The present study is carried out to examine the behavior of magnetohydrodynamic Williamson nanofluid flow and heat transfer over a non-linear stretching sheet embedded in a porous medium. In the current work, the influence of heat generation and viscous dissipation has been taken into account. The considered phenomenon in the form of partial differential equations is transformed into ordinary differential equations by utilizing an appropriate similarity transformation. The reduced form is solved by using rigorous MATLAB built-in solver bvp4c. The numerical solutions for the velocity field, temperature field, and mass concentration along with the skin friction coefficient, Nusselt number, and Sherwood number are computed. The obtained solutions are shown in graphs and are discussed with physical reasoning. It is noted that by increasing Williamson fluid parameter W, the velocity decreases and concentration profile increases. It is deduced that increasing Eckert number Ec leads to a rise in temperature and mass concentration. It has been viewed that with the increment in heat generation parameter Q, the temperature field increases and concentration decreases. The results show that an increasing magnetic field parameter M leaves a decreasing trend in the velocity field and an increasing trend in the temperature field and concentration profile. The present results are compared with the existing solution which shows good agreement and endorses the validation of current solutions.

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 goal of this paper is to establish a mathematical framework for Casson liquid flow along a perforated capillary of tri-hybrid nanofluid including nonlinear thermal radiation, energy omission/absorption, vicious dispassion and Ohmic heat. The current mathematical approach is designed for blood-based nanofluids considering three distinct nanomaterials such as CuO, TiO₂ and Al₂O₃. The nonlinear thermal radiations were set to modify the objective of this framework in order to examine the changes in temperature. Therefore, streamlining the dominating transport equations pertinent adaptations are used to transform the derived partial differential equations to ordinary differential equations. Analytically the restored mathematical formulations are handled using a successful technique known as the HAM scheme. The findings show that heat transfer increases with the radiation parameter while velocity decreases the magnetic strength. The findings of the present research are useful in determining the impact of several key design factors on heat transmission and therefore in optimizing the manufacturing processes. This mathematical model is comprised of thermal radiation and EMHD situations via blood circulation, which has a substantial impact on magnetized blood delivery, hyperthermia therapy, magnetizer endoscopy, blood stream knowledge, transport of complicated bio-waste liquids and heat exchange in capillaries.

The mathematical analysis of time-independent mobility of a modified blood-based Casson hybrid nanofluid including dissimilar nanomaterials in a convergent/di?vergent channel with stretchable/shrinkable walls is investigated. The cumulative impact of magnetic and electric fields governs the flow of modified hybrid nanofluids. In this study, a mediated hybrid fluid containing three unique nanomaterials (titania oxide, alumina oxide, and silver nanoparticles) is used to evaluate the efficiency of hybrid nanofluids in collaboration with blood as a base fluid. The flow analysis is performed using long-wavelength estimations and creeping processes. Such computational innovation will also be used to investigate the transmission of biofluids from big to smaller arteries and intestines. The homotopy analysis method is used to generate the analytical solutions for a system of non-dimensional boundary value problems. Utilizing MATHEMATICA software, the impacts of model physical parameters on rheological phenomena are visually illustrated. The mathematical model can be used to transmit complex biofluids and control fluid transit by employing electro-kinetic modification technologies. To verify the current findings, a comparable investigation is developed.

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.

Breast cancer ranks among the most prevalent malignancies affecting the female population and is a prominent contributor to cancer-related mortality. Mathematical modeling is a significant tool that can be employed to comprehend the dynamics of breast cancer progression and dissemination and to formulate novel therapeutic approaches. This paper introduces a mathematical model of breast cancer that utilizes the Caputo–Fabrizio fractal-fractional derivative. The aim is to elucidate and comprehend the intricate dynamics governing breast cancer cells and cytotoxic T lymphocytes in the context of the fractional derivative. The derivative presented herein offers a broader perspective than the conventional derivative, as it incorporates the intricate fractal characteristics inherent in the process of tumor proliferation. The significance of this study lies in its contribution to a novel mathematical model for breast cancer, which incorporates the fractal characteristics of tumor development. The present model possesses the capability to investigate the impacts of diverse treatment strategies on the proliferation of breast cancer, as well as to formulate novel treatment strategies that exhibit enhanced efficacy.