1 Introduction

Vision is an essential sense in humans and many animals. It is a fundamental issue for better understanding of biology, as well as the physics of sight for dealing with different diseases during treatment. The eye in the human body has a significant role of sight; two-thirds of its refractive power are due to its Cornea, which is commonly known as the front part of the eye. Researchers have contributed significantly to have a detailed knowledge of corneal anatomy and its optics (see [1]). The Cornea is considered to be highly sensitive, and many sight disorders may be due to a change in the corneal shape and its geometry. Possibly, an incorrect corneal geometry is partly responsible for some common diseases like astigmatism, hyperopia and myopia. The accuracy of refractive surgery and adjustment of contact lens is usually subject to a deep understanding of the corneal topography and models [2,3,4,5]. Therefore, for a better ophthalmological use, a detailed knowledge and information of the corneal shape are of paramount importance.

Researchers have conducted several investigations on mathematical models for corneal shape [6]. Some studies address the variable eccentricity ratio of the conics based on a complex shell theory model that describes its mathematical interpretation [7]. The corneal biomechanical behaviour is investigated through a finite element modelling [8]. Zernike polynomial approximations are used for corneal geometry modelling [9]. Models with rational functions for video keratoscopy compression have been used [10], and a Fourier series method (FSM) to identify the crystalline lens shape has been considered [11]. In general, the efficiency of the physical models of the eye is greatly improved through the use of theoretical and mathematical considerations [12, 13]. These models help to develop the equipments used in medicine without the involvement of patients. Recently, a few scientists have investigated collagen fibrils in soft tissues and have developed a class of models that illustrates the biomechanics of the Cornea [14].

In this study, a mathematical model of the corneal shape based on a nonlinear differential equation is studied, which provides an effective solution to address different diseases of the Cornea. The nonlinear corneal shape model (NCSM) is represented by means of a two-point boundary value problem (BVP) of second-order ordinary differential equation (ODE) as:

$$ \begin{aligned} & \frac{\text{d}}{{{\text{d}}x}}\left( {{{\frac{{{\text{d}}u}}{{{\text{d}}x}}} \mathord{\left/ {\vphantom {{\frac{{{\text{d}}u}}{{{\text{d}}x}}} {\sqrt {1 + \left( {\frac{{{\text{d}}u}}{{{\text{d}}x}}} \right)^{2} } }}} \right. \kern-0pt} {\sqrt {1 + \left( {\frac{{{\text{d}}u}}{{{\text{d}}x}}} \right)^{2} } }}} \right) - au(x) + {b \mathord{\left/ {\vphantom {b {\sqrt {1 + \left( {\frac{{{\text{d}}u}}{{{\text{d}}x}}} \right)^{2} } }}} \right. \kern-0pt} {\sqrt {1 + \left( {\frac{{{\text{d}}u}}{{{\text{d}}x}}} \right)^{2} } }} = 0, \\ & x \in [0,1],a > 0,b > 0,\,\,\,u^{\prime}(0) = 0,u(1) = 0, \\ \end{aligned} $$
(1)

where the curve u(x) = 0 represents a meridian of a surface of revolution related to the corneal geometry, while a and b are positive real constants. The N-dimensional counterpart of model (1) is given as follows:

$$ \left\{ {\begin{array}{*{20}l} {u = 0} \hfill & {{\text{on}}{\mkern 1mu} {\mkern 1mu} \varOmega } \hfill \\ {{\text{div}}\left( {\frac{\nabla u}{{\sqrt {1 + \left| {\nabla u} \right|^{2} } }}} \right) = au - \frac{b}{{\sqrt {1 + \left| {\nabla u} \right|^{2} } }}{\mkern 1mu} ,} \hfill & {{\text{in}}\,\varOmega } \hfill \\ \end{array} } \right. $$
(2)

The models in Eqs. (1–2) have been proposed as suitable models of the geometry of the human Cornea in [1,2,3,4]. However, in these studies, a simplified version of the above model has been investigated with the replacement of the mean curvature term \( {\text{div}}\left( {{{\nabla u} \mathord{\left/ {\vphantom {{\nabla u} {\sqrt {1 + \left| {\nabla u} \right|^{2} } }}} \right. \kern-0pt} {\sqrt {1 + \left| {\nabla u} \right|^{2} } }}} \right) \) by its linearization, \( {\text{div}}(\nabla u) \), in a neighbourhood of \( 0 \).

As a particular case, when \( b \in \left] {0,\frac{{3\sqrt {3a} }}{2\tanh \sqrt a }} \right[ \), the NCSM equation in (1) is modified as:

$$ \left\{ {\begin{array}{*{20}c} {u^{\prime\prime} - au + \frac{b}{{\sqrt {1 + u^{'2} } }} = 0} \\ {u^{\prime}\left( 0 \right) = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} u\left( 1 \right) = 0} \\ \end{array} } \right. { }\,x \in [0,1] , $$
(3)

The existence of a unique solution for the problem in (3) has been proven in [4].

To obtain the best refractive properties of the Cornea by using such type of models (3), the Cornea geometry must be shell-like with diameter 11 mm, inner thickness 0.5 mm and peripheral part 0.7 mm [15]. Moreover, the Cornea constructs a mechanical shield inside the eye for its durability [16, 17]. A schematic diagram of the human eye anatomy showing the corneal shape is displayed in Fig. 1.

Fig. 1
figure 1

Human eye anatomy

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The Cornea has five different layers from the outermost to the innermost; these layers are epithelium, Bowman’s layer, stroma, fourth Descement’s membrane and the last one known as endothelium. Each layer differs from another due to its own biological properties. For example, the 90% of the corneal thickness is due to stroma layer, which has great importance in optics and for identifying purposes. The reported studies to solve the corneal shape model are based on renewed deterministic numerical procedures. Nevertheless, stochastic computational techniques based on artificial neural networks (ANNs) optimized by genetic algorithms (GA), i.e. based on global search methods, have been used to solve broad nonlinear systems [18, 19] but they have not been applied yet for solving the nonlinear corneal shape models arising in bioinformatics. In addition, these approaches have many advantages over traditional numerical schemes, since they provide a continuous solution over the domain of training inputs, their computational requirements do not dependent upon sample size, and through interpolation one can find the solution at any point outside the trained interval.

ANNs have been applied to many problems modelled by linear and nonlinear systems [20,21,22,23,24,25]. Recently, stochastic solvers were used broadly in nanotechnology problems [26], optimization of nonlinear prey–predator systems [27], nonlinear pantograph systems [28], distribution of heat in porous fin models [29], solution for micropolar fluid flow problem [30], models of electrical conducting solids [31], mathematical problem arising in electromagnetic theory [32], Bratu systems arising in fuel ignition model [33], nonlinear Troesch’s problems [34, 35], astrophysics [36], nonlinear electrical circuit [37], atomic physics [38], thermodynamics [39], nonlinear optics [40], heartbeat models [41], HIV infection studies [42], mathematical model for wire coating analyses [43], nonlinear Painlevé II systems [44], nonlinear elliptic BVPs [45], fractional order Bagley–Torvik systems [46], nonlinear Falkner–Skan systems [47], nonlinear Flierl–Petviashivili system [48] and fractional order system of Riccati equation [49]. These are motivating factors to study the dynamics of the nonlinear corneal shape problem using ANNs and their optimization with standalone solver including interior-point technique (IPT), active-set technique (AST), sequential quadratic programming (SQP), genetic algorithm (GA), simulating annealing (SA) and pattern search (PS), as well as their hybrid combinations through the SA–AST, PS–AST and GA–AST. The efficiency of these techniques is evaluated through detailed statistical analyses that include one-way analysis of variance (ANOVA). The presented study is a novel investigation in intelligent computing paradigm to determine the approximate solution of the nonlinear corneal shape models. The salient features of this study are summarized as follows:

  • Novel application of bio-inspired computational techniques has been presented for the numerical solution of BVPs arising in the modelling of the corneal shape.

  • The computational process of modelling and optimization is exploited to obtain accurate solutions for governing relations of NCSM through ANNs, GAs, PS, SA, IPT, AST, SQP and their hybrids PS–AST, GA–AST and SA–AST.

  • The convergence study through multiple autonomous trials is conducted to authenticate the performance of the stochastic methodologies to obtain an effective, reliable and stable solution of the corneal geometry.

  • Statistical assessments through ANOVA testing further validate a better precision and convergence of computing solvers.

The rest of the paper is organized as follows. Design methodology for NCSM by means of ANN-based differential equation models and learning methodologies is presented in Sect. 2. Results of numerical experimentations for NCSM along with necessary interpretations are presented in Sect. 3. Some concluding remarks along with future recommendations are given in Sect. 4.

2 Design methodology for NCHM

The proposed designed scheme for solving the NCHM is presented here in two parts; firstly, the mathematical modelling of NCHM is presented with the help of neural networks, while in the second part, optimization schemes are introduced, which are used for finding the decision variables of the networks. The workflow of the proposed design procedure is shown in Fig. 2.

Fig. 2
figure 2

Generic workflow of hybrid computing methods for nonlinear corneal shape model

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2.1 Mathematical modelling for NCHM

In this section, mathematical modelling used for the solution of the two-point nonlinear BVP arising in the model of the corneal shape is presented. The approximate ANN solution is developed by using the continuous mapping of the solution u(x) and its derivatives as follows:

$$ \left\{ {\begin{array}{*{20}l} {\hat{u}\left( x \right) = \sum\limits_{i = 1}^{m} {\delta_{i} } f\left( {\beta_{i} + \omega_{i} x} \right)} \hfill \\ {\frac{{{\text{d}}\hat{u}\left( x \right)}}{{{\text{d}}x}} = \sum\limits_{i = 1}^{m} {\delta_{i } } \frac{\text{d}}{{{\text{d}}x}}f\left( {\beta_{i} + \omega_{i} x} \right)} \hfill \\ {{\mkern 1mu} \vdots } \hfill \\ {\frac{{{\text{d}}^{n} \hat{u}\left( x \right)}}{{{\text{d}}x^{n} }} = \sum\limits_{i = 1}^{m} {\delta_{i } } \frac{{{\text{d}}^{n} }}{{{\text{d}}x^{n} }}f\left( {\beta_{i} + \omega_{i} x} \right)} \hfill \\ \end{array} } \right. , $$
(4)

where \( \hat{u} \)(x) denotes the estimated solutions and \( W = [\delta_{i} ,\beta_{i} ,\omega_{i} ],\,\,i = 1,2,3 \ldots ,m. \) are weights of the ANN model with \( m \) neurons and \( f \) is the log-sigmoid activation function. The weights W of the ANN in (4) are the decision variables of the optimization mechanism used for training. Additionally, these weights W are optimization variables represented with arbitrary real numbers in a bounded domain. The effectiveness of log-sigmoid as an activation function in hidden layers of ANNs (4) is well proven over other counterparts due to its rapid convergence rates, better windowing characteristics and stability [50, 51].

The networks in (4) up to second-order derivatives can be written as follows:

$$ \left\{ {\begin{array}{*{20}l} {\hat{u}(x) = \sum\limits_{i = 1}^{m} {\frac{{\delta_{i} }}{{1 + {\text{e}}^{{( - \omega_{i} x - \beta_{i} )}} }}} } \hfill \\ {\hat{u}^{\prime } (x) = \sum\limits_{i = 1}^{m} {\frac{{\delta_{i} \omega_{i} }}{{2\left( {1 + \cosh (\omega_{i} x + \beta_{i} )} \right)}}} } \hfill \\ {\hat{u}^{\prime \prime } (x) = \sum\limits_{i = 1}^{m} {\delta_{i} \omega_{i}^{2} \left( { - 2{\text{csch}}(\omega_{i} x + \beta_{i} )^{3} \sinh \left( {\frac{{\omega_{i} x + \beta_{i} }}{2}} \right)^{4} } \right).} } \hfill \\ \end{array} } \right. $$
(5)

The networks in the set of Eqs. (5) are arbitrarily combined to represent the differential equation along with its boundary conditions, where the decision variables of the system depend upon the weights of ANNs. The fitness function is computed by defining two mean-square errors of the two-point BVP for the corneal shape model as

$$ \in \,\, = \,\, \in_{1} + \in_{2} \, . $$
(6)
$$ \in_{1} = \frac{1}{N + 1}\mathop \sum \limits_{k = 0}^{N} \left( {\hat{u}^{\prime\prime}_{k} - a\hat{u}_{k} + \frac{b}{{\sqrt {1 + \hat{u}_{k}^{'2} } }}} \right)^{2} ,{\kern 1pt} $$
(7)
$$ \in_{2} = \frac{1}{2}\left[ {\hat{u}(1)^{2} + \hat{u^{\prime}}(0)^{2} } \right] $$
(8)
$$ \begin{aligned} \in & = \frac{1}{N + 1}\mathop \sum \limits_{k = 0}^{N} \left( {\hat{u}^{\prime\prime}_{k} - a\hat{u}_{k} + \frac{b}{{\sqrt {1 + \hat{u}_{k}^{'2} } }}} \right)^{2} \\ & \,\, + \frac{1}{2}\left[ {\hat{u}(1)^{2} + \hat{u^{\prime}}(0)^{2} } \right]. \\ \end{aligned} $$
(9)

Our main goal is to minimize the value \( \in \) by searching suitable weights of the network in (5) for which the value \( \in \) approaches zero, i.e. \( \in \to 0 \). Thus, accordingly, the obtained solution \( \hat{u} \) will approximate the reference exact solution \( u \) of the BVP that models the corneal shape. The fitness function represented in Eq. (9) is presented in the layered structure of neural networks as shown in Fig. 3.

Fig. 3
figure 3

Layered structure of neural network models for nonlinear corneal shape model

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2.2 Learning techniques

In this section, a brief overview of optimization schemes AST, IPA, SQP, PS, SA and GA for ANN models is presented.

AST, SQP and IPTs belong to a class of local search methodologies that were exploited effectively for constrained and unconstrained optimization tasks [52]. In the procedure of both AST and SQP algorithms, the given optimization task is segmented/transformed into relatively easier sub-problems and procedures mainly based on Karush–Kuhn–Tucker conditions, which are adopted for iterative refinements. The IPT iteratively updates the weights by exploiting the feasible interior region of the optimization problem. The AST, SQP and IPT have been widely applied to many stiff and non-stiff optimization problems of practical interest (see [53, 54] and references cited therein).

The SA technique is a simple computational technique based on two characteristics of material, i.e. controlled cooling process and meticulous heating [55, 56]. The main objective of SA method is to adjust the approximate solutions successfully and effectively within a controlled interval of time. Complex optimization problems can be solved through the SA technique, and thus, it is widely applied by researchers in optimization of vehicle routing problem [57], capacitated vehicle routing problem with 2D loading constraints [58] and optimization of machine scheduling problem [59].

The PS technique is known for working without prior knowledge of the gradient of the function. This technique was proposed firstly by Hooke and Jeeves [60], while Yu was the first one to provide the convergence of the PS method with the help of the theory of positive bases [61]. The PS technique process through a set of patterns called mesh points in the close vicinity of the optimal location [62]. The PS method has been also used as a convenient procedure for the solution of optimization problems with bounded as well as linear constraints (see [60,61,62,63] and references cited therein).

A GA was firstly presented to simulate a simple picture of the natural selection by Holland [64], and its performance was subject to appropriate selection of data of the initial population, suitable choices for chromosomes to the next generation, survival of valuable genes to provide the seed for the next generation (called mutation). The GAs work on the evolutionary principle and are widely used by many researchers in stiff optimization tasks by exploiting its characteristic of robustness, avoidance of local minimum with moderate, divergence-free adaptation and consistent as compared to other counterparts [65,66,67,68].

In the present study, the optimization task is performed in the MATLAB environment using algorithms available in the graphical user interface optimization toolbox based on IPT, AST, SQP, PS, SA, GA, PS–SQP, SA–SQP and GA–SQP. The procedural process blocks of the proposed scheme are presented in Fig. 2, while the settings of the parameters are given in Table 1. Additionally, the pseudocode of the optimization procedure is given in Algorithm 1.

figure a
Table 1 Parameter settings of the local and global search algorithms
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3 Results and discussion

The ANN-based proposed method is applied for computing the solution of the corneal shape model (3) with unit value of coefficient a and b as

$$ \left\{ {\begin{array}{*{20}c} {u^{\prime\prime} - au + \frac{b}{{\sqrt {1 + u^{'2} } }} = 0,} \\ {u^{\prime}\left( 0 \right) = 0 ,\,\,\,u\left( 1 \right) = 0,} \\ \end{array} } \right. x \in [0,1], $$

The proposed standalone computing algorithms, i.e. SQP, AST, IPT, PS, GA and SA, as well as hybrid procedures, i.e. GA–AST, PS–AST and SA–ASTs, are applied to find the weights to solve the corneal shape model with the parameter settings given in Table 1 and procedure presented in Algorithm 1. The graphical representation of trained weights in terms of three-dimensional and two-dimensional plots is shown in Figs. 4 and 5, respectively, while numerical results are shown in Table 2.

Fig. 4
figure 4

Three-dimensional representation of weights of ANNs optimized with the proposed schemes

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Fig. 5
figure 5

Two-dimensional representation of weights of ANN optimized with proposed schemes

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Table 2 A set of weight vectors of ANN models of differential equation for each optimization solver
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The approximate solutions \( \hat{u}_{\text{GA - AST}} \), \( \hat{u}_{\text{PS - AST}} \) and \( \hat{u}_{\text{SA - AST}} \) are obtained for a number of 10 neurons in ANN and with the help of weights trained by hybrid optimization procedures based on GA–AST, PS–AST and SA–AST as listed in Table 2. These solutions are given by

$$ \begin{aligned} \hat{u}_{\text{GA - AST}} \left( x \right) & = \frac{1.947736}{{1 + e^{{\left( {1.49876 + 2.020087x} \right)}} }} + \,\frac{1.396893}{{1 + e^{{\left( {.47495 + .39565x} \right)}} }} \\ & \,\, + \cdots + \frac{ - 2.59037}{{1 + e^{{\left( {.94018 - 1.969659x} \right)}} }} \\ \end{aligned} $$
(10)
$$ \begin{aligned} \hat{u}_{\text{PS - AST}} \left( x \right) & = \frac{ - 0.625902}{{1 + e^{{\left( {.25572 - 7.853314x} \right)}} }} + \,\frac{ - 0.40317}{{1 + e^{{\left( {4.35784 + 8.957554x} \right)}} }} \\ & \,\, + \cdots + \,\frac{ - .86932}{{1 + e^{{\left( {.28391 - 3.368182x} \right)}} }} \\ \end{aligned} $$
(11)
$$ \begin{aligned} \hat{u}_{\text{SA - AST}} \left( x \right) & = \frac{ - 5.5354}{{1 + e^{{\left( {0.95399 + 0.12536x} \right)}} }} + \frac{ - .78324}{{1 + e^{{\left( { - 5.3268 + 4.116746x} \right)}} }} \\ & \,\, + \cdots + \,\frac{1.153815}{{1 + e^{{\left( { - 1.47945 + 6.04943x} \right)}} }} \\ \end{aligned} $$
(12)

The solutions provided by Eqs. (10–12) can be calculated with reasonable accuracy for continuous values in [0, 1] and are presented numerically in Table 3, while their graphical representations are shown in Fig. 6.

Table 3 Comparative analysis from the reference numerical solution
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Fig. 6
figure 6

Graphical representation of the approximate solutions along with reference solution

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The values of absolute errors (AEs), i.e. \( \left| {u(x) - \hat{u}(x)} \right| \), are also determined for the proposed algorithms from the reference solution provided with an implicit RK method with the help of Mathematica using the built-in software package “NDSolve” with default values of adaptive step size, accuracy goal and tolerances. The results are given in Table 4, while their graphical representation is shown in Fig. 7 for inputs in the interval [0,1] with step size 0.1. The detailed statistical observations for 100 independent runs of each solver are tabulated in Table 5, while the fitness values are shown in Fig. 8 for the considered problem for each stochastic numerical solver.

Table 4 Comparative analysis on AEs from the reference numerical solution
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Fig. 7
figure 7

Absolute errors for the proposed computing algorithms

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Table 5 Statistical indicators of the different approaches for the corneal shape model
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Fig. 8
figure 8

Results of statistics on fitness for multiple runs of the proposed computing algorithms

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From the results of detailed simulations, it is observed that the AEs are in the range 10−03–10−08, for AST; 10−03–10−07 for SQP; 10−03–10−06 for IPT; 10−02–10−03 for PS; 10−02–10−03 for GA; 10−02–10−03 for SA; 10−03–10−09 for PS–AST; 10−03–10−08 for GA–AST and 10−03–10−09 for SA–AST. The mean value of AEs (MAE) is calculated by the formula

$$ {\text{MAE}} = \frac{1}{11}\sum\limits_{i = 0}^{10} {\left| {u(x_{i} ) - \hat{u}(x_{i} )} \right|} , $$
(13)

and the values of MAE for each optimization solver are shown in Fig. 9. It is seen that the MAEs for standalone approaches AST, SQP, IPT, PS, GA and SA are in the range 10−01–10−04, while for the hybrid methodologies PS–AST, GA–AST and SA–AST are in the range 10−05–10−07.

Fig. 9
figure 9

Results of MSE for 100 runs

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The efficiency of the proposed solutions is further justified by using Monte Carlo simulations and their statistical analysis. The statistics by means of standard deviation (STD) and mean parameters are calculated in order to check the behaviour of the proposed solvers. Another metric based on the mean-square error is defined for 100 observations as

$$ {\text{MSE}} = \frac{1}{100}\sum\limits_{i = 0}^{100} {(\left| {u(x) - \hat{u}_{i} (x)} \right|} )^{2} $$
(14)

where‘\( i \)’ is the index of independent runs of the solvers.

The descriptive statistical analyses for 100 runs of each scheme to solve the corneal shape model are conducted, and the results are tabulated in Table 5. It is seen that the results of the proposed scheme achieved reasonably accurate values based on minimum, maximum, mean, MAE, MSE and RMSE indicators.

The reliable and effective statistical method based on analysis of variance (ANOVA) is exploited to analyse the small variation in the results of proposed methodologies. Therefore, the average efficiency based on MAE values for each optimization technique has been analysed via descriptive statistical operators in terms of plots of the confidence intervals as well as ANOVA outcomes. Results of ANOVA analyses are graphically presented in Figs. 10 and 11 for 95% confidence interval and numerically tabulated in Table 6. It is seen that in case of 95% confidence interval plot of ANOVA, the performance of the hybrid scheme is slightly better than that of PS and GA performance on both accuracy operators of MAEs and AEs. The comparative study on the basis of histogram illustrations is presented in Fig. 12, while two types of probability plots are shown in Figs. 13 and 14. Results presented in all these graphs show that each algorithm is effective with reasonable accuracy, but the performance of integrating computing solvers is relatively superior to the rest.

Fig. 10
figure 10

Confidence intervals of 95% for ANOVA

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Fig. 11
figure 11

Comparison on AE values calculated for 100 runs

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Table 6 Outcomes of the ANOVA test for each solver
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Fig. 12
figure 12

Histograms illustrating the proposed approaches

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Fig. 13
figure 13

Comparison through probabilities

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Fig. 14
figure 14

Comparison through normal distribution of data for the proposed approaches

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The computational complexity of six standalone algorithms AST, SQP, IPT, PS, GA and SA, as well as hybrid heuristic of PS–AST, GA–AST and SA–AST is analysed to determine the efficiency on the basis of multiple autonomous runs (100 runs). The computational averaged data in terms of execution times, iterations/generations executed and fitness functions evaluated are calculated for the analysis. The results are provided in Table 7 for all nine algorithms. These results show that AST algorithm is the most efficient in standalone methodologies with computation time around 6.45 ± 3 s, iterations around 340.5 ± 175 and fitness functions executed around 20,844 ± 10,783, while in combined computational heuristic GA–AST outperformed the rest with complexity indices around 189 ± 50, 2029 ± 635 and 508,267 ± 133,710. The results of all simulations are conducted on Dell Latitude E0420, Intel(R) Core(TM) i5-2520 M CPU 2.50 GHz, 6 GB RAM, running MATLAB 2018a on operating system Microsoft Window 10.

Table 7 Data on computational complexity
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4 Conclusions

Integrated neuro-heuristic computing paradigm is presented to calculate the approximate results of BVPs arising in the modelling of the corneal shape using neural networks and their optimization with standalone methods AST, GA, SQP, IPT, PS and SA, along with the approximate solutions of hybrid schemes GA–AST, PS–AST and SA–AST. The comparison of the approximated results with the numerical solution obtained through an implicit Runge–Kutta method illustrates the validity of the proposed stochastic numerical solvers; however, hybrid methodologies achieved relatively better precision. The convergence analyses based on 100 autonomous runs for each solver also certify the performance of the stochastic solvers for reliable, viable and robust solutions of the mathematical model of the corneal shape. The ANOVA-based statistics show a better precision and performance of the PS- and GA-based standalone proposed stochastic methods on the basis of 95% confidence interval as well as error bar plots. Similarly, the hybrid scheme based on PS–AST and GA–AST provides better accuracy and convergence.

In future, one may exploit the strength of the proposed stochastic solvers as good alternatives for solving governing differential equations representing stiff and non-stiff applications of linear and nonlinear physical systems, particularly the complex mathematical models arising in bioinformatics [69,70,71].