Integrated Stochastic Investigation of Singularly Perturbed Delay Differential Equations for the Neuronal Variability Model

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Introduction
Te perturbed theory is a huge collection of algebraic methods that approximate the outcomes of problems that have no analytical solution in the closed form.Tis method reduces a hard problem to a comparatively simple problem of infnite sequence that may be solved logically.Tese problems are based on a small parameter.Te perturbed theory is normally based on two forms [1,2].One is a regular perturbed theory whose series is a power series within ε, which has no vanished radius of convergent and defned as a singularly perturbed theory whose sequence either does not appear as a power series or if it appears, the power series has a vanished radius of convergence [3,4].
A special feature of all regular perturbed theories is the accurate solution.However, a small nonzero |ε| efciently proceeds the unperturbed neither zeroth-order solution when ε ⟶ 0. Sometimes, there is no solution to unperturbed problems in singularly perturbed theory.Te solution whose function ε may stop existing when ε � 0, but when a solution to the unperturbed problem does exist, its numerical features vary from the real result for arbitrarily little nonzero ε [5][6][7].In these cases, the solution ε � 0 is essentially distinct in quality from the corresponding solutions attained in the boundary ε ⟶ 0. We should categorize the problem as the regular perturbed theory if no change exists in quality [8][9][10].
A singularly perturbed ordinary diferential equation (ODE) whose highest derivative is multiplied with a small parameter is termed as a perturbed parameter.Te solution of SPDDEs was initiated in the year 1968 [11].In this era, diferent surveys were conducted by researchers [12][13][14].Kadalbajoo and Reddy carried out asymptotic as well as numerical techniques for the solution of the singularly perturbed theory [15].Te solution of SPDDEs difers rapidly in the region which is known as layers that may be obvious in the solution or its gradient and frequently seem at the boundary region.Several problems in science as well as in engineering, elasticity, control theory, biosciences, and fuid mechanics are created by SPDDEs, such as those present in red cell models [16][17][18][19][20].
In the literature, the delay diferential equations (DDEs) were considered since the 1940s, inspired by control problems; for instance, balancing the location of a ship with pushing water from a tank at one verge of the vessel to the tank on other verges, see also [21,22].In 1949, Myshkis described DDEs [23].In early 1963, Cooke and Bellman described DDEs, which appeared outside SU [24], which represent the basic theory more than the dynamical system's standpoint about semigroups as well as semifows, further parallel towards the ODE theory as long as possible.
Te numerical solutions at the earlier times are used to determine the value of terms in SPDDEs.Te delay in the process emerges because of the necessity of positive time to detect the guidance and respond to it [8,[25][26][27][28].Recently, some researchers presented physical examples of delay diferential equations, like the periodic oscillation of respiration frequency within constant conditions [29][30][31][32][33]. Tis delay is produced by cardiorespiratory inefciency in the biological circuit commanding the CO 2 level in the blood [34,35].Furthermore, some applications of DDEs are in biological sciences.Te DDE can be categorized into the retarded delay diferential equation and the neutral diferential equation.DDEs applications arise in the feld of the control theory, explanation in human pupil refex [36,37], as well as on numerical modeling in biological sciences [38], HIV infection [39], and so on.
In early 1968, Kadalbajoo and Reddy approximated the solution of SPDDEs, and several surveys and reviews of various researchers have been presented.Te study on diferent asymptotic and numerical methods for explaining singular perturbation queries is discussed [15].
Using a parametric cubic spline, the nonlinear singularly perturbed DDEs are changed into linear singularly perturbed DDEs by the quasilinearization method [40].If the smaller order of the singular perturbation parameter in the delay is not sufcient, the approach of increasing the delay term in Taylor's series may lead to a bad approximation.We construct a special type of mesh in such a way that the term containing delay lies on the nodal points after discretization.Te fnite diference technique on Shishkin mesh is utilized to determine SPDDEs of convection difusion kind with the integral border condition [41,42].Tis technique approximately belongs to frst order convergent.Te numerical solution for second-order SPDDEs is provided by the uniform fnite diference method.Te solution to the problem is utilized by a hybrid diference method that lies on a Shishkin-type mesh.Te interior and boundary layer occurs in the exact solution because of the delay term [43,44].
Pramod Chakravarthy et al. utilized the fnite diference technique with ftted operators by Numerov's method to approximate the solution of SPDDE [45].Te parameterized SPDDE system using a uniformly convergent numerical scheme is discussed in [46].To approximate the numerical solution of SPDDE, Chakravarthy with his team utilized an exponentially ftted fnite diference method to handle the large delay [47].
Shishkin mesh utilized a hybrid initial value technique to approximate the numerical solution of SPDDE for boundary value problems with a noncontinuous convection factor and a source term [48].To approximate the numerical solution and its absolute error for the boundary value problem for the linear and nonlinear singular perturbed DDE, we utilize the fxed point method [49].
Te singularly perturbed DDEs use the terminal boundary value method to obtain the solution such as exhibiting layer behaviour [50].By presenting a terminal point, the unique problem is distributed into internal and external region problems.To solve both the internal and external region problems, the second-order fnite diference method has been used.Te technique is iterative to the terminal point [51,52].
Te SPDDE used reproducing kernel technique (RKM) and cannot obtain better approximate solutions.Now, the singularly perturbed DDE used a piecewise reproducing kernel technique to approximate the numerical solution [35].Te linear singular perturbed diferential equation with delay in the convection term is changed into a linearized delay term by utilizing two-term Taylor series expansion.Te SPDDE uses an asymptotic numerical hybrid technique to uniformly approximate the solution [53,54].Usually, the singularly perturbed nonlinear DDEs of the boundary value problems play an important role in clarifying diferent uses such as the theory of nonpremixed combustion [55] and chemical reactions [56].Kadalbajoo and Sharma built a fnite-diference technique to approximate the numerical solution of SP nonlinear DDEs [57,58].
A fnite diference technique and a B-spline collocation technique have been recommended for small delay queries, respectively [59,60].An initial value technique and a uniformly valid fnite diference technique for convection diffusion problems with smooth data have been recommended, respectively [61], while the authors have recommended a singularly perturbed problem with nonsmooth data to be used as an initial value method.Boundary value problems containing DD calculations arose in reading the mathematical display of several practical occurrences, such as a microscale heat transfer, reaction-difusion equations, stability, and control including control of chaotic systems [62,63].

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International Journal of Intelligent Systems Due to their enormous importance in the control of ships, biological sciences, light absorption using stellar objects, chemistry, discrete mathematics, the medical industry, particle physics, object models, fnance, engineering disciplines, community composition, medicine, electromagnetics, contagious diseases, telecommunications equipment, reaction mechanisms, and control models, delay diferential mathematical systems have gained enormous signifcance for scientists/researchers [64].Finding the solutions to singular systems, which have enormous signifcance and are thought to be difcult to replicate because of the solitary point at the origin, is never easy.Lane Emden, which has enormous relevance and a long history, is among the important singular types of the models.Many applications of SPDDEs can be found in astronomy, quantum mechanics, and gas cloud-based systems [11,65].
Tis work presents a novel singular model that is rigid and more intricate due to the delay and perturbed factors.To solve the model, a stochastic numerical computational strategy is developed.Te solvers for stochastic processing can be used to solve systems with numerical approximation, delayed, and fractional order.Some of the key features and highlights of the proposed study are listed as follows: (1) A novel intelligent formulation is implemented to analyze the second-order SPDDE through ANN by using diferent solvers such as the GA, SQP, and PS, and hybridized with the AST and IPT.(2) Te accuracy, stability, and efciency of the proposed approach are validated by presenting the detailed statistical analysis and mean residual error analysis.(3) A comparative study is presented based on residual errors by comparing the results obtained from the GA, GA-IPT, and SQP for diferent problems in the form of tables and graphs.(4) Te hybrid optimum solvers have obtained the solution of SPDDE in less time and reduced the computational complexity and ensured the accuracy and rapid conversion of the obtained numerical solution.(5) Some notable benefts of the technique include consistency, reliability, improved workfow, simplicity of comprehension, and encompassing pertinency, in addition to the accurate projections of the GA, SQP, and IPT framework.
Te organization of the manuscript is designed as follows: Section 2 discusses the overall mathematical modeling, and it explains the modeling of neural networks, ftness function, learning techniques, and psuedocode for GA-IPT.Section 3 presents the results and discusses explaining the statistical analysis briefy.Section 4 explains the conclusions, remarks, and future recommendations.

Mathematical Designs of the Model
Te SPDDE with its BVP is given as with the boundary conditions where 0 < ε ≪ 1, δ is the delay parameter and particular suitably fat functions on [0, 2], φ(y) is a smooth function on [− δ, 0], and B is a given constant which is free of ε; the BVP (1) along with (2) shows a robust boundary level y � 0 [41].

Mathematical Modeling.
Here, we have designed the neural networks in the form of linear second-order SPDDEs, along with their boundary conditions.

Neural Networks
Modeling.Te mathematical model of SPDDEs is presented by "feed-forward" ANNs through the following continual mapping founded in the form of a single "input, hidden, and output" layer by solution  u(y) and its corresponding derivatives are given as follows: where p is the specifed number of neurons, f is the activation function, and α, β, and c represent real-valued component bounded weights and defned as follows: International Journal of Intelligent Systems Te networking in equation ( 3) utilized the log-sigmoid function f(x) � 1/1 + e − x ; furthermore, its certain derivatives for the activation function of the network displayed in (3) are written in equation ( 4). Figure 1 expresses the complete structure of the neural networks model for SPDDEs in the form of a structural diagram.It shows the complete features, layers, weights, derivatives, and functions involved in the construction of the input, hidden, and output layers, respectively.

Fitness Function Expression.
Te suited combination for the equations through a set of equation ( 4) is utilized to design the SPDDE for equations (1) and (2).A ftness function is an expression for (1) with boundary conditions to defne an error which depends on the sum of two mean square error functions, given as where h denotes the step size for the whole domain. Where

Learning Techniques.
Here, a detailed review of optimization solvers GA, SQP, IPT, and PS for ANN is expressed.
Te IPT and SQPs are among a group of local search techniques that have been successfully applied to both unconstrained and constrained optimization problems.
Te proposed optimization problem is transformed into more manageable subproblems in the process of SQP algorithms, and methods mostly based on Karush-Kuhn-Tucker conditions are used for iterative revisions.By taking advantage of the viable interior region of the optimization problem, the IPT iteratively updates the weights.In numerous stif and nonstif optimization problems of practical interest, the SQP and IPT have been frequently employed [66].
Te GA is inspired by Darwin's well-known theory of evolution.It was established properly in the early 1970s with great work by John Holland known as the pioneer of GA solver.Initially, it was designed for problem solving and analysis.Te GA is working with a set of weights in the population sample.Solutions of single best ft are taken in the next step and utilized to create a new population that is inspired by the whole process, and a new population will be ftter as compared to the previous selection.In 1992, for the frst time, John Koza (JK) developed a program for diferent tasks through the GA.He named his technique genetic programming.GAs are optimization techniques and stochastic search, encouraged by natural development.
"Pattern search" (also called direct search) is a sort of numeric optimization technique that does not need a gradient."Pattern search" was invented by Hooke and Jeeves.Most of the literature about this topic is referred to as "Hooke and Jeeves" along with the theory published in their article.Te title PS has been used as a collective term for all methods which search concerning the current point in a measured direction for best function value.When the search gets to the best point, that point is modifed as a new base and this point is a restart point for searching.In case of an unsuccessful search, either the search route is changed or the search border is cut below by diminishing the step size.PS techniques tend to have very easy tactics and hence are easy to utilize as the initial optimization method.Moreover, the PS techniques are also fexile and reliable in their use.4 International Journal of Intelligent Systems Te "interior-point technique" (also known as "barrier techniques") is a group of algorithms to solve nonlinear as well as linear convexity optimization models."John von Neumann" [67] recommended an IPT for linear programming that was neither an efcient nor a "polynomialtime technique" in practice.It revolved around being slower than the commonly utilized simplex technique [68]."Narendra Karmarkar" established a system for linear programming named Karmarkar's computation, in 1984, that runs likely in polynomial time which is also efcient for practice.It allowed solutions for linear programming methods that are behind the talents of the simplex technique.
Rather, the simplex technique attains the best result by passing through the interior of the feasible area.Eventually, modern IPTs have been instilled virtually in all felds for continuous optimization and have forced excellent improvements into the earlier procedures.In [69][70][71][72], the authors have presented the applications of higher-order diferential equations in fnancial and business predictions/forecasts based on the nonlinear and linear types of models.Figure 2 represents the complete structure of the hybrid solver for the singularly perturbed delay model.

Results and Discussion
In this section, the proposed technique is implemented on three diferent problems for singularly perturbed delay differential equations and the results and discussions are presented.Te numerical results of the SPDDE equation are obtained, analyzed, and presented by using "SQP," "GA," "PS," and hybrid techniques "GA-AST," "PS-AST," "GA-IPT," and "PS-IPT."Te outcome of these three SPDDEs optimized the results by local and global techniques in neural network designs which are given.Algorithm 1 elaborates a thorough pseudocode for GA-IPT to fnd weights based on ANN for solving the SPDDE, which plays an essential role in this study.

Problem 1. Consider the following BVP for an SPDDE as
with boundary conditions at ε � 2 − 4 and δ � 0.03.
We need to obtain the solution to the problem by utilizing the proposed methodology, as it depends upon the ftness function  E along with step size h � 0.1.Here, we determine the ftness function  E taking N � 10.International Journal of Intelligent Systems

GA-IPT technique Start. STEP: Initialization
Initially, the population is selected arbitrarily through the entries on the real line (number) toward express weights along with the equivalent elements identical toward the unknown weights of ANN paradigms.

STEP: (Fitness calculation)
To estimate the ftness value of each member of the population in equation ( 5) utilizing the values for the network specifed set on equations ( 3) and (4).

STEP: Ranking
In order of minimum value for ftness functions to the models, rank each individual concerning the populations.Members who performed well often had lower ftness values and vice versa.

STEP: Terminate criteria
Termination of the algorithm: (ii) Obtained the objective level (ii) Required generations achieved (iii) Obtained tolerance and generations Function tolerance: 1e − 13 , ftness limit: 1e − 13  Other settings: By default.If the terminate criterion fts, then the hybridization level is.

Reproductive:
Create the following population by each generation.Selection: Calls stochastic uniform selection as a function.
Crossover: Call for constraint dependent as function.
Mutation: Call for constraint dependent as function.Others: By default.

Hybridization:
Te interior point technique is incorporated for fne tuning of parameters by taking the best chromosome of GAs as a starting point.Setting up the parameters for IPT: Finite diference: Forward diferences.X-tolerance: 1e − 13 Function-tolerance: 1e − 16 ; constraint tolerance: By default.

STEP: Recurrence
Repeat the whole hybrid procedure for hidden layers.GA-IPT procedure end.
ALGORITHM 1: Pseudocode for GA-IPT to fnd weights based on ANN for solving the SPDDE.
Te proposed technique of sorted, scattered numbers of runs and residual error of four individual solvers for problem 1 is shown in Figure 5, and the y-axes show that y lies in the selected interval from 0 to 1, and the residual error's graphical representation shows that the behaviour of the PS technique lies from 10 − 5 to 10 − 2 and the hybrid PS technique with IPT lies from 10 − 9 to 10 − 7 , GA technique lies from 10 − 5 to 10 − 3 , and hybrid GA technique with IPT lies from 10 − 9 to 10 − 7 .Te 3-D display of best weights α, β, c with the GA, GA-IPT, PS, and PS-IPT for problem 1 is displayed in Figure 6, and the learning curves of the proposed techniques PS, PS-IPT, and GA, GA-IPT for problem 1 are shown in Figures 3 and 4.
3.2.Problem 2. We consider the following BVP for a SPDDE: With boundary conditions at ε � 10 − 2 , we need to obtain the solution to the problem by utilizing the proposed methodology, which depends upon ftness function  E along with the step size h � 0.1.Here, we have determined the ftness function  E, taking N � 10.

+ (1.47340)
1 + e − (2.52201 y+1.57181) 1 + e − (2.55527 y+1.54266) . ( To fnd the results for equation (15) with the given conditions, we applied the GA and SQP for refnement, and we have hybridized it with IPT, i.e., GA-IPT by utilizing MATLAB with diferent parameter settings, shown in Table 1.Obtained weights with individual ftness for SQP, GA, and GA-IPT algorithms are shown in Figure 7 as ftness functions given in equation (16).Tables 4 and 5 present values of "number of weights for GA, GA-IPT, and SQP" and residual errors of the proposed technique.Te suggested solution  u(y) is given in equation ( 4); the proposed model is designed by utilizing optimal weights, and equations ( 17)- (19) shows the optimal solutions obtained from the GA, GA-IPT, and SQP optimization solvers.Step size 0 500 1000

International Journal of Intelligent Systems
Step Size: 0.0109548 for PS-IPT 10 0 Iteration 1 + e − (5.16724 y+7.08939) . ( Te proposed technique of sorted, scattered numbers of runs and residual error of four individual solvers for problem 2 are shown in Figure 8, and the y-axes display that y lies in the selected interval from 0 to 1, and the residual error of obtained graphical representation shows that the behaviour of the SQP technique lies from 10 − 5 to 10 − 4 , GA technique lies from 10 − 4 to 10 − 2 , and hybrid GA technique with IPT lies from 10 − 5 to 10 − 4 .Te 3-D display of best weights α, β, c with GA, GA-IPT, and SQP for problem 2 is displayed in Figure 9, and the learning curve of the proposed techniques GA, GA-IPT, and SQP for problem 2 is shown in Figure 7.

􏽢
To obtain the solution for equation (20) with boundary conditions, we have utilized the GA, PS, and hybrid technique of GA-IPT, PS-IPT by using the MATLAB built-in functions with the parameters set, given in Table 1 as IPT, PS, and GA, respectively.Te residual error is presented in Figure 10 for problem 3. Te group of prepared weights with individual ftness for PS, GA, PS-IPT, and GA-IPT algorithms are shown in Figure 11 as ftness functions are given in equation (21).Tables 6 and 7 present the values as "number of weights for PS, GA, PS-IPT, and GA-IPT" and residual errors of the proposed technique.Te suggested solution  u(y) is given in equation ( 4), the proposed model is designed by utilizing optimal weights, and equations   1 + e − ((− 8.34331)y+(− 1.66416)) u GA (y) � (6.96770) 1 + e − (8.43611 y+8.93885) 1 + e − (5.62569 y+8.63041) , ( u PS− IPT (y) � (6.87138) 1 + e − ((− 1.28699)y+4.84758)+ (− 8.99257) 1 + e − ((− 0.84986)y+(− 2.86486)) 1 + e − ((− 0.11311)y+(− 0.70787)) , ( u GA− IPT (y) � (− 0.12362) 1 + e − ((− 0.23698)y+(− 0.92593)) + (0.15182) 1 + e − ((− 0.97499)y+(− 0.32822)) 1 + e − ((− 0.16695)y+0.03775) . ( Te proposed technique of the residual error of four individual solvers for problem 3 is shown in Figure 9 and the y-axes show that y lies in selected interval from 0 to 1 and the obtained graphical representation shows that the behaviour of the PS technique lies from 10 − 3 to 10 − 2 , and hybrid PS technique with the IPT lies from 10 − 5 to 10 − 4 , GA technique lies from 10 − 4 to 10 − 2 , and hybrid GA technique with the IPT lies from 10 − 8 to 10 − 7 .Te sorted and scattered numbers of runs for problem 3 are presented in Figure 9, and the y-axes displays that y lies in the selected interval from 0 to 100.Te 3-D display of best weights α, β, and c with GA, GA-IPT, PS, and PS-IPT for problem 3 is displayed in Figure 10, and best weights together with parameters of GA and GA-IPT are presented in Figure 11.Furthermore, the learning curve which presents the scheme systematic performance for hybrid solver is shown in Figures 12 and 13.Te interval confdence levels are expressed for the problems in Figures 14-16      Step size Step Size: 3.19816e-05 for PS-IPT

Conclusions and Recommendations
In this research work, the numerical solution of second-order SPDDE is presented by an innovative "artifcial neural network" (ANN).Tis study involves the use of a log-sigmoid function incorporated with optimization solvers such as the GA, PS, and SQP and with hybridized solvers such as GA-IPT, GA-AST, PS-IPT, and PS-AST.It is concluded that the proposed optimum technique has estimated the numerical results efciently and accurately, and the results are also very fast convergent.A comparative study is presented based on residual errors that are compared with the GA, GA-IPT, and SQP for problems 1, 2, and 3 in the form of tables and graphs.To further enhance and ensure the stability and accuracy of the presented results, HSU multiple comparisons with the best (MCB) simultaneous tests for problems 1, 2, and 3 are presented in tabular forms, strengthening the statistical analysis.
It is observed that the presented research work has better optimum accuracy than other numerical techniques at present.Te hybrid optimum solvers have approximated the numerical solution of the proposed problem in lesser time, ensuring the accuracy and reliability of the obtained results.Trough the statistical analysis, it is concluded that GA-IPT hybridization has achieved better results as compared to PS, GA, SQP, and PS-IPT.

Figure 1 :
Figure 1: Structural diagram of the neural network model for SPDDEs.

Figure 2 :
Figure 2: Flowchart of hybrid solver for the singularly perturbed delay model.

Figure 3 :Figure 4 :
Figure 3: Learning curve of the proposed techniques PS and PS-IPT for problem 1.

Figure 5 :
Figure 5: Sorted and scattered numbers of runs and the residual error for problem 1.

Figure 12 :
Figure 12: Learning curve of the proposed techniques PS and PS-IPT for problem 3.

Figure 13 :Figure 14 :Figure 15 :Figure 16 :
Figure 13: Learning curve of the proposed techniques GA and GA-IPT for problem 3.

Table 1 :
Setting of parameters for optimization solvers.

Table 2 :
Optimized weights towards ANN through the designed technique in problem 1.

Table 3 :
Residual error for the proposed techniques in problem 1.

Table 4 :
Optimized weights toward ANN through the designed technique in problem 2.

Table 5 :
Residual error for the proposed techniques in problem 2.

Table 6 :
Optimized weights toward ANN through the designed technique in problem 3.

Table 7 :
Residual error for the proposed techniques in problem 3.

Table 8 :
HSU multiple comparisons with the best (MCB) simultaneous tests for problem 1. value and T-values, ensuring the CI on or above 95%.Moreover, the diferences of means and SE are tabulated for all the cases of hybridized solvers.

Table 9 :
HSU multiple comparisons with the best (MCB) simultaneous tests for problem 2.

Table 10 :
HSU multiple comparisons with the best (MCB) simultaneous tests for problem 3.