This work aims to analyze the transfer of heat through new fractional-order convective straight fins by using the Riemann-Liouville type fractional derivatives. The convection through the fins is considered in such a way that the thermal conductivity depends on the temperature. The transformed fractional-order problems are constituted through an optimization problem in such a way that the L2 norm remains minimal. The objective functions are further analyzed with the hybrid Cuckoo search (HCS) algorithm that use the artificial neural network (ANN) mechanism. The impacts of the fractional parameter β, the thermo-geometric parameter of fin ψ, and dimensionless thermal conductivity α are explained through figures and tables. The fin efficiency during the whole process is explained with larger values of ψ. It is found that the larger values of ψ decline the fin efficacy. The fractional parameter declines the thermal profile as we approach the integer order. The convergence of HCS algorithm is performed in each case study. The residual error touches E-14 for the integer order of α. The present results are validated through Table 6 by comparing with HPM, VIM and LHPM, while the error for HCS-ANN touches E-13. This proves that the proposed HCS is efficient.

In this study, a new fractional-order model for human skull heat conduction is tackled by using a neural network, and the results were further modified by using the hybrid cuckoo search algorithm. In order to understand the temperature distribution, we introduced memory effects into our model by using fractional time derivatives. The objective function was constructed in such a way that the $L_2-$error remained at a minimum. The fractional order equation was then calculated by using the proposed biogeography-based hybrid cuckoo search (BHCS) algorithm to approximate the solution. When compared to earlier simulations based on integer-order models, this method enabled us to examine the fractional-order (FO) cases, as well as the integer order. The results are presented in the form of figures and tables for the different case studies. The results obtained for the various parameters were validated numerically against the available literature, where our proposed methodology showed better performance when compared to the least squares method (LSM).

This research analysis explores the mathematical model of the Van der Pol (VdP) heartbeat. The dependability of a mathematical model that shows how the heart beats back and forth has been looked into. This mathematical model is examined using a novel neuro-evolutionary methodology. The properties of the heart’s cardiac pulse are investigated by taking two different problems, each problem has two scenarios, and each scenario has two different cases. For the heartbeat model, artificial neural networks (ANNs) are built to find optimal results. The physics-informed random projection neural networks (PIRPNNs) is used to tune unknown weights. The well-conditioned nature of the model under examination is shown by the results achieved after taking various scenarios into account. Our proposed technique’s accuracy is proven by obtaining the lowest residual errors across all scenarios in our results and by making comparisons with MATLAB solvers. To further illustrate the accuracy of our method, graphical analyses are added.