Swarm Intelligence Procedures Using Meyer Wavelets as a Neural Network for the Novel Fractional Order Pantograph Singular System

Abstract

The purpose of the current investigation is to find the numerical solutions of the novel fractional order pantograph singular system (FOPSS) using the applications of Meyer wavelets as a neural network. The FOPSS is presented using the standard form of the Lane–Emden equation and the detailed discussions of the singularity, shape factor terms along with the fractional order forms. The numerical discussions of the FOPSS are described based on the fractional Meyer wavelets (FMWs) as a neural network (NN) with the optimization procedures of global/local search procedures of particle swarm optimization (PSO) and interior-point algorithm (IPA), i.e., FMWs-NN-PSOIPA. The FMWs-NN strength is pragmatic and forms a merit function based on the differential system and the initial conditions of the FOPSS. The merit function is optimized, using the integrated capability of PSOIPA. The perfection, verification and substantiation of the FOPSS using the FMWs is pragmatic for three cases through relative investigations from the true results in terms of stability and convergence. Additionally, the statics’ descriptions further authorize the presentation of the FMWs-NN-PSOIPA in terms of reliability and accuracy.

1. Introduction

The differential systems signified with fractional and integer orders are used extensively in several applications of physics, engineering and mathematics. Fractional calculus operators (FCOs) have been famous for scientists during the last three to four decades [1]. Some noteworthy applications of the FCOs are Weyl–Riesz [2], Grnwald–Letnikov [3], Riemann–Liouville [4] and Erdlyi–Kober [5]. Many scientists have described the importance of FCOs in diverse areas of fractional viscoplasticity models [6], Earth-based dynamical investigations [7], reaction networks of surface–volume [8], electromagnetic studies [9], detection of road edges [10], comprehensive performances in authentic supplies [11], mathematical nanofluids [12], viscoelastic systems [13] and LC-electric fractal circuit systems [14].

It is always considered tough to solve singular models with the use of numerical or analytical approaches. Singular models arise in spherical surfaces of gas clouds, astrophysics studies and quantum mechanics and are always considered difficult for researchers due to the harder nature and the occurrence of singular points. A variety of deterministic schemes have been executed to solve singular based models [15,16,17,18]. The literature form of the singular second-order system is described as [19,20,21]:

$$\{\begin{array}{l}\frac{d^{2}k}{d{\eta }^2}+\frac{\gamma }{\eta }\frac{dk}{d\eta }+g(k)=h(\eta ),\hspace{1em}\hspace{1em}\hspace{1em}\gamma \ge 1\\ k(0)=i_1,\hspace{1em}{k}^{\prime }(0)=i_2,\end{array}$$

(1)

where $\gamma$ represents the shape vector value in Equation (1), the singularity arises at $\eta =0$, h(η) is the forcing term and $g(k)$ is some known function of $k$, while i₁ and i₂ are the constants representing the initial conditions.

The differential form of the pantographs is a specific state of the functional model that contains proportional delay factors. Tayler and Ockendon introduced the “pantograph” word in the 7th decade of the 19th century by working on the collection of the pantograph electric head [22]. The pantograph differential system (PDS) has achieved huge importance due to its well-known applications in the biological system of cell growth [23], asymptotic constancy characteristics [24] and control networks [25]. A number of approaches have been applied to solve singular models, e.g., the Chebyshev spectral method [26], spectral tau technique [27], intricate homotopy optimal method [28], Genocchi scheme of operation matrix [29], Epsilon–Ritz-based least-square approach [30] and Taylor method [31]. The novelty of this study is described in two steps as follows:

• The design of a novel fractional order pantograph singular system (FOPSS) is presented using the suitable derivation process.
• The computing process based on machine learning or soft computing knacks is implemented to solve the novel FOPSS using the applications of the Meyer wavelets based fractional neural network.

The current investigations are relevant to design a novel FOPSS using the applications of Meyer wavelets as a neural network by implementing the concepts of the traditional singular second-order differential equation and pantograph differential model [32]. The numerical discussions of the FOPSS are provided based on the fractional Meyer wavelets (FMWs) as a neural network (NN) with the optimization procedures of global/local search procedures of particle swarm optimization (PSO) and interior-point algorithm (IPA), i.e., FMWs-NN-PSOIPA. The stochastic schemes based on numerical measures is applied to solve a variety of applications [33,34,35,36,37,38,39,40], and a few potential recently reported applications include the solution of nonlinear Lane–Emden multi-pantograph delay based ordinary differential equations (ODEs) [41], Gudermannian neural networks for sODEs [42], neuro-swarming approach to singular with multiple delay ODEss [43], intelligent backpropagated networks for solving Lene–Emden singular ordinary differential systems with pantograph delays [44], novel design of Morlet wavelet neural networks for solving singular pantograph nonlinear differential models [45], third kind of multi-singular nonlinear systems [46], novel design of evolutionary integrated heuristics for singular systems [47], Morlet wavelet neural networks for solving higher order singular nonlinear ODEs [48] and wavelet analysis on some surfaces of revolution [49]. All these applications inspire the authors to investigate the design of FOPSS, which has never been implemented nor treated, by using the proposed heuristics of FMWs-NN-PSOIPA.

FMWS-NN-PSOIPA is implemented to solve the novel FOPSS using the applications of Meyer wavelets as neural networks. The FOPSS is stiff in nature, involving singular points, pantographs and fractional order nature. A few novel features of the FMWs-NN-PSOIPA are provided as follows:

• A novel FOPSS is presented using the pantograph differential system (PDS) and fundamental form of the second-order singular model.
• The numerical performance of the novel FOPSS is obtained by using the designed approach FMWs-NN-PSOIPA, which is used to compare the obtained results and to perform the values of the absolute error (AE).
• The Meyer computing solvers via FMWs-NN-PSOIPA is applied to solve three examples based on the novel FOPSS to authenticate the convergence, precision and stability.
• The reliability of the proposed FMWS-NN-PSOIPA is accessible using the statistical procedures in terms of semi-interquartile range (S.I.R), Theil’s inequality coefficient (T.I.C) and variance account for (VAF).

Alongside the precise performance of the novel FOPSS, its easy understandable process, smooth operations, steadiness and sturdiness are other valued compensations of the fractional Meyer intelligent computing solver.

The other paper parts are presented as follows: Section 2 represents the design of the novel FOPSS using the applications of Meyer wavelets as a neural network. Section 3 presents the proposed procedure using the FMWs-NN-PSOIPA. Section 4 indicates the statistical performance. Section 5 defines the concluding remarks and future research reports.

2. Construction of the Novel FOPSS

This part of the study shows the construction of the novel FOPSS along with the comprehensive details of the shape factor (SF), singular point and fractional order factor. The necessary procedural steps to construct the novel FOPSS are drawn on the flow chart, while the construction of the novel FOPSS is provided as:

$${\eta }^{-p}\frac{d^r}{d{\eta }^r}\left({\eta }^p\frac{d^u}{d{\eta }^u}\right)k\left(\frac{\eta }{2}\right)+g(k)=h(\eta ),$$

(2)

For the novel FOPSS, the values of p and r are provided as follows:

$$r=1,\hspace{1em}\hspace{1em}u=\Re , \mathrm{where} 0<\Re <1.$$

(3)

The restructured form for the above two systems is given as:

$${\eta }^{-p}\frac{d}{d\eta }\left({\eta }^p\frac{d^{\Re }}{d{\eta }^{\Re }}\right)k\left(\frac{\eta }{2}\right)+g(k)=h(\eta ).$$

(4)

The simplification of Equation (4) is written as:

$$\frac{d}{d\eta }\left({\eta }^p\frac{d^{\Re }}{d{\eta }^{\Re }}\right)k\left(\frac{\eta }{2}\right)={\eta }^p\frac{d^{\Re +1}}{d{\eta }^{\Re +1}}k\left(\frac{\eta }{2}\right)+p{\eta }^{p-1}\frac{d^{\Re }}{d{\eta }^{\Re }}k\left(\frac{\eta }{2}\right).$$

(5)

The achieved form of the novel FOPSS is provided as:

$$\{\begin{array}{l}\frac{d^{\Re +1}}{d{\eta }^{{}^{\Re +1}}}k\left(\frac{\eta }{2}\right)+\frac{p}{\eta }\frac{d^{\Re }}{d{\eta }^{\Re }}k\left(\frac{\eta }{2}\right)+g(k)=h(\eta ),\\ k(0)=0,\hspace{1em}\hspace{1em}\hspace{1em}k(1)=0.\hspace{1em}\hspace{1em}\hspace{1em}\end{array}$$

(6)

The novel FOPSS is achieved above in Equation (6) with the singular point occurring at $\eta =0$; The fractional terms are noticed as $\Re$ and $\Re +1$, respectively, whereas the SF is at $p=1$. The flow chart based on the novel FOPSS describing the essential phases is provided in the block structure in Figure 1. These procedures are used to design a novel FOPSS system described in terms of mathematical relation given in Equations (2) to (6).

3. Methodology: FMWs-NN-PSOIPA

This section shows the proposed methodology using the FMWs as a NN along with the optimal methods of PSOIPA for solving the novel FOPSS. The process flow diagram of FMWs-NN-PSOIPA is portrayed in Figure 2 in terms of five blocks for the problem, modeling, learning, storage and results. The error function is constructed using the differential form and boundary conditions (BCs) together with the optimization procedure of PSOIPA provided here.

3.1. Objective Function: FMWs-NN

The ANNs systems are familiar to obtain the numerical performances of numerous systems based on the fractional order [42,43]. In the below system, $\widehat{k}(\eta )$ is the proposed solution form of the network, $D^{(n)}\widehat{k}(\eta )$ and $D^{\Re }\widehat{k}(\eta )$ indicate the nth order derivative and the fractional order form, respectively. These systems’ terminologies take the following forms:

$$\begin{array}{c}\widehat{k}(\eta )={\displaystyle \sum _{m=1}^z{r}_{m}p(c_m\eta +b_m)},\\ D^{(n)}\widehat{k}(\eta )={\displaystyle \sum _{m=1}^z{r}_m{p}^{(n)}(c_m\eta +b_m)},\\ D^{\Re }\widehat{k}(\eta )={\displaystyle \sum _{m=1}^z{r}_m{p}^{\Re }(c_m\eta +b_m)}\end{array}$$

(7)

where z represents the neurons. Similarly, r, c and b represent the components of weight vector (W), shown as:

$$\mathit{W}=[\mathit{r},\mathit{c},\mathit{b}], \mathrm{for} \mathit{r}=[r_1,r_2,\dots ,r_z],\hspace{1em}\hspace{1em}\hspace{1em}c=[c_1,c_2,\dots ,c_z]\hspace{1em}\hspace{1em}\mathrm{and}\hspace{1em}\hspace{1em}\mathit{b}=[b_1,b_2,\dots ,b_z]\hspace{1em}.$$

The mathematical representations the activation kernel based on the Meyer wavelet function is shown as:

$$p(\eta )=35{\eta }^4-84{\eta }^5+70{\eta }^6-20{\eta }^7.$$

(8)

The combination of the network (7) and (8) becomes:

$$\begin{array}{c}\widehat{k}(\eta )={\displaystyle \sum _{m=1}^z{r}_m\left(\begin{array}{l}35{(c_m\eta +b_m)}^4-84{(c_m\eta +b_m)}^5+\\ 70{(c_m\eta +b_m)}^6-20{(c_m\eta +b_m)}^7\end{array}\right)},\\ D^{(n)}\widehat{k}(\eta )={\displaystyle \sum _{m=1}^z{r}_m\left(\begin{array}{l}35D^{(\mathrm{n})}{(c_m\eta +b_m)}^4-84D^{(\mathrm{n})}{(c_m\eta +b_m)}^5\\ +70D^{(\mathrm{n})}{(c_m\eta +b_m)}^6-20D^{(\mathrm{n})}{(c_m\eta +b_m)}^7\end{array}\right)},\\ D^{\Re }\widehat{k}(\eta )={\displaystyle \sum _{m=1}^z{r}_m\left(\begin{array}{l}35D^{\Re }{(c_m\eta +b_m)}^4-84D^{\Re }{(c_m\eta +b_m)}^5+\\ 70D^{\Re }{(c_m\eta +b_m)}^6-20D^{\Re }{(c_m\eta +b_m)}^7\end{array}\right)}.\end{array}$$

(9)

The procedures of the arbitrary FMWs-NN are implemented for the novel FOPSS associated to the obtainability of suitable W. To assess the weights of FMWs-NN, one may calculate the theory of approximation with the mean squared error terminology to find an error function ${\epsilon }_{Fit}$, given as:

$${\epsilon }_{Fit}={\epsilon }_{Fit-}{}_1+{\epsilon }_{Fit-}{}_2.$$

(10)

where ${\epsilon }_{Fit-}{}_1$ and ${\epsilon }_{Fit-2}$ are the error functions related to the differential system and its BCs, shown as:

$${\epsilon }_{Fit-}{}_1=\frac{1}{N}{{\displaystyle \sum _{m=1}^z\left(\frac{d^{\Re +1}}{d{\eta }^{{}^{\Re +1}}}{\widehat{k}}_m+\frac{p}{{\eta }_m}\frac{d^{{}^{\Re }}}{d{\eta }^{\Re }}{\widehat{k}}_m+g({\widehat{k}}_m)-h_m\right)}}^2,$$

(11)

$${\epsilon }_{Fit-2}=\frac{1}{2}\left({({\widehat{k}}_0)}^2+{({\widehat{k}}_N)}^2\right),$$

(12)

for $Nh=1,\hspace{1em}\hspace{1em}{\widehat{k}}_m=\widehat{k}\left(\frac{{\eta }_m}{2}\right),\hspace{1em}\hspace{1em}{h}_m=h\left({\eta }_m\right),\hspace{1em}\hspace{1em}{\eta }_m=mh$.

3.2. Optimization of the Network

In this section, the parameter optimization procedures for the FMWs-NN are considered using the computing constructions of PSOIPA for solving the novel FOPSS.

Particle swarm optimization is a global search approach implemented to solve optimization problems. It is applied as an alternate of the genetic algorithm approach introduced at the end of the 19th century. PSO is a nature-based metaheuristic due to its immense optimization abilities in the large search spans. PSO executes efficiently as compared to the genetic algorithm due to its small amount of memory. In the process of PSO, the primary swarm escalates in the substantial domain. For the PSO improvement, the procedure produces iteratively optimal outcomes ${\mathit{P}}_{LB}^{h-1}$ for swarm’s position and ${\mathit{P}}_{GB}^{h-1}$ for swarm’s velocity, given as:

$${\mathit{X}}_i^h={\mathit{X}}_i^{h-1}+{\mathit{V}}_i^{h-1},$$

(13)

$$\begin{array}{l}{\mathit{V}}_i^h=\Re {\mathit{V}}_i^{h-1}+h_1({\mathit{P}}_{LB}^{h-1}-{\mathit{X}}_i^{h-1}){\mathit{r}}_1\\ \hspace{1em}\hspace{1em}\hspace{1em}+h_2({\mathit{P}}_{GB}^{h-1}-{\mathit{X}}_i^{h-1}){\mathit{r}}_2,\end{array}$$

(14)

where the inertia vector based on weight is $\Re$, the position is X_i and the velocity is V_i, while, $h_1$ and $h_2$ are acceleration constant factors. A few prominent applications of the PSO are optimal reactive power dispatch [50], fusion of features for detection of brain tumor [51], energy-efficient routing mechanism for mobile sink in wireless sensor networks [52], optimal power flow problems [53], dynamic service composition focusing on quality-of-service evaluations under hybrid networks [54] and enhancing the production of biodiesel from Microalga [55].

The convergence of the PSO scheme is more reliable using the hybridization process with the local search interior-point algorithm, which is used to find the fine-tuning of the outcomes. IPA is a valued approach, which is used to confine the system for improved understanding together with the optimization procedures of the designed system. In recent decades, IPA has been applied in optimal operation of interconnected energy hubs [56], economic load dispatch [57], a nonlinear well-determined model for power system observability [58] and power control of multiple interfering D2D communications underlaying cellular networks [59].

3.3. Performance Indices

In this work, the mathematical formulations of the performances based on the TIC, ENSE and EVAF along with the global illustrations of these indices to solve the novel FOPSS are provided as:

$$\{\begin{array}{l}\mathrm{VAF}=\left(1-\frac{\mathrm{var}\left(k_j-{\widehat{k}}_j\right)}{\mathrm{var}\left(k_j\right)}\right)\times 100,\\ \mathrm{EVAF}=\left|VAF-100\right|.\end{array}$$

(15)

$$\mathrm{T}.\mathrm{I}.\mathrm{C}=\frac{\sqrt{\frac{1}{n}{{\displaystyle \sum _{j=1}^q\left(k_j-{\widehat{k}}_j\right)}}^2}}{\left(\sqrt{\frac{1}{n}{\displaystyle \sum _{j=1}^q{k}_j^2}}+\sqrt{\frac{1}{n}{\displaystyle \sum _{j=1}^q{\widehat{k}}_j^2}}\right)},$$

(16)

$$\{\begin{array}{l}\mathrm{NSE}=\{1-\frac{{{\displaystyle \sum _{j=1}^q\left(k_j-{\widehat{k}}_j\right)}}^2}{{{\displaystyle \sum _{j=1}^q\left(k_j-{\overline{k}}_j\right)}}^2},\hspace{1em}\hspace{1em}{\overline{k}}_j=\frac{1}{n}{\displaystyle \sum _{j=1}^q{k}_j}\\ ENSE=1-NSE,\end{array}$$

(17)

where $\widehat{k}$ and k are the proposed and exact solutions. The necessary comparison of the proposed FMWs-NN-PSOIPA is conducted with respect to magnitudes of VAF, TIC and NSE for perfect modeling scenarios with values 100, 0 and 1, respectively.

4. Simulations and Results

In this section, the numerical implementations to solve three examples of the novel FOPSS are provided. The proposed outcomes along FMWs-NN-PSOIPA that depend upon 40 executions to solve the novel FOPSS are provided with essential graphical and numerical depictions to evaluate the accurateness and convergence.

Suppose a novel FOPSS is shown as:

$$\{\begin{array}{l}\eta \frac{d^{\Re +1}}{d{\eta }^{\Re +1}}k\left(\frac{\eta }{2}\right)+\frac{d^{\Re }}{d{\eta }^{\Re }}k\left(\frac{\eta }{2}\right)+\eta g(k)=\eta h(\eta )=F(\eta ),\\ k(0)=k(1)=0,\hspace{1em}\hspace{1em}\hspace{1em}\end{array}$$

(18)

where

$$\begin{array}{l}F(\eta )=\eta \left(\frac{\sqrt{(1+w)}}{\sqrt{1-\Re +w}}{\left(\frac{\eta }{2}\right)}^{w-\Re }-\frac{\sqrt{1+y}}{\sqrt{1-\Re +y}}{\left(\frac{\eta }{2}\right)}^{z-\Re }\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}+\frac{\sqrt{w+1}}{\sqrt{w-\Re +1}}{\left(\frac{\eta }{2}\right)}^{w-\Re }-\frac{\sqrt{1+y}}{\sqrt{1-\Re +y}}{\left(\frac{\eta }{2}\right)}^{z-\Re }+{\eta }^{w+1}-{\eta }^{y+1}\end{array}$$

(19)

where w and y are selected as positive. The modernized form using the above equations is given as:

$$\{\begin{array}{l}\eta \frac{d^{\Re +1}}{d{\eta }^{\Re +1}}k\left(\frac{\eta }{2}\right)+\frac{d^{\Re }}{d{\eta }^{\Re }}k\left(\frac{\eta }{2}\right)+\eta h(k)=\eta \left(\begin{array}{l}\frac{\sqrt{(1+w)}}{\sqrt{1-\Re +w}}{\left(\frac{\eta }{2}\right)}^{w-\Re }\\ -\frac{\sqrt{1+y}}{\sqrt{1-\Re +y}}{\left(\frac{\eta }{2}\right)}^{z-\Re }\end{array}\right)\\ +\frac{\sqrt{w+1}}{\sqrt{w-\Re +1}}{\left(\frac{\eta }{2}\right)}^{w-\Re }-\frac{\sqrt{1+y}}{\sqrt{1-\Re +y}}{\left(\frac{\eta }{2}\right)}^{z-\Re }+{\eta }^{w+1}-{\eta }^{y+1},\\ k(0)=k(1)=0.\hspace{1em}\end{array}$$

(20)

The true solution is given as:

$$k(\eta )={\eta }^w-{\eta }^y$$

(21)

For the specific performances of w = 3 and y = 2, the true solution is accomplished as:

$$k(\eta )={\eta }^3-{\eta }^2.$$

(22)

An error function is given as:

$$\begin{array}{l}{\epsilon }_{Fit}=\frac{1}{N}{{\displaystyle \sum _{m=1}^z\left(\begin{array}{l}{\eta }_m\frac{d^{{}^{\Re +1}}}{d{\eta }_m^{{}^{\Re +1}}}\widehat{k}\left(\frac{{\eta }_m}{2}\right)+\frac{d^{{}^{\Re }}}{d{\eta }_m^{{}^{\Re }}}\widehat{k}\left(\frac{{\eta }_m}{2}\right)+{\eta }_{m}h({\widehat{k}}_m)\\ -{\eta }_m^{w+1}+{\eta }_m^{z+1}-{\eta }_m\sqrt{\frac{w+1}{w-\Re +1}}\hspace{1em}\hspace{1em}{\left(\frac{{\eta }_m}{2}\right)}^{w-\Re }\\ +{\eta }_m\sqrt{\frac{y+1}{y-\Re +1}}\hspace{1em}\hspace{1em}{\left(\frac{{\eta }_m}{2}\right)}^{y-\Re }-\sqrt{\frac{w+1}{w-\Re +1}}\hspace{1em}\hspace{1em}{\left(\frac{{\eta }_m}{2}\right)}^{w-\Re }\\ +\sqrt{\frac{y+1}{y-\Re +1}}\hspace{1em}\hspace{1em}{\left(\frac{{\eta }_m}{2}\right)}^{y-\Re }\end{array}\right)}}^2\\ \hspace{1em}\hspace{1em}+\frac{1}{2}\left({\left({\widehat{k}}_0\right)}^2+{\left({\widehat{k}}_N\right)}^2\right).\end{array}$$

(23)

Three different types of the novel FOPSS are provided using the $\alpha$ values, respectively given as $\alpha =0.2,\hspace{1em}\hspace{1em}0.4$ and 0.6.

In order to examine the performance of each type of novel FOPSS, optimization is performed using the local and global search techniques, i.e., PSOIPA. The entire procedure is repeated for 40 independent runs to create a larger dataset of the parameters of FMWs-NN. These trained FMWs-NN weights are given in Equation (9) to evaluate the outcomes of the novel FOPSS. The mathematical form of the FMWs-NN-PSOIPA for each type of novel FOPSS is provided as:

$$\begin{array}{l}{\widehat{k}}_{E-1}=-0.113\left(\begin{array}{l}35{(-0.971\eta +0.570)}^4-84{(-0.971\eta +0.5708)}^5\\ +70{(-0.971\eta +0.5708)}^6-20{(-0.971\eta +0.5708)}^7\end{array}\right)\\ \hspace{1em}\hspace{1em}+0.3448\left(\begin{array}{l}35{(0.0641\eta +1.0772)}^4-84{(0.0641\eta +1.0772)}^5\\ +70{(0.0641\eta +1.0772)}^6-20{(0.0641\eta +1.0772)}^7\end{array}\right)+\cdots \\ \hspace{1em}\hspace{1em}+0.0267\left(\begin{array}{l}35{(0.2990\eta +1.2633)}^4-84{(0.2990\eta +1.2633)}^5\\ +70{(0.2990\eta +1.2633)}^6-20{(0.2990\eta +1.2633)}^7\end{array}\right),\end{array}$$

(24)

$$\begin{array}{l}{\widehat{k}}_{E-2}=-0.426\left(\begin{array}{l}35{(0.430\eta -0.4234)}^4-84{(0.4300\eta -0.4234)}^5\\ +70{(0.4300\eta -0.4234)}^6-20{(0.4300\eta -0.4234)}^7\end{array}\right)\\ \hspace{1em}\hspace{1em}+\hspace{1em}2.1279\left(\begin{array}{l}35{(-0.117\eta -0.028)}^4-84{(-0.117\eta -0.0284)}^5\\ +70{(-0.117\eta -0.0284)}^6-20{(-0.117\eta -0.0284)}^7\end{array}\right)+\cdots \\ \hspace{1em}\hspace{1em}-0.5029\left(\begin{array}{l}35{(0.950\eta -0.2247)}^4-84{(0.95000\eta -0.2247)}^5\\ +70{(0.9500\eta -0.2247)}^6-20{(0.9500\eta -0.2247)}^7\end{array}\right),\end{array}$$

(25)

$$\begin{array}{l}{\widehat{k}}_{E-3}=-0.2573\left(\begin{array}{l}35{(0.304\eta -0.2064)}^4-84{(0.3041\eta -0.2064)}^5\\ +70{(0.3041\eta -0.2064)}^6-20{(0.3041\eta -0.2064)}^7\end{array}\right)\\ \hspace{1em}\hspace{1em}-0.1217\left(\begin{array}{l}35{(0.0244\eta +1.3782)}^4-84{(0.0244\eta +1.3782)}^5\\ +70{(0.0244\eta +1.3782)}^6-20{(0.0244\eta +1.3782)}^7\end{array}\right)+\cdots \\ \hspace{1em}\hspace{1em}+0.406\left(\begin{array}{l}35{(-0.2985\eta -0.046)}^4-84{(-0.5903\eta -0.5980)}^5\\ +70{(-0.5903\eta -0.598)}^6-20{(-0.590\eta -0.598)}^7\end{array}\right).\end{array}$$

(26)

The estimated results through the FMWs-NN are indicated in systems (24)–(26) with the graphical plots illustrated in Figure 3a–c for each class of novel FOPSS. The mean, best and worst results comparison is drawn in Figure 3d–f for each class of novel FOPSS. It is observed that these outcomes are matched to each other. This accuracy of the numerical results shows the quality of the designed FMWs-NN-PSOIPA. The AE performance is obtained in Figure 3g for each class of novel FOPSS. It is observed that the performance of AE is calculated around 10⁻¹ to 10⁻³, 10⁻² to 10⁻³ and 10⁻¹ to 10⁻³ for Examples 1, 2 and 3, respectively. The convergence is assessed using the FIT, ENSE, TIC and EVAF measures, drawn in Figure 3h for each class of novel FOPSS. It is indicated that the best performance instances of the FIT measure are found around 10⁻⁵–10⁻⁶, 10⁻⁴–10⁻⁵ and 10⁻³–10⁻⁵ for each example of the novel FOPSS. The EVAF performance instances are calculated around 10⁻⁴ to 10⁻⁵ for each example of the novel FOPSS. The TIC measures’ performance instances are calculated around 10⁻³ to 10⁻⁵ for each example of the novel FOPSS. The ENSE is calculated around 10⁻⁴ to 10⁻⁵ for each variant of the novel FOPSS.

The TIC, ENSE, FIT and EVAF performance instances with the histogram and boxplots are drawn in Figure 4, Figure 5, Figure 6 and Figure 7 for each class of the novel FOPSS. It is demonstrated that the FIT performance is calculated around 10⁻³ to 10⁻⁶, 10⁻⁴ to 10⁻⁵ and 10⁻³ to 10⁻⁵ for Examples 1, 2 and 3. TIC lies around 10⁻³ to 10⁻⁵ for Example 1, whereas the other two examples of TIC values are found around 10⁻³ to 10⁻⁵. The EVAF and ENSE values for each case of the novel FOPSS lie in the ranges of 10⁻¹ to 10⁻² and 10⁻² to 10⁻³, respectively. These best measures, calculated through statistical gages, authenticate the correctness of FMWs-NN-PSOIPA.

In order to check the precision and exactness, the statistical measures, through standard deviation (STD), minimum (Min), S.I.R, mean, maximum (Max) and median (MED), are found for 40 accomplishments of FMWs-NN-PSOIPA as shown in

Table 1. Statistics illustrations through FMWs-NN-PSOIPA for solving the novel FOPSS.

Index	Mode	Proposed Outcomes $\mathit{k}(\mathit{\eta })$
		0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
1	Min	4 × 10−4	3 × 10−4	1 × 10−3	8 × 10−4	4 × 10−3	4 × 10−4	1 × 10−3	1 × 10−2	3 × 10−3	3 × 10−5
	Max	4 × 10−2	3 × 10−2	3 × 10−2	3 × 10−2	4 × 10−2	6 × 10−2	8 × 10−2	9 × 10−2	7 × 10−2	5 × 10−3
	MED	6 × 10−3	1 × 10−2	1 × 10−2	2 × 10−2	3 × 10−2	4 × 10−2	5 × 10−2	5 × 10−2	3 × 10−2	1 × 10−3
	Mean	4 × 10−3	1 × 10−2	1 × 10−2	2 × 10−2	3 × 10−2	5 × 10−2	6 × 10−2	6 × 10−2	3 × 10−2	3 × 10−4
	S.I.R	6 × 10−3	6 × 10−3	6 × 10−3	7 × 10−3	8 × 10−3	1 × 10−2	1 × 10−2	1 × 10−2	1 × 10−2	1 × 10−3
	STD	1 × 10−3	2 × 10−3	3 × 10−3	4 × 10−3	4 × 10−3	5 × 10−3	6 × 10−3	7 × 10−3	5 × 10−3	8 × 10−5
2	Min	3 × 10−4	8.7 × 10−5	1 × 10−3	2 × 10−3	8 × 10−3	2 × 10−2	2 × 10−2	2 × 10−2	8 × 10−3	6 × 10−6
	Max	7 × 10−2	8 × 10−2	8 × 10−2	8 × 10−2	7 × 10−2	7 × 10−2	8 × 10−2	1 × 10−1	9 × 10−2	1 × 10−2
	MED	9 × 10−3	1 × 10−2	1 × 10−2	2 × 10−2	3 × 10−2	4 × 10−2	6 × 10−2	6 × 10−2	4 × 10−2	1 × 10−4
	Mean	7 × 10−3	6 × 10−3	1 × 10−2	2 × 10−2	3 × 10−2	4 × 10−2	6 × 10−2	6 × 10−2	3 × 10−2	1 × 10−3
	S.I.R	1 × 10−2	1 × 10−2	1 × 10−2	1 × 10−2	1 × 10−2	1 × 10−2	1 × 10−2	1 × 10−2	2 × 10−2	2 × 10−3
	STD	2 × 10−3	6 × 10−3	1 × 10−2	1 × 10−2	1 × 10−2	1 × 10−2	1 × 10−2	1 × 10−2	1 × 10−2	6 × 10−4
3	Min	1 × 10−5	4 × 10−5	2 × 10−3	3 × 10−3	1 × 10−2	9 × 10−3	6 × 10−3	3 × 10−2	1 × 10−2	7 × 10−4
	Max	7 × 10−2	1 × 10−1	1 × 10−1	1 × 10−1	1 × 10−1	1 × 10−1	1 × 10−1	1 × 10−1	1 × 10−1	4 × 10−3
	MED	9 × 10−3	1 × 10−2	2 × 10−2	3 × 10−2	5 × 10−2	6 × 10−2	7 × 10−2	7 × 10−2	5 × 10−2	1 × 10−3
	Mean	6 × 10−3	5 × 10−3	1 × 10−2	2 × 10−2	3 × 10−2	5 × 10−2	7 × 10−2	8 × 10−2	6 × 10−2	1 × 10−3
	S.I.R	1 × 10−2	2 × 10−2	2 × 10−2	2 × 10−2	2 × 10−2	3 × 10−2	2 × 10−2	1 × 10−2	1 × 10−2	6 × 10−4
	STD	2 × 10−3	9 × 10−3	1 × 10−2	1 × 10−2	2 × 10−2	2 × 10−2	1 × 10−2	6 × 10−3	1 × 10−2	1 × 10−5

for solving the novel FOPSS. The Max and Min values indicate the worst and best executions, while S.I.R represents one half of the 3rd minus 1st quartiles. The values based on Min, Max, MED, Mean, S.I.R and S.T.D for Example 1 are found around 10⁻³–10⁻⁵, 10⁻²–10⁻³, 10⁻²–10⁻³, 10⁻²–10⁻⁴, 10⁻²–10⁻³ and 10⁻³–10⁻⁵. In Example 2, the performance lies around 10⁻²–10⁻⁶, 10⁻¹–10⁻², 10⁻²–10⁻⁴, 10⁻²–10⁻³, 10⁻²–10⁻³ and 10⁻²–10⁻⁴. Likewise, in Example 3, the performance lies around 10⁻³ to 10⁻⁵, 10⁻¹ to 10⁻³, 10⁻² to 10⁻³, 10⁻² to 10⁻³, 10⁻² to 10⁻³ and 10⁻² to 10⁻⁵. These calculated consistent and small performance instances of each operative authenticate the accuracy and constancy of FMWs-NN-PSOIPA for solving the novel FOPSS.

For the convergence of FMWs-NN-PSOIPA, the global operators using the EVAF, ENSE, FIT, and TIC for 40 executions for the novel FOPSS are provided below in

Table 2. Global values through FMWs-NN-PSOIPA for solving the novel FOPSS.

Index	G.FIT		G.TIC		G.ENSE		G.EVAF
	Min	SI.R	Min	SIR	MIN	SI.R	Min	SI.R
1	2.556 × 10−6	2.285 × 10−5	7.728 × 10−6	2.673 × 10−7	2.291 × 10−3	3.330 × 10−5	1.819 × 10−2	2.636 × 10−2
2	5.527 × 10−7	3.006 × 10−5	1.007 × 10−5	3.272 × 10−7	3.844 × 10−3	6.650 × 10−5	2.938 × 10−2	5.920 × 10−4
3	5.399 × 10−5	7.898 × 10−7	1.130 × 10−5	3.491 × 10−8	2.291 × 10−4	3.330 × 10−4	7.908 × 10−3	7.100 × 10−2

. It is noticeable that the Min global FIT, TIC, ENSE and EVAF values are found around 10⁻⁵–10⁻⁷, 10⁻⁵–10⁻⁶, 10⁻³–10⁻⁴ and 10⁻²–10⁻³, whereas the S.I.R gages for these measures are found around 10⁻⁵–10⁻⁷, 10⁻⁷–10⁻⁸, 10⁻⁴–10⁻⁵ and 10⁻²–10⁻⁴ to solve the novel FOPSS. The classic global performance validates the clarity of FMWs-NN-PSOIPA.

5. Conclusions

A novel design of the fractional order pantograph singular system using the applications of Meyer wavelets as a neural network is presented using the perceptions of standard forms of second-order singular and pantograph differential systems. The novel FOPSS is presented using the standard Lane–Emden equation and detailed discussions of the singularity, shape factor terms along with the fractional order forms. The singularity is noticed for a single time at $\eta =0$, while the fractional terms appear twice as $\Re$ and $\Re +1$. The perfection and exactness of the novel FOPSS is described using fractional Meyer wavelets as a neural network with the optimization procedures of global/local search procedures of the particle swarm optimization (PSO) and interior-point algorithm. The proposed FMWs-NN-PSOIPA is generally applied to solve the novel FOPSS to authenticate the stability, robustness, convergence and accuracy. To authenticate the correctness of the proposed FMWs-NN-PSOIPA, a comparison of the obtained outcomes with the true solutions is performed. The statistics using the operators TIC, Min, EVAF, S.I.R, Max, ENSE, Mean MED and STD are achieved using 40 executions to authenticate the consistency of the proposed FMWs-NN-PSOIPA. One can also prove that a variety of trials showed a greater accuracy level for the novel FOPSS. The novel FOPSS comprises pantographs, fractional terms and singular points, which shows the stiffness of the system and is considered complex to solve with conventional schemes. However, FMWs-NN-PSOIPA is an excellent choice to solve these types of intricate models.

In future, the FMWs-NN-PSOIPA can be implemented to solve fractional systems, nonlinear models and fluid systems [60,61,62,63,64,65].