Design of a Computational Heuristic to Solve the Nonlinear Liénard Differential Model

In this study,the design of a computational heuristic based on the nonlinear Liénard model is presented using the efficiency of artificial neural networks(ANNs)along with the hybridization procedures of global and local search approaches.The global search genetic algorithm(GA)and local search sequential quadratic programming scheme(SQPS)are implemented to solve the nonlinear Liénard model.An objective function using the differential model and boundary conditions is designed and optimized by the hybrid computing strength of the GA-SQPS.The motivation of the ANN procedures along with GA-SQPS comes to present reliable,feasible and precise frameworks to tackle stiff and highly nonlinear differentialmodels.The designed procedures of ANNs along with GA-SQPS are applied for three highly nonlinear differential models.The achieved numerical outcomes on multiple trials using the designed procedures are compared to authenticate the correctness,viability and efficacy.Moreover,statistical performances based on different measures are also provided to check the reliability of the ANN along with GASQPS.

Fractional Order Nonlinear Bone Remodeling Dynamics Using the Supervised Neural Network

This study aims to solve the nonlinear fractional-order mathematical model(FOMM)by using the normal and dysregulated bone remodeling of themyeloma bone disease(MBD).For themore precise performance of the model,fractional-order derivatives have been used to solve the disease model numerically.The FOMM is preliminarily designed to focus on the critical interactions between bone resorption or osteoclasts(OC)and bone formation or osteoblasts(OB).The connections of OC and OB are represented by a nonlinear differential system based on the cellular components,which depict stable fluctuation in the usual bone case and unstable fluctuation through the MBD.Untreated myeloma causes by increasing the OC and reducing the osteoblasts,resulting in net bone waste the tumor growth.The solutions of the FOMM will be provided by using the stochastic framework based on the Levenberg-Marquardt backpropagation(LVMBP)neural networks(NN),i.e.,LVMBPNN.The mathematical performances of three variations of the fractional-order derivative based on the nonlinear disease model using the LVMPNN.The static structural performances are 82%for investigation and 9%for both learning and certification.The performances of the LVMBPNN are authenticated by using the results of the Adams-Bashforth-Moulton mechanism.To accomplish the capability,steadiness,accuracy,and ability of the LVMBPNN,the performances of the error histograms(EHs),mean square error(MSE),recurrence,and state transitions(STs)will be provided.

An Intelligence Computational Approach for the Fractional 4D Chaotic Financial Model

The main purpose of the study is to present a numerical approach to investigate the numerical performances of the fractional 4-D chaotic financial system using a stochastic procedure.The stochastic procedures mainly depend on the combination of the artificial neural network(ANNs)along with the Levenberg-Marquardt Backpropagation(LMB)i.e.,ANNs-LMB technique.The fractional-order term is defined in the Caputo sense and three cases are solved using the proposed technique for different values of the fractional orderα.The values of the fractional order derivatives to solve the fractional 4-D chaotic financial system are used between 0 and 1.The data proportion is applied as 73%,15%,and 12%for training,testing,and certification to solve the chaotic fractional system.The acquired results are verified through the comparison of the reference solution,which indicates the proposed technique is efficient and robust.The 4-D chaotic model is numerically solved by using the ANNs-LMB technique to reduce the mean square error(MSE).To authenticate the exactness,and consistency of the technique,the obtained performances are plotted in the figures of correlation measures,error histograms,and regressions.From these figures,it can be witnessed that the provided technique is effective for solving such models to give some new insight into the physical behavior of the model.

Stochastic Investigations for the Fractional Vector-Host Diseased Based Saturated Function of Treatment Model

The goal of this research is to introduce the simulation studies of the vector-host disease nonlinear system(VHDNS)along with the numerical treatment of artificial neural networks(ANNs)techniques supported by Levenberg-Marquardt backpropagation(LMQBP),known as ANNs-LMQBP.This mechanism is physically appropriate,where the number of infected people is increasing along with the limited health services.Furthermore,the biological effects have fadingmemories and exhibit transition behavior.Initially,the model is developed by considering the two and three categories for the humans and the vector species.The VHDNS is constructed with five classes,susceptible humans Sh(t),infected humans Ih(t),recovered humans Rh(t),infected vectors Iv(t),and susceptible vector Sv(t)based system of the fractional-order nonlinear ordinary differential equations.To solve the number of variations of the VHDNS,the numerical simulations are performed using the stochastic ANNs-LMQBP.The achieved numerical solutions for solving the VHDNS using the stochastic ANNs-LMQBP have been described for training,verifying,and testing data to decrease the mean square error(MSE).An extensive analysis is provided using the correlation studies,MSE,error histograms(EHs),state transitions(STs),and regression to observe the accuracy,efficiency,expertise,and aptitude of the computing ANNs-LMQBP.

Stochastic Computational Heuristic for the Fractional Biological Model Based on Leptospirosis

The purpose of these investigations is to find the numerical outcomes of the fractional kind of biological system based on Leptospirosis by exploiting the strength of artificial neural networks aided by scale conjugate gradient,called ANNs-SCG.The fractional derivatives have been applied to get more reliable performances of the system.The mathematical form of the biological Leptospirosis system is divided into five categories,and the numerical performances of each model class will be provided by using the ANNs-SCG.The exactness of the ANNs-SCG is performed using the comparison of the reference and obtained results.The reference solutions have been obtained by using theAdams numerical scheme.For these investigations,the data selection is performed at 82%for training,while the statics for both testing and authentication is selected as 9%.The procedures based on the recurrence,mean square error,error histograms,regression,state transitions,and correlation will be accomplished to validate the fitness,accuracy,and reliability of the ANNs-SCG scheme.

Numerical Procedure for Fractional HBV Infection with Impact of Antibody Immune

The current investigations are presented to solve the fractional order HBV differential infection system(FO-HBV-DIS)with the response of antibody immune using the optimization based stochastic schemes of the Levenberg-Marquardt backpropagation(LMB)neural networks(NNs),i.e.,LMBNNs.The FO-HBV-DIS with the response of antibody immune is categorized into five dynamics,healthy hepatocytes(H),capsids(D),infected hepatocytes(I),free virus(V)and antibodies(W).The investigations for three different FO variants have been tested numerically to solve the nonlinear FO-HBV-DIS.The data magnitudes are implemented 75%for training,10%for certification and 15%for testing to solve the FO-HBV-DIS with the response of antibody immune.The numerical observations are achieved using the stochastic LMBNNs procedures for soling the FO-HBV-DIS with the response of antibody immune and comparison of the results is presented through the database Adams-Bashforth-Moulton approach.To authenticate the validity,competence,consistency,capability and exactness of the LMBNNs,the numerical presentations using the mean square error(MSE),error histograms(EHs),state transitions(STs),correlation and regression are accomplished.

A Neural Study of the Fractional Heroin Epidemic Model

This works intends to provide numerical solutions based on the nonlinear fractional order derivatives of the classical White and Comiskey model(NFD-WCM).The fractional order derivatives have provided authentic and accurate solutions for the NDF-WCM.The solutions of the fractional NFD-WCM are provided using the stochastic computing supervised algorithm named Levenberg-Marquard Backpropagation(LMB)based on neural networks(NNs).This regression approach combines gradient descent and Gauss-Newton iterative methods,which means finding a solution through the sequences of different calculations.WCM is used to demonstrate the heroin epidemics.Heroin has been on-growth world wide,mainly in Asia,Europe,and the USA.It is the fourth foremost cause of death due to taking an overdose in the USA.The nonlinear mathematical system NFD-WCM discusses the overall circumstance of different drug users,such as suspected groups,drug users without treatment,and drug users with treatment.The numerical results of NFD-WCM via LMB-NNs have been substantiated through training,testing,and validation measures.The stability and accuracy are then checked through the statistical tool,such asmean square error(MSE),error histogram,and fitness curves.The suggested methodology’s strength is demonstrated by the high convergence between the reference solutions and the solutions generated by adding the efficacy of a constructed solver LMB-NNs,with accuracy levels ranging from 10?9 to 10?10.

An Artificial Approach for the Fractional Order Rape and Its Control Model

The current investigations provide the solutions of the nonlinear fractional order mathematical rape and its controlmodel using the strength of artificial neural networks(ANNs)along with the Levenberg-Marquardt backpropagation approach(LMBA),i.e.,artificial neural networks-Levenberg-Marquardt backpropagation approach(ANNs-LMBA).The fractional order investigations have been presented to find more realistic results of the mathematical form of the rape and its control model.The differential mathematical form of the nonlinear fractional order mathematical rape and its control model has six classes:susceptible native girls,infected immature girls,susceptible knowledgeable girls,infected knowledgeable girls,susceptible rapist population and infective rapist population.The rape and its control differential system using three different fractional order values is authenticated to perform the correctness of ANNs-LMBA.The data is used to present the rape and its control differential system is designated as 70%for training,14%for authorization and 16%for testing.The obtained performances of the ANNs-LMBA are compared with the dataset of the Adams-Bashforth-Moulton scheme.To substantiate the consistency,aptitude,validity,exactness,and capability of the LMBA neural networks,the obtained numerical values are provided using the state transitions(STs),correlation,regression,mean square error(MSE)and error histograms(EHs).

Numerical Computation of SEIR Model for the Zika Virus Spreading

The purpose of this study is to present the numerical performancesand interpretations of the SEIR nonlinear system based on the Zika virusspreading by using the stochastic neural networks based intelligent computingsolver. The epidemic form of the nonlinear system represents the four dynamicsof the patients, susceptible patients S(y), exposed patients hospitalized inhospital E(y), infected patients I(y), and recovered patients R(y), i.e., SEIRmodel. The computing numerical outcomes and performances of the systemare examined by using the artificial neural networks (ANNs) and the scaledconjugate gradient (SCG) for the training of the networks, i.e., ANNs-SCG.The correctness of the ANNs-SCG scheme is observed by comparing theproposed and reference solutions for three cases of the SEIR model to solvethe nonlinear system based on the Zika virus spreading dynamics throughthe knacks of ANNs-SCG procedure based on exhaustive experimentations.The outcomes of the ANNs-SCG algorithm are found consistently in goodagreement with standard numerical solutions with negligible errors. Moreover,the procedure’s constancy, dependability, and exactness are perceived by usingthe values of state transitions, error histogram measures, correlation, andregression analysis.

Fractional Order Environmental and Economic Model Investigations Using Artificial Neural Network

The motive of these investigations is to provide the importance and significance of the fractional order(FO)derivatives in the nonlinear environmental and economic(NEE)model,i.e.,FO-NEE model.The dynamics of the NEE model achieves more precise by using the form of the FO derivative.The investigations through the non-integer and nonlinear mathematical form to define the FO-NEE model are also provided in this study.The composition of the FO-NEEmodel is classified into three classes,execution cost of control,system competence of industrial elements and a new diagnostics technical exclusion cost.The mathematical FO-NEE system is numerically studied by using the artificial neural networks(ANNs)along with the Levenberg-Marquardt backpropagation method(ANNs-LMBM).Three different cases using the FO derivative have been examined to present the numerical performances of the FO-NEE model.The data is selected to solve the mathematical FO-NEE system is executed as 70%for training and 15%for both testing and certification.The exactness of the proposed ANNs-LMBM is observed through the comparison of the obtained and the Adams-Bashforth-Moulton database results.To ratify the aptitude,validity,constancy,exactness,and competence of the ANNs-LMBM,the numerical replications using the state transitions,regression,correlation,error histograms and mean square error are also described.

A Novel Stochastic Framework for the MHD Generator in Ocean

This work aims to study the nonlinear ordinary differential equations(ODEs)system of magnetohydrodynamic(MHD)past over an inclined plate using Levenberg-Marquardt backpropagation neural networks(LMBNNs).The stochastic procedures LMBNNs are provided with three categories of sample statistics,testing,training,and verification.The nonlinear MHD system past over an inclined plate is divided into three profiles,dimensionless momentum,species(salinity),and energy(heat)conservations.The data is applied 15%,10%,and 75%for validation,testing,and training to solve the nonlinear system of MHD past over an inclined plate.A reference data set is designed to compare the obtained and proposed solutions for the MHD system.The plots of the absolute error(AE)are provided to check the accuracy and precision of the considered nonlinear system of MHD.The obtained numerical solutions of the nonlinear magnetohydrodynamic system have been considered to reduce the mean square error(MSE).For the capability,dependability,and aptitude of the stochastic LMBNNs procedure,the numerical performances are provided to authenticate the relative arrangements of MSE,error histograms(EHs),state transitions(STs),correlation,and regression.

Novel Approach for a van der Pol Oscillator in the Continuous Time Domain

We investigate the continuous time domain numerical treatment of a van der Pol oscillator,applying the trial solution as an artificial feed-forward neural network model containing unknown adjustable parameters.The optimization of the network is performed by simulated annealing in an unsupervised method.The proposed scheme is tested successfully by its application in both non-stiff and stiff conditions.Its reliability and effectiveness is validated through comprehensive statistical analyses.The obtained results are in good agreement with the classical RK45 method.

Stochastic Computational Approach for Complex Nonlinear Ordinary Differential Equations

We present an evolutionary computational approach for the solution of nonlinear ordinary differential equations(NLODEs).The mathematical modeling is performed by a feed-forward artificial neural network that defines an unsupervised error.The training of these networks is achieved by a hybrid intelligent algorithm,a combination of global search with genetic algorithm and local search by pattern search technique.The applicability of this approach ranges from single order NLODEs,to systems of coupled differential equations.We illustrate the method by solving a variety of model problems and present comparisons with solutions obtained by exact methods and classical numerical methods.The solution is provided on a continuous finite time interval unlike the other numerical techniques with comparable accuracy.With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.

Numerical Solutions of a Novel Designed Prevention Class in the HIV Nonlinear Model

The presented research aims to design a new prevention class(P)in the HIV nonlinear system,i.e.,the HIPV model.Then numerical treatment of the newly formulated HIPV model is portrayed handled by using the strength of stochastic procedure based numerical computing schemes exploiting the artificial neural networks(ANNs)modeling legacy together with the optimization competence of the hybrid of global and local search schemes via genetic algorithms(GAs)and active-set approach(ASA),i.e.,GA-ASA.The optimization performances through GA-ASA are accessed by presenting an error-based fitness function designed for all the classes of the HIPV model and its corresponding initial conditions represented with nonlinear systems of ODEs.To check the exactness of the proposed stochastic scheme,the comparison of the obtained results and Adams numerical results is performed.For the convergence measures,the learning curves are presented based on the different contact rate values.Moreover,the statistical performances through different operators indicate the stability and reliability of the proposed stochastic scheme to solve the novel designed HIPV model.

Numerical Computational Heuristic Through Morlet Wavelet Neural Network for Solving the Dynamics of Nonlinear SITR COVID-19

The present investigations are associated with designing Morlet wavelet neural network(MWNN)for solving a class of susceptible,infected,treatment and recovered(SITR)fractal systems of COVID-19 propagation and control.The structure of an error function is accessible using the SITR differential form and its initial conditions.The optimization is performed using the MWNN together with the global as well as local search heuristics of genetic algorithm(GA)and active-set algorithm(ASA),i.e.,MWNN-GA-ASA.The detail of each class of the SITR nonlinear COVID-19 system is also discussed.The obtained outcomes of the SITR system are compared with the Runge-Kutta results to check the perfection of the designed method.The statistical analysis is performed using different measures for 30 independent runs as well as 15 variables to authenticate the consistency of the proposed method.The plots of the absolute error,convergence analysis,histogram,performancemeasures,and boxplots are also provided to find the exactness,dependability and stability of the MWNN-GA-ASA.

Swarming Computational Techniques for the Influenza Disease System

The current study relates to designing a swarming computational paradigm to solve the influenza disease system(IDS).The nonlinear system’s mathematical form depends upon four classes:susceptible individuals,infected people,recovered individuals and cross-immune people.The solutions of the IDS are provided by using the artificial neural networks(ANNs)together with the swarming computational paradigm-based particle swarmoptimization(PSO)and interior-point scheme(IPA)that are the global and local search approaches.The ANNs-PSO-IPA has never been applied to solve the IDS.Instead a merit function in the sense of mean square error is constructed using the differential form of each class of the IDS and then optimized by the PSOIPA.The correctness and accuracy of the scheme are observed to perform the comparative analysis of the obtained IDS results with the Adams solutions(reference solutions).An absolute error in suitable measures shows the precision of the proposed ANNs procedures and the optimization efficiency of the PSOIPA.Furthermore,the reliability and competence of the proposed computing method are enhanced through the statistical performances.

An Advanced Stochastic Numerical Approach for Host-Vector-Predator Nonlinear Model

A novel design of the computational intelligent framework is presented to solve a class of host-vector-predator nonlinear model governed with set of ordinary differential equations.The host-vector-predator nonlinear model depends upon five groups or classes,host plant susceptible and infected populations,vectors population of susceptible and infected individuals and the predator population.An unsupervised artificial neural network is designed using the computational framework of local and global search competencies of interior-point algorithm and genetic algorithms.For solving the hostvector-predator nonlinear model,a merit function is constructed using the differential model and its associated boundary conditions.The optimization of this merit function is performed using the computational strength of designed integrated heuristics based on interior point method and genetic algorithms.For the comparison,the obtained numerical solutions of networks models optimized with efficacy of global search of genetic algorithm and local search with interior point method have been compared with the Adams numerical solver based results or outcomes.Moreover,the statistical analysis will be performed to check the reliability,robustness,viability,correctness and competency of the designed integrated heuristics of unsupervised networks trained with genetic algorithm aid with interior point algorithm for solving the biological based host-vector-predator nonlinear model for sundry scenarios of paramount interest.

Intelligent Networks for Chaotic Fractional-Order Nonlinear Financial Model

The purpose of this paper is to present a numerical approach based on the artificial neural networks(ANNs)for solving a novel fractional chaotic financial model that represents the effect of memory and chaos in the presented system.The method is constructed with the combination of the ANNs along with the Levenberg-Marquardt backpropagation(LMB),named the ANNs-LMB.This technique is tested for solving the novel problem for three cases of the fractional-order values and the obtained results are compared with the reference solution.Fifteen numbers neurons have been used to solve the fractional-order chaotic financial model.The selection of the data to solve the fractional-order chaotic financial model are selected as 75%for training,10%for testing,and 15%for certification.The results indicate that the presented approximate solutions fit exactly with the reference solution and the method is effective and precise.The obtained results are testified to reduce the mean square error(MSE)for solving the fractional model and verified through the various measures including correlation,MSE,regression histogram of the errors,and state transition(ST).

Dynamics of Fractional Differential Model for Schistosomiasis Disease

In the present study,a design of a fractional order mathematical model is presented based on the schistosomiasis disease.To observe more accurate performances of the results,the use of fractional order derivatives in the mathematical model is introduce based on the schistosomiasis disease is executed.The preliminary design of the fractional order mathematical model focused on schistosomiasis disease is classified as follows:uninfected with schistosomiasis,infected with schistosomiasis,recovered from infection,susceptible snail unafflicted with schistosomiasis disease and susceptible snail afflicted with this disease.The solutions to the proposed system of the fractional order mathematical model will be presented using stochastic artificial neural network(ANN)techniques in conjunction with the LevenbergMarquardt backpropagation(LMBP),referred to as ANN-LMBP.To illustrate the preciseness of the ANN-LMBP method,mathematical presentations of three different values focused on fractional order will be performed.These statics performances are taken in these investigations are 78%and 11%for both learning and certification.The accuracy of the ANN-LMBP method is determined by comparing the values obtained by the database Adams-Bash forth-Moulton scheme.The simulation-based error histograms(EHs),MSE,recurrence,and state transitions(STs)will be offered to achieve the capability,accuracy,steadiness,abilities,and finesse of the ANN-LMBP method.

Swarming Computational Efficiency to Solve a Novel Third-Order Delay Differential Emden-Fowler System

The purpose of this research is to construct an integrated neuro swarming scheme using the procedures of the artificial neural networks(ANNs)with the use of global search particle swarm optimization(PSO)along with the competent local search interior-point programming(IPP)called as ANN-PSOIPP.The proposed computational scheme is implemented for the numerical simulations of the third order nonlinear delay differential Emden-Fowler model(TON-DD-EFM).The TON-DD-EFM is based on two types along with the particulars of shape factor,delayed terms,and singular points.A merit function is performed using the optimization of PSOIPP to find the solutions to the TON-DD-EFM.The effectiveness of the ANN-PSOIPP is certified through the comparison with the exact results for solving four examples of the TON-DD-EFM.The scheme’s efficiency is observed by performing the absolute error in suitable measures found around 10−04 to 10−07.Furthermore,the statistical-based assessments for 100 trials are provided to compute the accuracy,stability,and constancy of the ANNPSOIPP for solving the TON-DD-EFM.