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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.

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In this research, the transmission and interaction of nonlinear electron acoustic waves (EAWs) in such an unmagnetized, homogeneous, collision-free plasma composed of hot and cold electrons together with stationary ions throughout in the background have been analyzed. For the small-amplitude limit, the Korteweg–de Vries (KdV) equation for (EAWs) have been extracted. For electron acoustic solitary waves (EASWs), using the new extended direct algebraic approach, soliton solutions have also documented. The parametric analysis demonstrated that the hot to cold electron ratio and hot electron superthermal play a key role in changing the (EASWs) amplitude. The family of semi-bright solitons, dark singular solitons, Type 1 as well as 2 single solitons, trigonometric, intermingled hyperbolic and rational solitons was constructed and tested with the assistance of the innovative package software of numerical computations. The results show that the method is clear and efficient, produces analytical results in a generalized form, and these findings can also help resolve the difficulties and predicaments in the relevant disciplines of plasma physics and may be useful for studying the relationship between two (EASWs) in astrophysical and laboratory plasma. The solutions presented in this prototype are the latest in a literature review. For physical interpretation, some randomly selected solutions are shown graphically. Conclusions are held at the end.

The dynamic behavior of ion-acoustic waves (IAWs) inside a magnetized auroral zone plasma including ions as well as two-temperature electrons is investigated. The Kadomtsev–Petviashvili (KP) and modified KP (mKP) equations are provided using the reductive perturbation method (RPT). In fluid dynamics, plasma physics, and electrodynamics, mKP equation can be utilized to characterize wave phenomena. It increases our interest due to its significant physical and mechanical value. The new extended direct algebraic approach is used to compute several traveling wave patterns for the nonlinear mKP equation. It is a simple, concise and general method in symbolic computations for solitary wave solutions for nonlinear problems. It yields dark solitary solutions, the community for semi-dark soliton results, dark singular soliton effects, kink soliton results, and soliton consequences encompassing trigonometric function, mixture hyperbolic function, as well as rational functions. Some arbitrarily chosen solutions are displayed visually for physical understanding. Moreover, the sensitivity of the perturbed dynamical structure equivalent to the mKP equation has been explored. Furthermore, the effect of IAWs spectral index on the parameters of the solitary-wave solutions and the sensitivity of the perturbed dynamical system of mKP equations is investigated in detail.