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The fractional neuro-evolution-based intelligent computing has substantial potential to solve fractional order systems represented with Lane–Emden equation arising in astrophysics including Newtonian self-gravitating, spherically symmetric and polytropic fluid. The present study aimed to present a neuro-swarm-based intelligent computing solver for the solution of nonlinear fractional Lane–Emden system (NFLES) using by exploitation of fractional Meyer wavelet artificial neural networks (FMW-ANNs) and global optimization mechanism of particle swarm optimization (PSO) combined with rapid local search of sequential quadratic programming (SQP), i.e., FMW-ANN-PSO-SQP. The motivation for the design of FMW-ANN-PSO-SQP intelligent computing comes with an objective of presenting an accurate, reliable, and viable framworks to deal with stiff nonlinear singular models represented with NFLES involving both fractional and integer derivative terms. The designed algorithm is tested for six different variants of NFLESs. The obtained numerical outcomes obtained by the proposed FMW-ANN-PSO-SQP are compared with the exact results to authenticate the correctness, efficacy, and viability, and these aspects are further endorsed statistical observations.

The present study is related to design a novel multi-fractional multi-singular Lane–Emden model (MFMS-LEM) by keeping the ideas of the literature LEM and by extension of the work of doubly singular multi-fractional LEM. This mathematical novel MFMS-LEM is numerically treated by applying the fractional Meyer neuro-evolution intelligent solver (FMNEICS). The optimization is performed using the mutual heuristics of fractional Mayer wavelet neural networks (FMW-NN), the global search aptitude of genetic algorithms (GAs) and interior-point algorithm (IPA), i.e., FMW-NN-GAIPA. The derivation steps, details of the singular points, fractional terms, shape factors and singular points are also provided. The modeling strength of MW-NN is implemented to characterize the novel model in the sagacity of mean squared error of objective function and network optimization is performed with the integrated capability of GAIPA. The authentication, perfection and verification of FMNEICS is checked for three diverse cases of the novel model which are conventional via relative studies through the reference solutions based on accuracy, stability, robustness and convergence procedures. Furthermore, the explanations via the statistical measures validate the value of the designed stochastic solver FMW-NN-GAIPA.

Optimization of reactive power dispatch (ORPD) problem is a key factor for stable and secure operation of the electric power systems. In this paper, a newly explored nature-inspired optimization through artificial ecosystem optimization (AEO) algorithm is proposed to cope with ORPD problem in large-scale and practical power systems. ORPD is a well-known highly complex combinatorial optimization task with nonlinear characteristics, and its complexity increases as a number of decision variables increase, which makes it hard to be solved using conventional optimization techniques. However, it can be efficiently resolved by using nature-inspired optimization algorithms. AEO algorithm is a recently invented optimizer inspired by the energy flocking behavior in a natural ecosystem including non-living elements such as sunlight, water, and air. The main merit of this optimizer is its high flexibility that leads to achieve accurate balance between exploration and exploitation abilities. Another attractive property of AEO is that it does not have specific control parameters to be adjusted. In this work, three-objective version of ORPD problem is considered involving active power losses minimization and voltage deviation and voltage stability index. The proposed optimizer was examined on medium- and large-scale IEEE test systems, including 30 bus, 118 bus, 300 bus and Algerian electricity grid DZA 114 bus (220/60 kV). The results of AEO algorithm are compared with well-known existing optimization techniques. Also, the results of comparison show that the proposed algorithm performs better than other algorithms for all examined power systems. Consequently, we confirm the effectiveness of the introducing AEO algorithm to relieve the over losses problem, enhance power system performance, and meet solutions feasibility. One-way analysis of variance (ANOVA) has been employed to evaluate the performance and consistency of the proposed AEO algorithm in solving ORPD problem.

In this study, bio-inspired computational techniques have been exploited to get the numerical solution of a nonlinear two-point boundary value problem arising in the modelling of the corneal shape. The computational process of modelling and optimization makes enormously straightforward to obtain accurate approximate solutions of the corneal shape models through artificial neural networks, pattern search (PS), genetic algorithms (GAs), simulated annealing (SA), active-set technique (AST), interior-point technique, sequential quadratic programming and their hybrid forms based on GA–AST, PS–AST and SA–AST. Numerical results show that the designed solvers provide a reasonable precision and efficiency with minimal computational cost. The efficacy of the proposed computing strategies is also investigated through a descriptive statistical analysis by means of histogram illustrations, probability plots and one-way analysis of variance.

In this study, an advance computational intelligence scheme is designed and implemented to solve third-order nonlinear multiple singular systems represented with Emden–Fowler differential equation (EFDE) by exploiting the efficacy of artificial neural networks (ANNs), genetic algorithms (GAs) and active-set algorithm (ASA), i.e., ANN–GA–ASA. In the scheme, ANNs are used to discretize the EFDE for formulation of mean squared error-based fitness function. The optimization task for ANN models of nonlinear multi-singular system is performed by integrated competency GA and ASA. The efficiency of the designed ANN–GA–ASA is examined by solving five different variants of the singular model to check the effectiveness, reliability and significance. The statistical investigations are also performed to authenticate the precision, accuracy and convergence.

In this paper, a novel meta-heuristic computing solver is presented for solving the singular three-point second-order boundary value problems using artificial neural networks (ANNs) optimized by the combined strength of global and local search ability of genetic algorithms (GAs) and interior point algorithm (IPA), i.e., ANN–GA–IPA. The inspiration for presenting this numerical work comes from the intention of introducing a consistent framework that combines the effective features of neural networks optimized with the contexts of soft computing to handle with such challenging systems. Three numerical variants of singular second-order system have been taken to examine the proficiency, robustness, and stability of the designed approach. The comparison of the proposed results of ANN–GA–IPA from available exact solutions shows the good agreement with 5 to 7 decimal places of the accuracy which established worth of the methodology through performance analyses on a single and multiple executions.

This study presents a novel implementation of evolutionary heuristics through backtracking search optimization algorithm (BSA) for accurate, efficient and robust parameter estimation of power signal models. The mathematical formulation of fitness function is accomplished by exploiting the approximation theory in mean squared errors between actual and estimated responses, as well as, true and approximated decision variables. Variants of BSA-based meta-heuristics are applied for parameter estimation problem of power signals for identification of amplitude, frequency and phase parameters for different scenarios of noise variation. Analysis of performance evaluation for BSAs is conducted through exhaustive statistical observations in terms of mean weight deviation, root mean square error and Thiel inequality coefficient-based assessment metrics, as well as, ANOVA tests for statistical significance.

In this study, a novel application of differential evolution (DE)-based computational heuristics is proposed for the identification of Hammerstein structures representing the electrically stimulated muscle (ESM) models as a part of rehabilitation interventions for the stock patient to prevent the post-spinal cord injury atrophy. The strength of approximation theory is incorporated for defining the fitness function for ESM system based on mean square deviation between actual and estimated responses. DE, genetic algorithms (GAs), particle swarm optimization (PSO), pattern search (PS), and simulated annealing (SA) are used as optimization mechanisms to identify the ESM models with input nonlinearities of sigmoidal, polynomial, and spline kernels for noiseless and noisy environments. Comparative studies based on detailed statistics establish the worth of DE-based heuristics over its counterparts GAs, PSO, PS, and SA in terms of accuracy, convergence, robustness, and efficiency for the identification of ESM models arising in rehabilitation of the stock patients.

Optimal reactive power dispatch (RPD) for reducing the real power losses of the transmission system is one of the paramount concerns for the research community to investigate the efficiency of power systems. In this paper, strength of meta-heuristic computing paradigm based on fractional-order Darwinian particle swarm optimization (FO-DPSO) is exploited for optimization of RPD problems in energy sector. The fitness functions including line loss minimization and voltage deviation (voltage profile index) are constructed to find the optimal reactive power flow for IEEE 30- and 57-bus test systems. The rich heritage of fractional evolutionary computing through variants of FO-DPSO is applied to minimization problem of optimal power flow by determination of control variables in terms of VAR compensators, bus voltages and transformer tap settings. Comparison of the results shows that fractional swarming intelligence outperformed the state-of-the-art counterparts by means of both line loss minimization and voltage deviation. Superiority of the proposed scheme is also validated for different degrees of freedom in the optimal RPD problems.

In this paper, a novel application of biologically inspired computing paradigm is presented for solving initial value problem (IVP) of electric circuits based on nonlinear RL model by exploiting the competency of accurate modeling with feed forward artificial neural network (FF-ANN), global search efficacy of genetic algorithms (GA) and rapid local search with sequential quadratic programming (SQP). The fitness function for IVP of associated nonlinear RL circuit is developed by exploiting the approximation theory in mean squared error sense using an approximate FF-ANN model. Training of the networks is conducted by integrated computational heuristic based on GA-aided with SQP, i.e., GA-SQP. The designed methodology is evaluated to variants of nonlinear RL systems based on both AC and DC excitations for number of scenarios with different voltages, resistances and inductance parameters. The comparative studies of the proposed results with Adam’s numerical solutions in terms of various performance measures verify the accuracy of the scheme. Results of statistics based on Monte-Carlo simulations validate the accuracy, convergence, stability and robustness of the designed scheme for solving problem in nonlinear circuit theory.

Fast and effective recommender systems are fundamental to fulfill the growing requirements of the e-commerce industry. The strength of matrix factorization procedure based on stochastic gradient descent (SGD) algorithm is exploited widely to solve the recommender system problem. Modern computing paradigms are designed by utilizing the concept of fractional gradient in standard SGD and outperform the standard counterpart. The performance of fractional SGD improves considerably by adaptively tuning the learning rate parameter. A nonlinear computing paradigm based on normalized version of fractional SGD is developed in this paper to investigate the adaptive behavior of learning rate with novel application to recommender systems. The accuracy of the proposed approach is verified through root mean square error metric by using different latent features, learning rates, fractional orders and datasets. The superiority of the designed method is validated through comparison with the state-of-the-art counterparts.

In this study, adaptive filtering paradigm-based kernel least mean square (KLMS) algorithm is developed for feed-forwarded active noise control (ANC) systems by exploiting the strength of activation functions of neural network (NN) as kernels. The transfer functions NN based on logistic, tan-sigmoid and inverse-tan kernels are introduced as a variant of KLMS, normalized KLMS and affine projection KLMS algorithms. All three proposed adaptive filtering strategies are implemented for optimization of design parameters of ANC system of a headset with nonlinear noise interference under several scenarios based on tonal, narrowband, broadband and varying acoustic path. Comparison studies on the basis of detailed numerical experimentation are conducted to establish the worth of the proposed methodologies.

Fractional calculus plays a fundamental role in understanding the physics of nonlinear systems due to its heritage of uncertainty, nonlocality and complexity. In this study, novel sign fractional least mean square (F-LMS) algorithms are designed for ease in hardware implementation by applying sign function to input data and estimation error corresponding to first and fractional-order derivative terms in weight update mechanism of the standard F-LMS method. Theoretical expressions are derived for proposed sign F-LMS and its variants; strength of methods for different fractional orders is evaluated numerically through computer simulations for parameter estimation problem based on nonlinear Hammerstein system for low and high signal–noise variations. Comparison of the results from true parameters of the model illustrates the worth of the scheme in terms of accuracy, convergence and robustness. The stability and viability of design methodologies are examined through statistical observations on sufficiently large number of independent runs through mean square deviation and Nash–Sutcliffe efficiency performance indices.

In the present study, a novel application of nature-inspired heuristics is presented for problems in nonlinear circuit analysis using neural networks, particle swarm optimization (PSO), and interior-point algorithm (IPA) as well as integrated approach PSO–IPA. The governing system models of resistor–capacitor circuits with nonlinear capacitance as well as resistor–inductor circuits with nonlinear inductance are mathematically modeled through competency of neural networks and weights of these networks are trained for global search with PSO hybrid with IPA for speedy refinements. The designed technique is applied on a number of scenarios by taking different values of resistance, current, voltage inductance, and capacitance parameters in nonlinear electrical circuit models. Comparative study with Adams numerical solvers having matching of the order 10−04–10−07 and consistently attaining near-optimal gauges of performance indices based on root-mean-squared error, Theil’s inequality coefficient, and Nash–Sutcliffe efficiency metrics validate and verify the efficacy of the scheme.

This article presents a methodology to solve a one-dimensional steady-state nonlinear reactive transport model (RTM) that is meant for fluid and solute transport model of soft tissues and microvessels. The methodology integrates the artificial neural network (ANN), genetic algorithms (GAs), and pattern search (PS) aided by active-set technique (AST) and interior-point technique (IPT). The RTM is represented with nonlinear second-order system based on the boundary value problem of ordinary differential equation. The ANN modeling is used for governing expression of RTM to form a fitness function in mean square sense, and optimization solvers based on the GA, PS, GA-AST, GA-IPT, PS-AST, PS-IPT are used for viable learning of weights. Proposed techniques are applied to different nonlinear RTMs based on variation in the characteristic reaction rate and half-saturation concentration. The proposed stochastic numerical solutions are compared with state-of-the-art solvers in order to check the accuracy and convergence based on sufficient large multiple runs of the algorithms.

In this work, novel application of integrated computational heuristics is presented for computational fluid mechanics problem arising in the study of heat transfer and thermal radiation in two-phase magnetohydrodynamic (MHD) fluid flow model involving nanoparticles using the accurate approximation ability of neural networks hybrid with global exploration of genetic algorithm aided with local search exploitation of sequential quadratic programming. The networks are designed and arbitrarily combined to formulate mean squared error-based objective function for solving and governing nonlinear nanofluidic system. The designed methodology is evaluated to study the dynamics of the system by means of velocities, temperature and concentration profiles for prevailing factors based on variation in Reynolds and Schmidt numbers, as well as, rotation, radiation, magnetic, thermophoretic and Brownian parameters. The pragmatic worth of the scheme is established through statistical inferences in terms of accuracy, convergence and complexity metrics.

The strength of evolutionary computational heuristic paradigms is exploited for parameter estimation of power signal modeling problems by incorporating differential evolution (DE), genetic algorithms (GAs) and pattern search (PS) methodologies. The objective function of power signal harmonics is constructed by utilizing the power of approximation theory in mean squared error sense. The stiff optimization task of signal harmonics is performed with heuristic solvers DE, GAs and PS that provide efficacy, fast convergence rate and avoid getting trapped in local minima. Statistics reveal that DE outperforms its counterparts in terms of accuracy, robustness and complexity measures.

Aim of this research is to explore the strength of evolutionary and swarm intelligence techniques for parameter identification of control autoregressive moving average (CARMA) systems. The fitness function for CARMA system identification problem is formulated through error function created in mean square sense, and learning of unknown parameters of the system model is carried out with an effective global search techniques based on genetic algorithms and particle swarm optimization algorithm. Comparative study of the design methodology is conducted from actual parameters of the systems for different values of noise variance and degree of freedom in CARMA identification model. The correctness of the proposed scheme is validated through the results of various performance measures based on mean absolute error, mean weight deviation, variance account for and Theil’s inequality coefficient, and their global variants for sufficiently large number of independent runs.

In the present study, strength of fractional-order adaptive signal processing through fractional Volterra least mean square (FV-LMS) algorithm is exploited for Hammerstein nonlinear control autoregressive model (HN-CAR) identification. The FV-LMS method is a generalization of standard V-LMS by taking usual gradient as well as fractional derivative of cost function in the optimization process. The adaptive scheme FV-LMS is applied to HN-CAR systems for different variations of step size parameter, noise and fractional order. Comparative study of the optimized design variables by FV-LMS from true values of HN-CAR model is carried out using performance metrics of fitness and mean square error, to establish its effectiveness. The performance of the proposed scheme is validated through comparison with standard V-LMS based on multiple independent runs of the scheme.

Parameter estimation of plane waves emitted by sources lying in Fraunhofer zone is one of the active areas of research for last few decades. In this study, Fractional Order Darwinian Particle Swarm Optimization (FO-DPSO) algorithm is designed for direction of arrival and amplitude estimation of plane waves impinging on uniform linear array representing the scenario of monostatic Multiple Input and Multiple Output (MIMO) radar. Approximation theory in the mean square error sense is exploited to develop a fitness function of the problem. Design parameters of the system model are optimized by utilizing the strength of the FO-DPSO algorithm in case of various numbers of non-coherent sources, and analysis is performed in terms of fitness, mean square error, Nash–Sutcliffe efficiency, and computational complexity operators. Worth of the proposed FO-DPSO based optimization mechanism is established by consistently achieving the near to optimal values of performance metrics in all three scenarios for monostatic MIMO radar systems.

In this study, nature-inspired computational intelligence is exploited for active noise control (ANC) systems using variants of particle swarm optimization (PSO) algorithm and its memetic combination with efficient local search technique based on active-set (AS), interior-point (IP), Nelder–Mead (NM) and sequential quadratic programming (SQP) algorithms. In ANC, filtered extended least mean square algorithm is normally used for finding the optimal parameters of the linear finite-impulse response filter, which is more likely to trap in local minima (LM). The issue of LM problem is effectively handled with competence of nature-inspired heuristics by developing four variants of memetic computing approaches based on PSO-NM, PSO-AS, PSO-IP, and PSO-SQP in order to adapt the design variables of ANC with linear and nonlinear primary and secondary paths by taking input noise interferences of pure sinusoidal, random and complex random types. The comparative studies of proposed schemes through statistical performance indices have established the worth of the schemes in terms of accuracy, convergence and complexity parameters.

In this study, new intelligent computing methodologies have been developed for highly nonlinear singular Flierl–Petviashivili (FP) problem having boundary condition at infinity by exploiting three different neural network models integrated with active-set algorithm (ASA). A modification in the modeling is introduced to cater the singularity, avoid divergence in results for unbounded inputs and capable of dealing with strong nonlinearity. Three models have been constructed in an unsupervised manner for solving the FP equation using log-sigmoid, radial basis and tan-sigmoid transfer functions in the hidden layers of the network. The training of adaptive adjustable variables of each model is carried out with a constrained optimization technique based on ASA. The proposed models have been evaluated on three variants of the two FP equations. The designed models have been examined with respect to precision, stability and complexity through statistics.

In this paper, a bio-inspired computational intelligence technique is presented for solving nonlinear doubly singular system using artificial neural networks (ANNs), genetic algorithms (GAs), sequential quadratic programming (SQP) and their hybrid GA–SQP. The power of ANN models is utilized to develop a fitness function for a doubly singular nonlinear system based on approximation theory in the mean square sense. Global search for the parameters of networks is performed with the competency of GAs and later on fine-tuning is conducted through efficient local search by SQP algorithm. The design methodology is evaluated on number of variants for two point doubly singular systems. Comparative studies with standard results validate the correctness of proposed schemes. The consistent correctness of the proposed technique is proven through statistics using different performance indices.

In the present study, bio-inspired computational heuristics are exploited for finding the solution of economic load dispatch (ELD) problem with valve point loading effect using variants of genetic algorithm (GA) hybrid with sequential quadratic programming (SQP) and interior-point algorithms (IPAs). Variants of GAs are constructed using different sets of routines for its fundamental operators in order to explore the entire search space for global optimum solutions while SQP and IPA are integrated with GAs for rapid local convergence. Nine variants of each design scheme based on GAs, GA-SQP and GA-IPAs are applied on three different ELD problems of thermal power plant systems. Comparative studies of the proposed schemes are performed through the results of statistical performance indices in order to establish the worth and effectiveness in terms of accuracy, convergence and complexity measures.

The present study investigate the numerical solution of nonlinear singular system represented with sixth Painleve equation by the strength of artificial intelligence using feed-forward artificial neural networks (ANNs) optimized with genetic algorithms (GAs), interior point technique (IPT), sequential quadratic programming (SQP), and their hybrids. The ANN provided a compatible method for finding nature-inspired mathematical model based on unsupervised error for sixth Painleve equation and adaptation of weights of these networks is carried out globally by the competency of GA aided with IPT or SQP algorithms. Moreover, a hybrid approach has been adopted for better proposed numerical results. An extensive statistical analysis has been performed through several independent runs of algorithms to validate the accuracy, convergence, and exactness of the proposed scheme.

In this article, strength of fractional neural networks (FrNNs) is exploited to find the approximate solutions of nonlinear systems based on Riccati equations of arbitrary order. The feed-forward artificial FrNN are used to develop the energy function of the system by defining an error function in mean square sense. Design parameters for optimization of the energy function are adapted using viable local search with interior point methods (IPMs). The performance of design methodology in terms of accuracy and convergence is analyzed for two different variants of the nonlinear system. Comparison of the results with the exact solutions, as well as approximate numerical results, illustrates the correctness of the methodology. The worth of the scheme is established through statistical inferences based on a large number of simulation runs.

This paper presents a novel hybrid technique for the solution of economic load dispatch problem with valve point loading effect using Nelder–Mead (NM) simplex method and pattern search (PS) algorithm. Strength of globalized NM optimization algorithm has been employed to explore the search space for near optimal solution, and PS algorithm is used in combination with a search space reduction strategy, incorporating the principles of selection and stochastic reproduction, to fine-tune the result. The proposed technique has been applied to three different systems having 3, 13 and 40 generating units to demonstrate the application for small to large load dispatch set-up. The efficacy of the design scheme is established from comparison of the results with the state-of-the-art solvers, and it is found that the proposed scheme gives the best result in terms of mean cost while the average computational time is less than most of the reported methods.

In this work, an intelligent computing algorithm is developed for finding the approximate solution of heart model based on nonlinear Van der Pol (VdP)-type second-order ordinary differential equations (ODEs) using feed-forward artificial neural networks (FF-ANNs) optimized with genetic algorithms (GAs) hybrid through interior-point algorithm (IPA). The mathematical modeling of the system is constructed using FF-ANN models by defining an unsupervised error and unknown weights; the networks are tuned globally with GAs, and local refinement of the results is made with IPA. Design scheme is applied to study the VdP heart dynamics model by varying the pulse shape modification factor, damping coefficients and external forcing factor while keeping the fixed value of the ventricular contraction period. The results of the proposed algorithm are compared with reference numerical solutions of Adams method to establish its correctness. Multiple independent runs are performed for the scheme, and results of statistical analyses in terms of mean absolute deviation, root-mean-square error and Nash---Sutcliffe efficiency illustrate its applicability, effectiveness and reliability.

In this study, we solve nonlinear initial value problems arising in circuit analysis by applying bio-inspired computational intelligence technique using feed-forward artificial neural networks (ANNs) optimized with genetic algorithms (GAs), sequential quadratic programming (SQP), and their combined scheme. The system of resister–capacitor (RC) circuit having nonlinear capacitance is mathematically modelled with unsupervised ANNs by defining an energy function in mean-square error (MSE) sense. The objectives are to minimize the MSE for which the parameters of the networks are estimated initially with GA-based global search and in steady state with SQP algorithm for efficient local search. We consider a set of scenarios to evaluate the performance of the proposed scheme for different resistance and capacitance values along with current variations in the nonlinear RC circuit system. The results are compared with well-established fully explicit Runge–Kutta numerical solver in order to verify the accuracy of the applied bio-inspired heuristics. To prove the worth of the scheme, a comprehensive statistical analysis is provided for the performance metrics based on root MSE, mean absolute error, Theil’s inequality coefficient, Nash–Sutcliffe efficiency, variance account for, and the coefficient of determination (R 2).

In the present work, strength of momentum least mean square (MLMS) algorithm is exploited for nonlinear system identification problems represented with Hammerstein model. The MLMS algorithm uses the previous gradient information to estimate the current weights instead of using only current value of gradient; thus, it is faster in the convergence and less probable to trap in local minima. The perfection of the design scheme is certified through effective parameter estimation of nonlinear Hammerstein control autoregressive models. The robustness of the scheme is established by examining the performance for different levels of noise variance. The performance comparison through mean square error and Nash–Sutcliffe efficiency parameters, calculated for sufficiently large number of multiple runs, proves the intrinsic worth of MLMS algorithm in system identification.

In this study, strength of evolutionary computational intelligence based on genetic algorithms (GAs) is exploited for parameter identification of nonlinear Hammerstein controlled autoregressive (NHCAR) systems. The fitness function is constructed for the NHCAR system by defining an error function in the mean square sense. Unknown adjustable weights of the system are optimized with GAs, used as an effective tool for effective global search. Comparative analysis of the proposed scheme is made from true parameters of the systems for a number of scenarios based on different levels of signal-to-noise ratios. The validation of the performance is made through statistics based on sufficiently large number of runs using indices of mean absolute error, variance account for, and Thiel's inequality coefficient as well as their global versions.

In this study, biologically inspired intelligent computing approached based on artificial neural networks (ANN) models optimized with efficient local search methods like sequential quadratic programming (SQP), interior point technique (IPT) and active set technique (AST) is designed to solve the higher order nonlinear boundary value problems arise in studies of induction motor. The mathematical modeling of the problem is formulated in an unsupervised manner with ANNs by using transfer function based on log-sigmoid, and the learning of parameters of ANNs is carried out with SQP, IPT and ASTs. The solutions obtained by proposed methods are compared with the reference state-of-the-art numerical results. Simulation studies show that the proposed methods are useful and effective for solving higher order stiff problem with boundary conditions. The strong motivation of this research work is to find the reliable approximate solution of fifth-order differential equation problems which are validated through strong statistical analysis.

In this study, a computational intelligence technique based on three different designs of artificial neural networks (ANNs) is presented to solve the nonlinear Troesch’s boundary value problem arising in plasma physics. The structure of three ANN models is formulated by incorporating log-sigmoid (ANN-LS), radial-base (ANN-RB) and tan-sigmoid (ANN-TS) kernel functions in the hidden layers. Mathematical modeling of the problem is constructed for all three feed-forward ANN models by defining an error function in an unsupervised manner. Sequential quadratic programming method is employed for the learning of unknown parameters for all three ANN-LS, ANN-RB and ANN-TS schemes. The proposed models are applied to solve variants of Troesch’s problems by taking the small and large values of critical parameter in the system. A comparison of the proposed solution obtained from these models has been made with the standard numerical results of Adams method. The accuracy and convergence of the proposed models are investigated through results of statistical analysis in terms of performance indices based on the mean absolute deviation, root-mean-square error and variance account for.

In the present study, nature-inspired computing technique has been designed for the solution of nonlinear systems by exploiting the strength of particle swarm optimization (PSO) hybrid with Nelder–Mead method (NMM). Fitness function based on least square approximation theory is developed for the systems, while optimization of the design variables is performed with PSO, an efficient global search method, refined with NMM for rapid local convergence. Sixteen variants of the proposed hybrid scheme PSO-NMM have been evaluated on five benchmark nonlinear systems, namely interval arithmetic benchmark model, kinematic application model, neurophysiology problem, combustion model and chemical equilibrium system. Reliability and effectiveness of the proposed solver have been validated after comparison with the results of statistical analysis based on massive data generated for sufficiently large number of independent executions.

In the present study, stochastic numerical computing approach is developed by applying artificial neural networks (ANNs) to compute the solution of Lane–Emden type boundary value problems arising in thermodynamic studies of the spherical gas cloud model. ANNs are used in an unsupervised manner to construct the energy function of the system model. Strength of efficient local optimization procedures based on active-set (AS), interior-point (IP) and sequential quadratic programming (SQP) algorithms is used to optimize the energy functions. The performance of all three design methodologies ANN-AS, ANN-IP and ANN-SQP is evaluated on different nonlinear singular systems. The effectiveness of the proposed schemes in terms of accuracy and convergence is established from the results of statistical indicators.

In the present study, a design of biologically inspired computing framework is presented for solving second-order two-point boundary value problems (BVPs) by differential evolution (DE) algorithm employing finite difference-based cost function. The DE has been implemented to minimize the combined residue from all nodes in a least square sense. The proposed methodology has been evaluated using five numerical examples in linear and nonlinear regime of BVPs in order to demonstrate the process and check the efficacy of the implementation. The assessment and validation of the DE algorithm have been carried out by comparing the DE-computed results with exact solution as well as with the corresponding data obtained using continuous genetic algorithms. These benchmark comparisons clearly establish DE as a competitive solver in this domain in terms of computational competence and precision.

In this paper, meta-heuristic intelligent approaches are developed for handling nonlinear oscillatory problems with stiff and non-stiff conditions. The mathematical modeling of these oscillators is accomplished using feed-forward artificial neural networks (ANNs) in the form of an unsupervised manner. The accuracy as well as efficiency of the model is subject to the tuning of adaptive parameters for ANNs that are highly stochastic in nature. These optimal weights are carried out with swarm intelligence and pattern search methods hybridized with an efficient local search technique based on constraints minimization known as active set algorithm. The proposed schemes are validated on various stiff and non-stiff variants of the oscillator. The significance, applicability and reliability of the proposed scheme are well established based on comparison made with the results of standard numerical solver.

In the present study, a novel intelligent computing approach is developed for solving nonlinear equations using evolutionary computational technique mainly based on variants of genetic algorithms (GA). The mathematical model of the equation is formulated by defining an error function. Optimization of fitness function is carried out with the competency of GA used as a tool for viable global search methodology. Comprehensive numerical experimentation has been performed on number of benchmark nonlinear algebraic and transcendental equations to validate the accuracy, convergence and robustness of the designed scheme. Comparative studies have also been made with available standard solution to establish the correctness of the proposed scheme. Reliability and effectiveness of the design approaches are validated based on results of statistical parameters.

In this paper, stochastic techniques have been developed to solve the 2-dimensional Bratu equations with the help of feed-forward artificial neural networks, optimized with particle swarm optimization (PSO) and sequential quadratic programming (SQP) algorithms. A hybrid of the above two algorithms, referred to as the PSO-SQP method is also studied. The original 2-dimensional equations are solved by first transforming them into equivalent one-dimensional boundary value problems (BVPs). These are then modeled using neural networks. The optimization problem for training the weights of the network has been addressed using particle swarm techniques for global search, integrated with an SQP method for rapid local convergence. The methodology is evaluated by applying on three different test cases of BVPs for the Bratu equations. Monte Carlo simulations and extensive analyses are carried out to validate the accuracy, convergence and effectiveness of the schemes. A comparative study of proposed results is made with available exact solution, as well as, reported numerical results.

In this paper, numerical techniques are developed for solving two-dimensional Bratu equations using different neural network models optimized with the sequential quadratic programming technique. The original two-dimensional problem is transformed into an equivalent singular, nonlinear boundary value problem of ordinary differential equations. Three neural network models are developed for the transformed problem based on unsupervised error using log-sigmoid, radial basis and tan-sigmoid functions. Optimal weights for each model are trained with the help of the sequential quadratic programming algorithm. Three test cases of the equation are solved using the proposed schemes. Statistical analysis based on a large number of independent runs is carried out to validate the models in terms of accuracy, convergence and computational complexity.

In this paper, numerical treatment is presented for the solution of boundary value problems of one-dimensional Bratu-type equations using artificial neural networks. Three types of transfer functions including Log-sigmoid, radial basis, and tan-sigmoid are used in the neural networks’ modeling. The optimum weights for all the three networks are searched with the interior point method. Various test cases of Bratu-type equations have been simulated using the developed models. The accuracy, convergence, and effectiveness of the methods are substantiated by a large number of simulation data for each model by taking enough independent runs.

In this paper, a stochastic technique is developed to solve 2-dimensional Bratu equations using feed-forward artificial neural networks, optimized with genetic and interior-point algorithms. The 2-dimensional equations are first transformed into a 1-dimensional boundary value problem, and a mathematical model of the transformed equation is then formulated with neural networks using an unsupervised error. Network weights are optimized to minimize the error. Evolutionary computing based on genetic algorithms is used as a tool for global search, integrated with an interior-point method for rapid local convergence. The methodology is applied to solve three cases of boundary value problems for the Bratu equations. The accuracy, convergence and effectiveness of the scheme is validated for a large number of simulations. Comparison of results is made with the exact solution derived using MATHEMATICA, and is found to be in good agreement.