In this study, a novel application of neurocomputing technique is presented for solving nonlinear heat transfer and natural convection porous fin problems arising in almost all areas of engineering and technology, especially in mechanical engineering. The mathematical models of the problems are exploited by the intelligent strength of Euler polynomials based Euler neural networks (ENN’s), optimized with a generalized normal distribution optimization (GNDO) algorithm and Interior point algorithm (IPA). In this scheme, ENN’s based differential equation models are constructed in an unsupervised manner, in which the neurons are trained by GNDO as an effective global search technique and IPA, which enhances the local search convergence. Moreover, a temperature distribution of heat transfer and natural convection porous fin are investigated by using an ENN-GNDO-IPA algorithm under the influence of variations in specific heat, thermal conductivity, internal heat generation, and heat transfer rate, respectively. A large number of executions are performed on the proposed technique for different cases to determine the reliability and effectiveness through various performance indicators including Nash–Sutcliffe efficiency (NSE), error in Nash–Sutcliffe efficiency (ENSE), mean absolute error (MAE), and Thiel’s inequality coefficient (TIC). Extensive graphical and statistical analysis shows the dominance of the proposed algorithm with state-of-the-art algorithms and numerical solver RK-4.

In this study, the intelligent computational strength of neural networks (NNs) based on the backpropagated Levenberg-Marquardt (BLM) algorithm is utilized to investigate the numerical solution of nonlinear multiorder fractional differential equations (FDEs). The reference data set for the design of the BLM-NN algorithm for different examples of FDEs are generated by using the exact solutions. To obtain the numerical solutions, multiple operations based on training, validation, and testing on the reference data set are carried out by the design scheme for various orders of FDEs. The approximate solutions by the BLM-NN algorithm are compared with analytical solutions and performance based on mean square error (MSE), error histogram (EH), regression, and curve fitting. This further validates the accuracy, robustness, and efficiency of the proposed algorithm.

In this paper, one dimensional mathematical model of convective-conductive-radiative fins is presented with thermal conductivity depending on temperature. The temperature field with insulated tip is determined for a fin in convective, conductive and radiative environments. Moreover, an intelligent soft computing paradigm named as the LeNN-WOA-NM algorithm is designed to analyze the mathematical model for the temperature field of convective-conductive-radiative fins. The proposed algorithm uses function approximating ability of Legendre polynomials based on artificial neural networks (ANN’s), global search optimization ability of Whale optimization algorithm (WOA), and local search convergence of Nelder-Mead algorithm. The proposed algorithm is applied to illustrate the effect of variations in coefficients of convection, radiation heat losses, and dimensionless parameter of thermal conductivity on temperature distribution of conductive-convective and radiative fins in convective and radiative environments. The experimental data establishes the effectiveness of the design scheme when compared with techniques in the latest literature. It can be observed that accuracy of approximate temperature increases with lower values of $N_{c}$ and $N_{r}$ while decreases with increase in $\lambda $ . The quality of solutions obtained by LeNN-WOA-NM algorithm are validated through performance indicators including absolute errors, MAD, TIC, and ENSE.