE performance for the singular systems 1–4. (b) Performance Measures for the singular systems 1–4.

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Figure 8. Convergence performances-based Fitness and EVAF for the singular systems 1–4. Convergence through Fitness for the singular systems 1–4. Convergence through EVAF for the singular systems 1–4.

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Figure 9. Convergence performances-based TIC and MAD for the singular systems 1–4. (a) Convergence through TIC for the singular systems 1–4. (b) Convergence through MAD for the singular systems 1–4.

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The convergence performance based on ${E_{Fit}}$ , EVAF, TIC and MAD is provided in and . It is observed that most of the trials achieved high level of fitness. One can conclude that an accurate, specific and precise ${E_{Fit}}$ , EVAF, TIC and MAD values have been achieved for the nonlinear singular LE systems, DD models, PD systems and PDSs.

The statistical performances have been presented using the Gudermannian function as a neural network using the optimization procedures of GA-SQP for solving the nonlinear singular LE systems, DD models, PD systems and PDSs. The statistical gages based on the Minimum (Min), Mean, Median (Med), standard deviation (STD) and semi–interquartile ranges (SIR) for 50 trials are provided in Table 2. The Min values indicate the best trials, while the formulation of the SIR is provided in system (13). It is observed that the Min, Mean, Med, STD and SIR values are found in good ranges and approve the accuracy/precision of the designed scheme. The global measures for the nonlinear singular LE systems, DD models, PD systems and PDSs using the proposed approach are provided in Table 3. The Min values using the [G. FIT], [G. TIC] and [G. EVAF] operators found around 10⁻⁹–10⁻¹², 10⁻⁸–10⁻¹⁰ and 10⁻⁷–10⁻¹¹ for the nonlinear singular LE system (1), DD models (2), PD system (3) and PDSs (4), respectively. The Med values using the [G. FIT], [G. TIC] and [G. EVAF] operators found around 10⁻⁵–10⁻⁹, 10⁻⁶–10⁻⁸ and 10⁻⁵–10⁻⁹ for the nonlinear singular LE system (1), DD models (2), PD system (3) and PDSs (4), respectively. These optimum performances through the global operators approve the accurateness of the designed scheme. The complexity studies for the nonlinear singular LE systems, DD models, PD systems and PDSs using the function calculations, implemented time and generations are provided in Table 4. One can conclude that the average function calculations, implemented time and generations are calculated around 30.53279,448.52 and 29761.935, respectively, for the nonlinear singular LE systems, DD models, PD systems and PDSs.

Table 2. Statistical performances for the nonlinear singular LE systems, DD models, PD systems and PDSs.

Problem	Gages	Statistical interpretations of the singular systems
Problem	Gages	0	0.10	0.20	0.30	0.40	0.50	0.60	0.70	0.80	0.90	1.0
1	Min	4×10^–7	2×10^–6	4×10^–6	3×10^–6	4×10^–6	4×10^–6	4×10^–6	3×10^–6	3×10^–6	3×10^–6	2×10^–6
	Mean	5×10^–5	1×10^–4	3×10^–4	4×10^–4	4×10^–4	3×10^–4	3×10^–4	3×10^–4	2×10^–4	2×10^–4	1×10^–4
	Med	7×10^–6	3×10^–5	5×10^–5	6×10^–5	6×10^–5	6×10^–5	6×10^–5	6×10^–5	5×10^–5	4×10^–5	3×10^–5
	STD	2×10^–4	2×10^–4	9×10^–4	9×10^–4	9×10^–4	8×10^–4	7×10^–4	5×10^–4	4×10^–4	3×10^–4	3×10^–4
	SIR	2×10^–5	9×10^–5	2×10^–4	2×10^–4	2×10^–4	2×10^–4	2×10^–4	2×10^–4	1×10^–4	1×10^–4	8×10^–5
2	Min	1×10^–7	1×10^–6	6×10^–8	4×10^–6	1×10^–6	9×10^–6	8×10^–6	3×10^–6	4×10^–7	2×10^–6	8×10^–7
	Mean	3×10^–2	4×10^–2	3×10^–3	6×10^–3	6×10^–3	6×10^–3	6×10^–3	6×10^–3	5×10^–3	5×10^–3	6×10^–3
	Med	7×10^–5	6×10^–3	2×10^–2	2×10^–2	2×10^–2	2×10^–2	2×10^–2	2×10^–2	1×10^–2	4×10^–3	1×10^–2
	STD	2×10^–1	1×10^–3	8×10^–3	1×10^–2	1×10^–2	1×10^–2	1×10^–2	1×10^–2	1×10^–2	1×10^–2	1×10^–2
	SIR	1×10^–3	1×10^–1	7×10^–1	1×10^–3	3×10^–3	1×10^–3	2×10^–3	2×10^–3	3×10^–3	3×10^–3	4×10^–3
3	Min	3×10^–9	1×10^–5	5×10^–5	1×10^–4	2×10^–4	2×10^–4	6×10^–5	7×10^–5	2×10^–4	3×10^–4	3×10^–4
	Mean	4×10^–5	2×10^–2	5×10^–2	6×10^–2	8×10^–2	9×10^–2	1×10^–1	1×10^–1	1×10^–1	1×10^–1	1×10^–1
	Med	5×10^–8	5×10^–4	1×10^–3	2×10^–3	3×10^–3	3×10^–3	4×10^–3	4×10^–3	4×10^–3	4×10^–3	4×10^–3
	STD	2×10^–4	7×10^–2	1×10^–1	2×10^–1	2×10^–1	3×10^–1	3×10^–1	3×10^–1	3×10^–1	3×10^–1	3×10^–1
	SIR	8×10^–8	4×10^–3	1×10^–2	2×10^–2	2×10^–2	3×10^–2	3×10^–2	3×10^–2	3×10^–2	3×10^–2	3×10^–2
4	Min	4×10^–7	2×10^–6	4×10^–6	3×10^–6	4×10^–6	4×10^–6	4×10^–6	3×10^–6	3×10^–6	3×10^–6	2×10^–6
	Mean	5×10^–5	1×10^–4	3×10^–4	4×10^–4	4×10^–4	3×10^–4	3×10^–4	3×10^–4	2×10^–4	2×10^–4	1×10^–4
	Med	7×10^–6	3×10^–5	5×10^–5	6×10^–5	6×10^–5	6×10^–5	6×10^–5	6×10^–5	5×10^–5	4×10^–5	3×10^–5
	STD	2×10^–4	2×10^–4	9×10^–4	9×10^–4	9×10^–4	8×10^–4	7×10^–4	5×10^–4	4×10^–4	3×10^–4	3×10^–4
	SIR	2×10^–5	9×10^–5	2×10^–4	2×10^–4	2×10^–4	2×10^–4	2×10^–4	2×10^–4	1×10^–4	1×10^–4	8×10^–5

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Table 3. Global measures for the nonlinear singular LE systems, DD models, PD systems and PDSs.

Problem	[G. FIT]		[G. TIC]		[G. EVAF]
Problem	Min	Med	Min	Med	Min	Med
1	4.2066E–10	1.2466E–08	7.5497E–10	1.4839E–08	2.0401E–11	5.3046E–09
2	2.0990E–09	2.5323E–05	2.9352E–09	5.0278E–06	2.7706E–10	2.0433E–06
3	6.8622E–12	3.9934E–09	2.2284E–08	1.2320E–06	2.0761E–07	3.6934E–05
4	5.3601E–10	4.7692E–08	3.8819E–09	3.5679–06	7.1462E–10	5.6785E–06

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Table 4. Complexity studies for the nonlinear singular LE systems, DD models, PD systems and PDSs.

Problem	Function Calculations		Executed Time		Generations
Problem	[Mean]	[STD]	[Mean]	[STD]	[Mean]	[STD]
1	29284.44000	12995.57912	447.58000	209.32819	35.53745	18.20444
2	35162.58000	12725.02770	542.18000	203.48592	34.83724	12.63487
3	27556.22000	13006.06683	410.02000	183.59933	26.14958	12.53492
4	27044.50000	18216.81644	394.30000	270.97987	25.60687	17.78549

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4. Neuron analysis

In this section, the detailed neuron analysis will be investigated for taking small and large neurons for solving the nonlinear singular LE systems, DD models, PD systems and PDSs through the optimization procedures of the GA–SQP. Two small and large values of the neurons are taken as 3 and 15, which are demonstrated in Figure 10(a), (b). The comparative investigations through the proposed and exact solutions are provided in Figure 10. It is observed in Figure 10(a) that the values of the AE for 3 neurons lie around 10^–1 to 10^–2 for problem 1, 10^–2 to 10^–4 for problems 2 and 3, while for the last problem, the AE found around 10^–3 to 10^–4. Figure 10(b) indicates that the values of the AE for 15 neurons lie around 10^–6 to 10^–8 for problem 1, 10^–4 to 10^–6 for problem 2, 10^–2 to 10^–10 for problem 3, while for the last problem, the AE found around 10^–2 to 10^–6. Based on these AE performances, one can accomplish that the large number of neurons perform better as small neurons, but the computational cost can be higher in case of large neurons.

Figure 10. Performances of the AE based on the nonlinear singular LE systems, DD models, PD systems and PDSs using 2 and 20 number of neurons. (a) AE for each problem of the singular system using 3 number of neurons. (b) AE for each problem of the singular system using 15 number of neurons.

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5. Conclusions

In this study, the design of the Gudermannian neural network is presented along with the optimization procedures of global and local search schemes. The nonlinear models based on the singular delay, prediction and the pantograph forms have been numerically investigated using the proposed solver. These kinds of nonlinear models are considered tough to solve due to the singularity at the origin, so it becomes more complicated when the delayed, prediction and pantograph terms are involved in the singular equations. Therefore, the designed scheme is considered as an impressive solver for these different types of the singular systems. The overlapping of mean and best solutions with the exact results indicates the correctness of the proposed scheme. The plots of AE are found in good measures, which are calculated around 10^–5 to 10^–7 in the singular delay, prediction and the pantograph forms using 10 numbers of neurons. The best calculated statistical performances based on EVAF, TIC and MAD shows the reliability of the proposed stochastic numerical scheme. The statistical explanations using 50 trials are also provided for the nonlinear singular LE systems, DD models, PD systems and PDSs using the Min, SIR and Med operators, which validates the exactness, robustness and perfection of the scheme. Moreover, neuron analysis by taking small and large neurons is provided to solve these nonlinear singular systems using the optimization procedures-based GA-SQP.

In future, the proposed scheme can be implemented to solve the nonlinear systems, fractional models and fluid mechanics models ^{[71,72,73,74,75,76,77,78,79,80,81,82]}. Moreover, time-varying delays and impulsive effects can be studied in future ^[83,84,85].

Conflict of interest

The authors state that they have no conflict of interest.

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沈阳化工大学材料科学与工程学院沈阳 110142

Mathematical Biosciences and Engineering

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Mathematical Biosciences and Engineering

A novel design of Gudermannian function as a neural network for the singular nonlinear delayed, prediction and pantograph differential models

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Abstract

1. Introduction

2. Methodology

2.1. Designed procedures through Gudermannian function as a neural network

2.2. Network optimization GA-SQP

2.3. Performance procedures

3. Simulations of the results and discussions

4. Neuron analysis

5. Conclusions

Conflict of interest

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Mathematical Biosciences and Engineering

A novel design of Gudermannian function as a neural network for the singular nonlinear delayed, prediction and pantograph differential models

Related Papers:

Abstract

1. Introduction

2. Methodology

2.1. Designed procedures through Gudermannian function as a neural network

2.2. Network optimization GA-SQP

2.3. Performance procedures

3. Simulations of the results and discussions

4. Neuron analysis

5. Conclusions

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

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