These investigations are to find the numerical solutions of the nonlinear smoke model to exploit a stochastic framework called gudermannian neural works (GNNs) along with the optimization procedures of global/local search terminologies based genetic algorithm (GA) and interior-point algorithm (IPA), i.e., GNNs-GA-IPA. The nonlinear smoke system depends upon four groups, temporary smokers, potential smokers, permanent smokers and smokers. In order to solve the model, the design of fitness function is presented based on the differential system and the initial conditions of the nonlinear smoke system. To check the correctness of the GNNs-GA-IPA, the obtained results are compared with the Runge-Kutta method. The plots of the weight vectors, absolute error and comparison of the results are provided for each group of the nonlinear smoke model. Furthermore, statistical performances are provided using the single and multiple trial to authenticate the stability and reliability of the GNNs-GA-IPA for solving the nonlinear smoke system.

The aim of this work is to design a stochastic framework to solve the fractional-order differential model based on the breast cancer progression during the immune-chemotherapeutic treatment phase, including certain control parameters such as anti-cancer medications, ketogenic diet and immune boosters. The developed model considers tumor density progression throughout chemotherapy treatment, as well as an immune response during normal cell–tumor cell interaction. This study’s subject seems to be to demonstrate the implications and significance of the fractional-order breast cancer mathematical model. The goal of these studies is to improve accuracy in the breast cancer model by employing fractional derivatives. This study also includes an integer, nonlinear mathematical system with immune-chemotherapeutic treatment impacts. The mathematical system divides the fractional-order breast cancer mathematical model among four manifestations: normal cell population (N), tumor cells (T), immune response class (I), and estrogen compartment (E), i.e., (NTIE). The fractional-order NTIE mathematical system is still not published previously, nor has it ever been addressed employing the stochastic solvers’ strength. To solve a fractional-order NTIE mathematical system, stochastic solvers based on the Levenberg–Marquardt backpropagation scheme (LMBS) and neural networks (NNs), namely, LMBNNs, are been constructed. To solve the fractional-order NTIE mathematical model, three cases with varying values for this same fractional order have been supplied. The statistics used to offer the numerical solutions of the fractional-order NTIE mathematical model are divided as follows: 75% in training, 15% in testing, and 10% in the authorization. The acquired numerical findings were compared using the reference solutions to determine the accuracy of the LMBNNs using Adams–Bashforth–Moulton. The numerical performances employing error histograms (EHs), state transitions (STs), regression, correlation, including mean square error (MSE) have been further supplied to authenticate overall capability, competence, validity, consistency, as well as exactness of such LMBNNs.

The present work provides the numerical solutions of the mathematical model based on the fractional-order Layla and Majnun model (MFLMM). A soft computing stochastic-based Bayesian regularization neural network approach (BRNNA) is provided to investigate the numerical accomplishments of the MFLMM. The nonlinear system is classified into two dynamics, whereas the correctness of the BRNNA is observed through the comparison of results. Furthermore, the reducible performance of the absolute error improves the exactitude of the computational BRNNA. Twenty neurons have been chosen, along with the data statics of training 74% and 13%, for both authorization and testing. The consistency of the designed BRNNA is demonstrated using the correlation/regression, error histograms, and transition of state values in order to solve the MFLMM.

In this study, a design of Morlet wavelet neural networks (MWNNs) is presented to solve the prediction differential model (PDM) by applying the global approximation capability of a genetic algorithm (GA) and local quick interior-point algorithm scheme (IPAS), i.e., MWNN-GAIPAS. The famous and historical PDM is known as a variant of the functional differential system that works as theopposite of the delay differential models. A fitness function is constructed by using the mean square error and optimized through the GA-IPAS for solving the PDM. Three PDM examples have been presented numerically to check the authenticity of the MWNN-GAIPAS. For the perfection of the designed MWNN-GAIPAS, the comparability of the obtained outputs and exact results is performed. Moreover, the neuron analysis is performed by taking 3, 10, and 20 neurons. The statistical observations have been performed to authenticate the reliability of the MWNN-GAIPAS for solving the PDM.