In this paper, we introduce a new type of integral transforms, called the ARA integral transform that is defined as: ????????[????(????)](????)=????(????,????)=????∫0∞????????−1????−????????????(????)????????,????>0. We prove some properties of ARA transform and give some examples. Also, some applications of the ARA transform are given.

Fins are widely used in many industrial applications, including heat exchangers. They benefit from a relatively economical design cost, are lightweight, and are quite miniature. Thus, this study investigates the influence of a wavy fin structure subjected to convective effects with internal heat generation. The thermal distribution, considered a steady condition in one dimension, is described by a unique implementation of a physics-informed neural network (PINN) as part of machine-learning intelligent strategies for analyzing heat transfer in a convective wavy fin. This novel research explores the use of PINNs to examine the effect of the nonlinearity of temperature equation and boundary conditions by altering the hyperparameters of the architecture. The non-linear ordinary differential equation (ODE) involved with heat transfer is reduced into a dimensionless form utilizing the non-dimensional variables to simplify the problem. Furthermore, Runge–Kutta Fehlberg’s fourth–fifth order (RKF-45) approach is implemented to evaluate the simplified equations numerically. To predict the wavy fin's heat transfer properties, an advanced neural network model is created without using a traditional data-driven approach, the ability to solve ODEs explicitly by incorporating a mean squared error-based loss function. The obtained results divulge that an increase in the thermal conductivity variable upsurges the thermal distribution. In contrast, a decrease in temperature profile is caused due to the augmentation in the convective-conductive variable values.

In this article a new approach in solving time fractional partial differential equations (TFPDEs) is introduced, that is, the ARA-residual power series method. The main idea of this technique , depends on applying the ARA-transform and using Taylor's expansion to construct approximate series solutions. The procedure of getting the approximate solutions for nonlinear TFPDEs is a difficult mission, the ARA-residual power series method over comes this trouble throughout expressing the solution in a series form then obtain the series coefficients using the idea of the residual function and the concept of the limit at infinity. This method is efficient and applicable to solve a wide family of TFPDEs. Four attractive applications are considered to show the speed and the strength of the proposed method in constructing solitary series solutions of the target equations.

In this article, the numerical adaptive predictor corrector (Apc-ABM) method is presented to solve generalized Caputo fractional initial value problems. The Apc-ABM method was utilized to establish approximate series solutions. The presented technique is considered to be an extension to the original Adams–Bashforth–Moulton approach. Numerical simulations and figures are presented and discussed, in order to show the efficiency of the proposed method. In the future, we anticipate that the provided generalized Caputo fractional derivative and the suggested method will be utilized to create and simulate a wide variety of generalized Caputo-type fractional models. We have included examples to demonstrate the accuracy of the present method.

The present study scrutinizes the significance of heat source/sink (HSS), thermophoretic particle deposition, and porous media on the time-dependent ternary nanofluid stream across a stretchable surface in the presence of Newtonian heating (NH) and common wall temperature (CWT) cases. The governing equations of the investigated model are changed into ordinary differential equations by using suitable similarity transformations. The resultant dimensionless equations are solved using the Laguerre polynomial collocation method. For comparison, the Runge Kutta Fehlberg’s fourth-fifth order (RKF-45) method is employed. Graphs are used to illustrate the significant parameters’ impacts on each profile, and relevant physical quantities such as the Sherwood number, skin friction, and Nusselt number are exhibited. The study reveals that the velocity profile drops with an increase in permeable parameters. The thermal profile increases with improvement in porous and HSS constraints. The concentration diminishes as the value of the thermophoretic parameter rises. For better solid volume fraction values, the rate of temperature dispersal is lower in the NH case associated with the CWT case. Additionally, the rate of thermal distribution is enhanced by approximately 2.90% surface drag force, 4.73% in the CWT case and 2.27% in the NH case, and the rate of mass transfer is enhanced by 2.99% when transitioning from ternary the ternary hybrid nanofluid to the (normal) nanofluid. The results of the study will help in heat exchangers, thermal management, chemical engineering, biomedical instruments, and design and optimization of electronic equipment.