In this paper, we intend to solve special kind of ordinary differential EQUATIONs which is called Heun EQUATIONs, by converting to a corresponding stochastic differential EQUATION (S.D.E.). So, we construct a stochastic linear EQUATION system from this EQUATION which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential EQUATIONs.

In this paper, we , nd an integral representation for the fundamental solution of the fractional Ostrovsky EQUATION in terms of the Airy and Bessel-Wright functions. The EQUATION is studied in the sense of the Weyl fractional derivative and the solu-tion is presented as the Airy transforms of Wright functions. Using the asymptotic expansion of Wright function the asymptotic behavior of solution is also discussed.

This paper presents the introduction of two novel EQUATION types: the partial hesitant fuzzy EQUATION and the half hesitant fuzzy EQUATION‎. Additionally, ‎ an efficient method is proposed to solve these EQUATIONs by defining four solution categories: Controllable‎, ‎Tolerable Solution Set (TSS)‎, Controllable ‎Solution Set (CSS)‎, ‎and Algebraic Solution Set (ASS)‎. ‎ Furthermore, ‎ the paper establishes eight theorems that explore different types of solutions and lay out the conditions for the existence and non-existence of hesitant fuzzy solutions‎. ‎ The practicality of the proposed method is demonstrated through numerical examples.

Fundamental hydrodynamic properties in the fluidized beds are the maximal pressure drop ΔPmax and the minimal fluidization velocity Umf. Accurately forecasting these variables is necessary to maximize the functionality and architecture of this system. This study introduces a fresh technique for multilayer modeling. Two dimensionless numbers were generated by utilizing the ERGUN EQUATION. The initial numerical value, π1, represents the energy dissipation from fluid friction within the fluidized bed. It demonstrates a clear relationship with the fluid velocity in the ERGUN EQUATION, denoted as "U." This characteristic is consistent with the properties of the dimensionless viscous (friction) number. In contrast, the second numerical value π2 represents the dissipation of kinetic energy within the bed and exhibits a direct proportionality to U2, akin to the dimensionless inertial number. Furthermore, the hydraulic diameter was created to differentiate between the conical and cylindrical beds. The ideal configuration for the MultiLayer Perceptron (MLP) model was identified as a two-layer feedforward architecture comprising fourteen hidden neurons. The study determined that the precision of ΔPmax and Umf was 6.11% and 9.77% using the Average Absolute Relative Difference (AARD) metric. Additionally, the Mean Squared Error (MSE) metric yielded values of 0.0146 and 2.9 × 10-4 for ΔPmax and Umf, respectively. Furthermore, the Root Mean Squared Error (RMSE) metric resulted in values of 0.120 and 0.0172 for ΔPmax and Umf, respectively. The R-squared values for the respective variables were 0.935 and 0.989. The correlation coefficients (R values) were also found to be 0.967 and 0.994, respectively. The findings obtained from the simulation were compared to the experimental data, indicating a significant level of agreement and confirming the correctness and dependability of the MLP technique.