Fractional calculus is considered a useful tool for gaining deeper insights into systems with memory effects or history. Fractional-order modeling of nonlinear systems may increase the stiffness and complexity of the system, but also provides better insights. This study introduces a swarm intelligence-based parameter estimation of the fractional Hammerstein autoregressive exogenous noise (fractional-HARX) model. The Grünwald–Letnikov finite difference formula is used to develop the fractional-HARX model from the standard HARX model. This study presents the design of a swarm intelligence-based electric eel foraging optimization algorithm (EEFOA) for parameter estimation of the fractional-HARX model under multiple noise scenarios for second- and third-order polynomial type nonlinearity. The key-term separation principle is also incorporated in the system model to reduce the occurrence of redundant parameters due to cross-product terms in the information vector. The designed methodology is examined, and the superiority of EEFOA is endorsed in terms of convergence, robustness, stiff parameter estimation, and deviation from the mean point in comparison with state-of-the-art optimization heuristics such as the whale optimization algorithm, the African vulture optimization algorithm, Harris hawk’s optimizer, and the reptile search algorithm. The statistical significance of the EEFOA for the estimation of fractional-HARX models is also established using statistical indices of best, mean, and worst fitness values along with standard deviation for multiple noise scenarios.

The incredible synergy between monoclonal anti- bodies and interferons in cancer chemotherapy signifies a stride forward in our battle against this inexorable disease. Through meticulous mathematical modeling that delineate the interplay between tumor growth and immune response, this helps in the development of immunomodulatory treatments and aids in counteracting the cost of drug discovery while minimizing the resource-intensive experimental iterations. This study develops a precise and reliable application of numerical as well as artificial intelligence-based treatment methodology via predictive super- vised eXegenious networks for calculable understanding of the movement of the immune response to treatment such as timing, dosing and forecasting therapy retorts to a specific dose. The out- comes of this work underscore the potency of these methodologies in clarifying the pivotal determinants essential to the dynamic of tumor-immune interactions, therapeutic efficacy and the for- mulation of rationalized therapeutic interventions. In the pursuit of unraveling the complexities inherent to the interactions within the tumor-immune-chemotherapy model, this study harnesses the predictive power of nonlinear autoregressive exogenous (NARX) networks, synergistically coalesced with stochastic-deterministic differential modeling, to unfold the hidden intricacies that hold significant potential within this intricate process. Reference data for training, testing and validation of the proposed methodology was generated using Adams numerical method by utilizing baseline parameters derived through experimental data. Error analysis was conducted to verify the authenticity and perfor- mance of the designed framework for different scenarios. The framework demonstrates impressive performance and accuracy, achieving a mean square error between $10^{-11}$ and $10^{-8}$ through iterative refinement.

Modelling of high-dimension chaotic, noisy, and non-stationary time series of complex fractal dynamics is a big challenge. In this research work , a novel approach of Leverage Convolution LC ARFIMA– GARCH model is presented for sequential learning of irregular, fractal dynamic patterns of stochastic time series dynamics. Different classical methods including the auto-regressive approach indecently are unable to capture fragile patterns and consequently convert some weak signals into random errors by creating white noise patterns. The Convolution Leverage provides an additional degree of freedom for critical noisy points and asymmetrical distribution dynamics to regulate the frequency of imbalance and skewed observation. The designed transformation in the ARFIMA model preserves the loss of information by shifting the skewed data distribution toward a normal pattern. The pattern of population parameters in the proposed leverage paradigm provides out of box approach to track and extract additional information on population variance patterns in the form of an additional degree of freedom to stabilize imbalance signals. The model can provide reliable performance for long-range dependence, particularly for mean reversion chaotic phase variation at critical non-differentiable points. The performance of the dynamic model is verified on chaotic real data of the Ireland Stock Market (ISEQ). The result statistics confirmed the optimized outcome with the addition of the GARCH heteroscedastic multimodal and radial basis neural network . The novel technique can help to address inherited challenges in the imbalance learning and extreme events of the time series modeling by monitoring the chaotic trajectory of fractal physical phenomena , particularly in, finance healthcare, climate, and intelligent sustainability .

This study designs the Morlet wavelet neural network (MWNN) for the numerical performance of the second-order delay differential perturbed singular model (DD-PSM). These stiff singular models are always challenging for the research community to numerically present their results. The DD-PSM is used as an objective function, and its boundary conditions are assembled and then optimised using the computing hybrid proficiency of the global genetic algorithm (GA) and local active-set approach (ASA). Details of the singularity, shape factor, perturbed and delay terms based on the DD-PSM are also provided. Three problems of the DD-PSM are presented and numerically solved using the MWNN–GA–ASA. The precision of the MWNN–GA–ASA is studied by comparing the proposed solution-based DD-PSM and exact solutions. Moreover, a comparison of the MWNN with the Meyer wavelet neural network is presented. The reliability, convergence, correctness and constancy of the numerical scheme are observed by different statistical performances.

In this work, a swarming computational procedure is presented for the numerical treatment of the dynamical model of the susceptible, exposed, infected, and recovered (SEIR) classes that portrayed the spreading of Zika virus. The artificial neural network procedures (ANNPs) have been applied to solve the SEIR mathematical model for spreading of the Zika virus together with the hybridization efficiency of global swarming and local search schemes. The global particle swarm optimization (PSO) and local search active-set algorithm (ASA) have been proposed to solve the model. An error based objective function is presented for the SEIR differential model and then optimized by the hybrid computing efficiency of PSO-ASA. Five neurons, fifteen variables of each class and ten numbers of trials have been used to solve the SEIR mathematical model for spreading of the Zika virus. The correctness of the proposed computing ANNPs-PSO-ASA is observed by using the comparison of the obtained and reference solutions along with the performances of the absolute error, ranges around 10−06 to 10−08. The reliability of the designed computing ANNPs-PSO-ASA technique is observed by using the statistical operator performances on single/multiple trials for the SEIR system for spreading of the Zika virus dynamics.

The proposed research utilizes a computational approach to attain a numerical solution for the singularly perturbed delay differential equation (SPDDE) problem arising in the neuronal variability model through artificial neural networks (ANNs) with different solvers. The log‐sigmoid function is used to construct the fitness function. The implementation of ANN on SPDDE problems is formulated for different solvers and trained with different weights. The optimization solvers such as the genetic algorithm (GA), sequential quadratic programming (SQP), and pattern search (PS) are hybridized with the active set technique (AST) and the interior‐point technique (IPT) and is used to check the accuracy and rapid convergence of the numerical results of the SPDDE model. The numerical outcomes demonstrate that the system is easy to handle and efficient to solve with boundary conditions. Moreover, we used the mean residual error for one hundred runs for each solver to validate the accuracy of the proposed scheme.

In this investigative study, the electro‐magneto hydrodynamic (EMHD) influence on a nano viscous fluid model is scrutinized by designing an artificial neural network (ANN) paradigm using a neuro‐heuristic approach (NHA) through the combination of GAs (genetic algorithms) and one of the most efficient locally searching solver SQP (sequential quadratic programming), i.e., NHA‐GA‐SQP. The fluid flow for the proposed problem is initially interpreted in the form of PDEs and then utilization of suitable similarity transformation on these PDEs yields in terms of a stiff nonlinear system of ODEs. The numerical results of the suggested fluidic model based on the variation of its physically existing parameters are calculated through the NHA‐GA‐SQP solver to detect the variation in velocity, thermal gradient, and concentration during the fluid flow. A detailed analysis of obtained outcomes through the NHA‐GA‐SQP algorithm and their comparison with the reference results estimated via the Adams method are presented. The calculation of the proposed solver’s accuracy, stability, and consistency through various statistical operators is also involved in the current inspection.

Abstract The aim of the current study is to present the numerical solutions of a nonlinear second-order coupled Emden–Fowler equation by developing a neuro-swarming-based computing intelligent solver. The feedforward artificial neural networks (ANNs) are used for modelling, and optimization is carried out by the local/global search competences of particle swarm optimization (PSO) aided with capability of interior-point method (IPM), i.e., ANNs-PSO-IPM. In ANNs-PSO-IPM, a mean square error-based objective function is designed for nonlinear second-order coupled Emden–Fowler (EF) equations and then optimized using the combination of PSO-IPM. The inspiration to present the ANNs-PSO-IPM comes with a motive to depict a viable, detailed and consistent framework to tackle with such stiff/nonlinear second-order coupled EF system. The ANNs-PSO-IP scheme is verified for different examples of the second-order nonlinear-coupled EF equations. The achieved numerical outcomes for single as well as multiple trials of ANNs-PSO-IPM are incorporated to validate the reliability, viability and accuracy.

In this paper, a method for solving grid mismatch or off-grid target is presented for direction of arrival (DOA) estimation problem using compressive sensing (CS) technique. Location of the sources are at few angles as compare to the entire angle domain, i.e., spatially sparse sources, and their location can be estimated using CS methods with ability of achieving super resolution and estimation with a smaller number of samples. Due to grid mismatch in CS techniques, the source energy is distributed among the adjacent grids. Therefore, a fitness function is introduced which is based on the difference of the source energy among the adjacent grids. This function provides the best discretization value for the grid through iterative grid refinement. The effectiveness of the proposed scheme is verified through extensive simulations for different number of sources.

A differential evolution based methodology is introduced for the solution of elliptic partial differential equations (PDEs) with Dirichlet and/or Neumann boundary conditions. The solutions evolve over bounded domains throughout the interior nodes by minimization of nodal deviations among the population. The elliptic PDEs are replaced by the corresponding system of finite difference approximation, yielding an expression for nodal residues. The global residue is declared as the root-mean-square value of the nodal residues and taken as the cost function. The standard differential evolution is then used for the solution of elliptic PDEs by conversion to a minimization problem of the global residue. A set of benchmark problems consisting of both linear and nonlinear elliptic PDEs has been considered for validation, proving the effectiveness of the proposed algorithm. To demonstrate its robustness, sensitivity analysis has been carried out for various differential evolution operators and parameters. Comparison of the differential evolution based computed nodal values with the corresponding data obtained using the exact analytical expressions shows the accuracy and convergence of the proposed methodology.

The demand for wireless mobile communication is growing exponentially with expectations that in the near future mobile device or user will be available in every corner of the globe. Alternatively, it increases the importance of antenna arrays which are responsible for transmission and reception of information. Every antenna array is projected to generate a desired pattern and, hence, failure of any antenna causes misrepresentation of the overall pattern in terms of increased side lobe levels and displacement of nulls from their original position. The aim of the study is to present viable, simple, and accurate stochastic solver based on backtracking search optimization algorithm (BSA) for the pattern correction of faulty antenna array in mobile communication systems. A fitness function is developed to optimize the weights of the remaining healthy antenna elements in the array. The fitness function consists of two parts: the first part is based on mean square error approach for the reduction of sidelobes level, while, in the second part, steering vectors are used for the repositioning of nulls. Simulation results establish the validity of the BSA from its counterparts based on genetic algorithm and its memetic combination with pattern search technique.

The aim of this study is to develop an autonomous epidemic virus model to depict the transmission of malicious computer code in active networks with pre-existing immunity and quarantine as effective control strategies. Due to the rapid spread of computer viruses and a delay in the update of antivirus signature database, the role of quarantine as a controlling mechanism has gained importance. The existence of disease free equilibrium point and its stability, as well as the existence of endemic equilibrium point and its stability are explored in terms of basic reproduction number R 0 . The model exhibits two equilibria points: disease free equilibrium and endemic equilibrium. Numerical simulations are performed to analyze the dynamics of the model in the presence of controlling mechanisms and in the absence of up-to-date antivirus software in terms of accuracy and convergence. The model interpretation invokes interesting inferences for effective quarantine strategy, with or without immunity and control mechanisms for security holes and zero-day vulnerabilities.

In this study, hybrid computational frameworks are developed for active noise control (ANC) systems using an evolutionary computing technique based on genetic algorithms (GAs) and interior-point method (IPM), following an integrated approach, GA-IPM. Standard ANC systems are usually implemented with the filtered extended least mean square algorithm for optimization of coefficients for the linear finite-impulse response filter, but are likely to become trapped in local minima (LM). This issue is addressed with the proposed GA-IPM computing approach which is considerably less prone to the LM problem. Also, there is no requirement to identify a secondary path for the ANC system used in the scheme. The design method is evaluated using an ANC model of a headset with sinusoidal, random, and complex random noise interferences under several scenarios based on linear and nonlinear primary and secondary paths. The accuracy and convergence of the proposed scheme are validated based on the results of statistical analysis of a large number of independent runs of the algorithm.

In this study, a new non‐linear recursive mechanism for Volterra least mean square (VLMS) algorithm is proposed in the domain of non‐linear adaptive signal processing and control. The proposed adaptive scheme is developed by applying concepts and theories of fractional calculus in weight adaptation structure of standard VLMS approach. The design scheme based on fractional VLMS (F‐VLMS) algorithm is applied to parameter estimation problem of non‐linear Hammerstein Box‐Jenkins system for different noise and step size variations. The adaptive variables of F‐VLMS are compared from actual parameters of the system as well as with the results of conventional VLMS for each case to verify its correctness. Comprehensive statistical analyses are conducted based on sufficient large number of independent runs and performance indices in terms of mean square error, variance account for and Nash–Sutcliffe efficiency establish the worth and effectiveness of the scheme.

We present a neuro-heuristic computing platform for finding the solution for initial value problems (IVPs) of nonlinear pantograph systems based on functional differential equations (P-FDEs) of different orders. In this scheme, the strengths of feed-forward artificial neural networks (ANNs), the evolutionary computing technique mainly based on genetic algorithms (GAs), and the interior-point technique (IPT) are exploited. Two types of mathematical models of the systems are constructed with the help of ANNs by defining an unsupervised error with and without exactly satisfying the initial conditions. The design parameters of ANN models are optimized with a hybrid approach GA–IPT, where GA is used as a tool for effective global search, and IPT is incorporated for rapid local convergence. The proposed scheme is tested on three different types of IVPs of P-FDE with orders 1–3. The correctness of the scheme is established by comparison with the existing exact solutions. The accuracy and convergence of the proposed scheme are further validated through a large number of numerical experiments by taking different numbers of neurons in ANN models.

In this research, the well-known non-linear Lane–Emden–Fowler (LEF) equations are approximated by developing a nature-inspired stochastic computational intelligence algorithm. A trial solution of the model is formulated as an artificial feed-forward neural network model containing unknown adjustable parameters. From the LEF equation and its initial conditions, an energy function is constructed that is used in the algorithm for the optimisation of the networks in an unsupervised way. The proposed scheme is tested successfully by applying it on various test cases of initial value problems of LEF equations. The reliability and effectiveness of the scheme are validated through comprehensive statistical analysis. The obtained numerical results are in a good agreement with their corresponding exact solutions, which confirms the enhancement made by the proposed approach.

In this paper, an efficient procedure based on the neural networks methodology is presented for the solution of the fuel ignition model in one dimension. The neural networks were optimised with the particle swarm optimisation algorithm hybridised with sequential quadratic programming. The accuracy and convergence of the scheme are analysed by Monte Carlo simulations and their statistical analyses for three test cases of the problem represented by Bratu-type equations. It was found that the hybrid approach converges in all cases, and can solve the problem with higher accuracy and reliability than most of the methodologies used so far to solve this problem.

This study presents a novel application of fractional adaptive algorithms for parameter identification of Box–Jenkins (BJ) systems. The idea is to adapt the unknown parameter vector of the BJ system by the fractional least mean square (FLMS) algorithm for three different values of the fractional order and then to compare the estimated results with state of the art Volterra least mean square and kernel least mean square adaptive algorithms to validate and verify the correctness of the design scheme. The reliability and effectiveness of the proposed scheme is analysed through the results of the statistical analysis based on sufficient large number of independent runs and it is found that the proposed FLMS algorithm provides consistently accurate and convergent results for BJ systems under different scenarios.