Design of Mayer Wavelet Neural Networks for Solving Functional Nonlinear Singular Differential Equation

Sabir, Zulqurnain; Zahoor Raja, Muhammad Asif; Guirao, Juan L. G.; Saeed, Tareq

doi:https://doi.org/10.1155/2022/1213370

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2022 | Article ID 1213370 | https://doi.org/10.1155/2022/1213370

Design of Mayer Wavelet Neural Networks for Solving Functional Nonlinear Singular Differential Equation

Zulqurnain Sabir,¹Muhammad Asif Zahoor Raja,²Juan L. G. Guirao,^3,4and Tareq Saeed⁴

Academic Editor: Junming Huang

Received17 Jun 2021

Revised16 Nov 2021

Accepted25 Mar 2022

Published21 Apr 2022

Abstract

In the present work, an advance computational intelligence paradigm based on functional Mayer artificial neural network (FM-ANN) is accessible for solving the singular nonlinear functional differential equation (NFDE) numerically. The solution of singular NFDE is performed by using the artificial neural networks (ANNs) optimized with global search genetic algorithm (GA) enhanced by local refinements of sequential quadratic (SQ) programming and the hybrid of GASQ programming. The proposed scheme is applied for solving three types of second-order singular NFDEs. In order to validate the correctness of the designed scheme, the comparison of the proposed and exact solutions has been performed. Moreover, the statistical interpretations are used to prove the worth, convergence, accuracy, stability, and robustness of FM-ANN-GASQP for the solution of singular NFDEs.

1. Introduction

In recent decades, the study of nonlinear functional differential equations (NFDEs) along with the applications of singularity has been widely discussed. These NFDEs have numerous submissions, some of them are electrodynamics [1], population growing systems [2], tumor growth model [3], biological infection model of HIV-1 [4], chemical kinetics [5], gene regulation [6], virus infection model of hepatitis-B [7], and epidemiologic infection systems [8]. There are only few numerical techniques for solving these singular NFDEs. Some of them are Mirzaee and Hoseini [9] used collocation approach for the nonlinear functional differential equations. Genga et al. [10] implemented a numerical kernel scheme to get the numerical solutions of these equations. Xu and Jin [11] found the results of the vector form of singular NFDE using the fractional steps and boundary function. Kadalbajoo and Sharma [12, 13] discussed a numerical scheme for solving these systems. The generic form of the singular NFDE is given as [14, 15]where , c, b, and are the constant values while t be the independent variable of functional differential (1) expressed with u dependent on t. The researcher’s community takes keen interest to tackle the challenge of singularity at the origin, and there are only few numerical/analytical techniques available in literature to solve the problems involving singular points. Few well-known applications of the singular models are thermal occurrences [16], stellar configuration [17], isothermal spheres of gas [18], oscillating magnetic ranges [19], and thermionic-based currents [20].

The heuristic numerical computing schemes have been broadly applied to many researchers to solve the singular and nonlinear systems by operational strength of neural networks (NNs) and operational versions with evolutionary standards. Few recent submissions of evolutionary solvers are HIV inflectional model [21], Thomas–Fermi singular systems [22], cell biology [23], doubly singular nonlinear systems [24], transistor-level uncertainty quantification [25], and control systems [26]. These contributions proved the significance, worth, and value of the stochastic numerical solvers in terms of accuracy and robustness. To consider the importance of these applications, authors are attracted to exploit the stochastic-based numerical solvers to find the numerical performances of the NFDEs efficiently. The intension of the current research is to solve the model via intelligent computing on the basis of functional Mayer wavelet artificial neural network (FM-ANN) with the optimization of global search genetic algorithm (GA) and local search sequential quadratic (SQ) programming, shortly FM-ANN-GASQP. The FM-ANN has never been applied to solve the singular NFDEs before. The prime contributions and innovative insights of the FM-ANN-GASQP are summarized as follows:(i)The novel design of FM-ANN-GASQP is introduced to solve different nonlinear problems represented with singular functional differential systems effectively(ii)The comparison of the proposed and existing exact solutions validates the exactness and reliability of the designed FM-ANN-GASQP approach to solve the singular NFDEs(iii)The performance of the designed FM-ANN-GASQP scheme is recognized with the favourable tendencies of statistical outcomes based on the “mean absolute error (MSE),” “semi-interquartile range (SIR),” “root mean square error (RMSE),” and “Theil’s inequality coefficient (TIC)”(iv)Singular NFDEs are difficult to handle because of its stiffer nature of nonlinear and functionality; however, the design FM-ANN-GASQP is a promising assortment to deal with these types of complex nonlinear problems, which are still challengeable for conventional deterministic schemes

The remaining parts of this study are given as follows. Section 2 narrates the design process of the FM-ANN-GASQP approach. Section 3 presents an impression of the performance studies. Section 4 shows the numerical values of suggested FM-ANN-GASQP together with the statistical explanations. The conclusion together with future guidance is listed in Section 5.

2. Designed Methodology

The FM-ANN is designed for solving the singular NFDEs. The construction to design the differential systems and error function along with the optimization practice using the GASQP is described.

2.1. Functional Mayer Wavelet Neural Network

The models based on ANN are acquainted to show the excellent results for the variety of submission in numerous fields. In the FM-ANN, denotes the proposed outcomes and represents the nth derivatives. The manifestation of these networks is written aswhere m is used for neurons. The vector components of weight matrix W are , and are expressed as

The Mayer wavelet function is defined as

Using the above Mayer wavelet function in set (2) asthe arbitrary form of the FM-ANN is applied to solve singular NFDE model (1) related to the accessibility of suitable weight matrix W. In order to find the appropriate weight vectors of singular FM-ANN, one can use mean square approximation in the form of sum of two mean squared error functions as follows:where the error function is associated to singular NFDE (1) and shows the boundary conditions of model (1), which are written aswhere

2.2. Optimization: GASQP

The FM-ANN parameter optimization is approved for the context of hybrid computing using the GASQP.

GA is known as a global technique and used to model the natural genetic procedures. GAs frequently variates population, i.e., consists of individuals or chromosomes, by adapting its reproduction processes through “crossover,” “mutation,” “elitism,” and “selection” operators. Some current addressed submissions of GAs including car-like robot control [27], HIV infection system [28, 29], identification and modelling of nonlinear multivariable systems [30], a fully customizable hardware implementation [31], classification of hyperplastic resources [32], prediction of biosorption capacity [33], and torque estimation problem [34]. GAs combined with local search technique, i.e., SQ programming, can upgrade its sluggishness through the optimization technique.

SQP is a well-organized, rapid, and quick local search optimization approach having different submissions arising in various areas. Sequential quadratic programming belongs to the class of convex optimization problem-solving technique exploit for both constrained and unconstrained optimization tasks. Few recent submissions of SQP are multiproduct economic production [35], economic load dispatch problems [36], bipedal dynamic walking robot [37], temporary hydrothermal organization [38], analysis of guidewire distortion in the vessels of blood [39], convex quadratic bi-level programming problems [40], Lane–Emden pantograph systems [41], simple LNG process [42], optimal power flow problems [43], and flight recovery for transference aircraft [44].

The combination of GASQP is oppressed for obtaining the designed variables of FM-ANN to solve singular NFDE. The detailed, expressive optimization performances through GASQP programming is provided in Table 1 in the form of pseudocode.

3. Performance Measures

The performance procedures are applied to analyze the weaknesses and strength of the proposed FM-ANN-GASQP methodology for solving the variants of singular NFDE, incorporated with mean absolute deviation (MAD), Nash Sutcliffe error (NSE), root mean square error (RMSE), and Theil’s inequality coefficient (TIC) and are mathematically given as

4. Simulation and Results

The detailed result simulations using FM-ANN-GASQP for 100 autonomous execution to solve singular NFDE are presented here for three different problems.

Problem I. Singular nonhomogeneous differential difference second-order equation is [14] is the exact form of the solution, and the merit function becomes as

Problem II. Singular NFDE of the Lane–Emden type is [14]The exact solution is , and the error function of the above equation becomes as

Problem III. Multiple singular NFDE involving trigonometric functions is [14] is the true solution, while the fitness formulation of model (16) becomes asThe proposed designed scheme FM-ANN-GASQP is executed for 100 independent implementations to achieve the system parameters for solving singular NFDE as given in (1). The best weight vectors are indicated to find the approximate results of (1). The mathematical representation of the approximate solutions is provided asThe proposed results are obtained using equations (15)–(17) and the graphical representation of Problems I, II, and III is provided in Figures 1–3 by using 10 neuron-based results of proposed design FM-ANN-GASQP computing paradigm. The set of trained weights of FM-ANN, result comparisons of solution dynamics, performance indices, and absolute error (AE) for proposed design FM-ANN-GASQP computing paradigm are plotted in Figure 1 for all three problems. The set of best weights is represented graphically in Figures 1(a) to 1(c). The obtained solutions of the present scheme are calculated on the basis of these weight vectors. The 2nd part of Figure 1 indicates the assessment of the proposed and exact results. The overlapping of best and exact solutions indicates the correctness of the FM-ANN-GASQP approach. The AE for Problems I to III is plotted in the third half of Figure 1. The best and mean solutions are plotted in this portion. The best measures are calculated 10–08 to 10–10, 10–04 to 10–06, and 10–06 to 10–08 for cases I, II, and III, while the values based on mean for all the problems lie near to 10–02. The final half of Figure 1 shows the performance measures based on statistical values MAD, RMSE, ENSE, and TIC. The best values of the FIT, MAD, RMSE, ENSE, and TIC proved very good results and found around 10–08 to 10–10, 10–04 to 10–06, and 10–05 to 10–10. However, the mean-based values are also lying in the good ranges. The perceived values enhance the efficiency and worth of the FM-ANN-GASQP approach.
Figures 2 and 3 protray the respective results of MAD and RMSE and ENSE and TIC, respectively, together with the illustrations of histograms for each problems of model (1) for 10 numbers of neuron-based model of proposed design FM-ANN-GASQP computing paradigm for all three problems. The normal distribution curves on the histograms’ illustrations are also plotted with red font curves in order to the mean trend of the results for each performance indices in case of all three problems solved by proposed design FM-ANN-GASQP computing paradigm. The best performances for all the presentations of each problem proved very good measures. It is seen that huge number of independent runs achieved best standards of the statistical values for all three problems. The algorithm performance is found to be precise for the statistical operators on RMSE, ENSE, TIC, and MAD indices. Thus, one can summarize that the converge to accurate solutions by minimizing the respective objective functions, i.e., mean square error models with FM-ANNs as presented in equations (13), (15), and (17) for three variants of functional nonlinear singular differential equation; GASQP is proved by getting consistent near to optimal values of FIT, MAD, RMSE, TIC, and ENSE operators by proposed design FM-ANN-GASQP computing paradigm, on the basis of single best runs as well as analysis on multiple autonomous runs.
The convergence of the proposed design FM-ANN-GASQP computing paradigm is further examined via statistical results of measure of central tendency and variation in the sense of minimum (Min), “median (Med),” and semi-interquartile range (SIR) indices. The operator SIR is one half of 3rd quartile (Q3 = 75% data) minus 1st quartile (Q1 = 25% data) for 100 trials of FM-ANN-GASQP for each problem of singular NFDE. The basic aim of these statistical values is to find accuracy of FM-ANN-GASQP approach. The independent trials using the MIN error values of the fitness are signified as the best run of proposed design FM-ANN-GASQP computing paradigm. The statistics by means of Min, Med, and SIR operator are tabulated in Table 2 of singular NFSE. It is cleared that the most of the Min-based values lie 10−08 to 10−10 range for each problem of singular NFDE. Correspondingly, the median and SIR performances are calculated precisely, which lie around 10−05 to 10−06 and 10−04 to 10−05 for each case of singular NFDE.
The computational cost of the FM-ANN-GASQP algorithm is scrutinized through average consumed time, in second, completed cycles, i.e., iterations/generations, and number of counts for the execution of fitness functions used to calculate the system’s decision variables, i.e., trained weights of FM-ANN. The outcomes of the complexity studies in terms of all three said operators are calculated for 100 execution of proposed design FM-ANN-GASQP computing paradigm for each problem of singular NFDE, and their results are provided in Table 3. One may see that the time, cycles, and function counts are around 650 ± 25, 150 ± 5, and 85000 ± 5000, respectively.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

5. Conclusion

The inspiration of the present work is to design a novel functional Mayer neural network to explore in numerical results of the nonlinear function differential model by operating the hybrid computing strength of ANNs’ models under the optimization of GAs together with SQ programming. Some key findings of the current research are presented as(i)The design of novel functional Mayer-based computing ANN-GASQP is presented effectively for the singular nonlinear differential models(ii)The correctness of the designed scheme FM-ANN-GASQP for the singular nonlinear functional differential system is verified by overlapping the proposed and exact solutions up to accuracy of order 4 to 7(iii)The AE values have been noticed in a good agreement of order 10–09 to 10–10, 10–04 to 10–06, and 10–06 to 10–08 for each problem of singular NFDEs(iv)Statistical gages based on Min, SIR, TIC, ENSE, and TIC represent that around 70% executions of the algorithm provide accurate and precise outcomes reliably

In future, the proposed FM-ANN-GASQP is a fast-convergent way, which can be applied for the singular, biological, and fluid dynamics systems governed with differential equation.

Data Availability

No data were used to support the findings the study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding this work.

Acknowledgments

This work was partial supported by the Ministerio de Ciencia, Innovación y Universidades (Grant no. PGC2018-097198-B-I00) and Fundacón Séneca. de la Región de Murcia (Grant no. 20783/PI/18).

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Copyright © 2022 Zulqurnain Sabir et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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