In this paper, a numerical method to solve a class of stochastic nonlinear differential equations is introduced. The proposed method is based on the Temimi-Ansari method. The special states of the four systems are studied to show that the proposed method is efficient and applicable. These systems are stochastic Langevin's equation, Ginzburg-Landau equation, Davis-Skodje, and Brusselator systems. The results clarify the accuracy and efficacy of the presented new method with no need for any restrictive assumptions for nonlinear terms.

Open-phase faults (OPFs) represent a major challenge in PMSM drive systems due to the induced current imbalance, torque ripple, and degradation in speed regulation performance. Although fractional-order controllers and model predictive control have been extensively investigated for PMSM drives, their individual application does not directly address the coordinated control requirements arising from post-fault operation. Therefore, this paper develops a coordinated fault-tolerant PMSM drive framework in which open-phase fault detection, neutral-point reconfiguration, and fractional-order PI (FO-PI) speed regulation are integrated within a unified post-fault control architecture. A mathematical PMSM model covering both healthy and open-phase operating conditions is established to capture the altered post-fault dynamics and provide the basis for fault diagnosis and controller design. In contrast to conventional approaches in which fractional-order speed control is primarily designed for healthy operation, the proposed controller parameters are obtained through bounded numerical optimization while explicitly considering the post-fault operating condition. The proposed framework is evaluated in MATLAB/Simulink under multiple operating speeds and compared with the conventional PI controller using the integral absolute error (IAE) as a quantitative performance metric. The optimized FO-PI controller reduces the post-fault IAE by 27.0% at 1000 rpm and approximately 51.0% at 800 rpm compared with the conventional PI controller. For further validation, an FO-PID controller is included as an advanced fractional-order benchmark, allowing the performance of the proposed FO-PI design to be assessed against a controller with additional tuning degrees of freedom under the corresponding operating condition. The results demonstrate that the coordinated diagnosis reconfiguration fractional controller design framework improves post-fault PMSM speed regulation and maintains stable operation following an open-phase fault.

This work presents a unified computational scheme for solving stochastic partial differential equations (SPDEs) governed by fractional dynamics. The proposed numerical technique combines the modified non-polynomial cubic spline technique and the conformable fractional derivative (CFD) for both noise and time within a single effective framework. By redefining fractional white noise through conformable derivatives, the proposed method overcomes the limitations of classical kernel-based formulations that often lead to singularities and high computational cost. The developed Conformable Fractional Noise Stochastic Ten Non-Polynomial Cubic Spline method integrates ten spline variants within a finite difference structure, achieving fourth-order spatial precision while efficiently capturing temporal fractional effects. A series of comparative simulations demonstrates that the proposed approach achieves accuracy comparable to that of more advanced wavelet-based techniques while requiring fewer random realizations approximately 80% fewer samples in practice. Its stability and consistency were further tested over various weights of the Hurst parameter. In addition, the flexibility of the proposed framework allows it to be adapted to a broad class of fractional stochastic models with different noise structures. This highlights its potential as a reliable and efficient computational tool for practical applications involving complex random phenomena. All computations were performed in Mathematica 12 to ensure consistency and reproducibility throughout the numerical results.

In this work, we propose a semi-analytical approach based on the Sumudu–Adomian Decomposition Method (SADM) for nonlinear fractional population models governed by the Caputo–Fabrizio derivative. The proposed approach integrates the Adomian decomposition method with the Sumudu transform. The solution of the nonlinear fractional initial-value problem is represented by a rapidly convergent decomposition series. Existence and uniqueness of the solution are demonstrated under Lipschitz-type conditions. The convergence of the derived decomposition series, an explicit truncation-error estimate, and uniform stability are established. Moreover, model-specific verification of the sufficient contraction condition is provided over appropriate biologically admissible intervals to link the theoretical analysis with the numerical applications. The method is applied to four representative population-growth models: the logistic model, Allee-effect model, Richards model and Gompertz model. Comparison of the SADM approximations with the corresponding numerical reference solutions confirms the reliability and accuracy of the proposed method.