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This study aims to derive a novel spatial numerical method based on multidimensional local Taylor series representations for solving high-order advection-diffusion (AD) equations.

The parabolic AD equations are reduced to the nonhomogeneous elliptic system of partial differential equations by utilizing the Chebyshev spectral collocation method (ChSCM) in the temporal variable. The implicit-explicit local differential transform method (IELDTM) is constructed over two- and three-dimensional meshes using continuity equations of the neighbor representations with either explicit or implicit forms in related directions. The IELDTM yields an overdetermined or underdetermined system of algebraic equations solved in the least square sense.

The IELDTM has proven to have excellent convergence properties by experimentally illustrating both h-refinement and p-refinement outcomes. A distinctive feature of the IELDTM over the existing numerical techniques is optimizing the local spatial degrees of freedom. It has been proven that the IELDTM provides more accurate results with far fewer degrees of freedom than the finite difference, finite element and spectral methods.

This study shows the derivation, applicability and performance of the IELDTM for solving 2D and 3D advection-diffusion equations. It has been demonstrated that the IELDTM can be a competitive numerical method for addressing high-space dimensional-parabolic partial differential equations (PDEs) arising in various fields of science and engineering. The novel ChSCM-IELDTM hybridization has been proven to have distinct advantages, such as continuous utilization of time integration and optimized formulation of spatial approximations. Furthermore, the novel ChSCM-IELDTM hybridization can be adapted to address various other types of PDEs by modifying the theoretical derivation accordingly.

In this paper, the author presents a hybrid method along with its error analysis to solve (1+2)-dimensional non-linear time-space fractional partial differential equations (FPDEs).

The proposed method is a combination of Sumudu transform and a semi-analytc technique Daftardar-Gejji and Jafari method (DGJM).

The author solves various non-trivial examples using the proposed method. Moreover, the author obtained the solutions either in exact form or in a series that converges to a closed-form solution. The proposed method is a very good tool to solve this type of equations.

The present work is original. To the best of the author's knowledge, this work is not done by anyone in the literature.

The numerical methods are of great importance for approximating the solutions of a system of nonlinear singular ordinary differential equations. In this paper, the authors present the biorthogonal flatlet multiwavelet collocation method (BFMCM) as a numerical scheme for a class of system of Lane–Emden equations with initial or boundary or four-point boundary conditions.

The approach is involved in combining the biorthogonal flatlet multiwavelet (BFM) with the collocation method. The authors investigate the properties and procedure of the BFMCM for first time on this class of equations. By using the BFM and the collocation points, the method is constructed and it transforms the nonlinear differential equations problem into a system of nonlinear algebraic equations. The unknown coefficients of the assuming solution are determined by solving the obtained system. Additionally, convergence analysis and numerical stability of the suggested method are provided.

According to the attained results, the proposed BFMCM has more accurate results in comparison with results of other methods. The maximum absolute errors are calculated by using the BFMCM for comparison purposes provided.

The key desirable properties of BFMCM are its efficiency, simple applicability and minimizes errors. Therefore, the proposed method can be used to solve nonlinear problems or problems with singular points.

This paper aims to explore the double diffusive magneto-hydrodynamic (MHD) squeezed flow of (Cu–water) nanofluid between two analogous plates filled with Darcy porous material in existence of chemical reaction and external magnetic field.

The governing nonlinear equations are transformed into ordinary differential equations by means of similarity transforms, and the coupled mass and heat transference equations are resolved analytically with the application of differential transform method (DTM). The effects of different relevant parameters on velocity, temperature and concentration, including the squeeze number, magnetic parameter, Biot number, Darcy number and chemical reaction parameter, are illustrated with figures. In addition, for various parameters, the local skin friction coefficient, local Nusselt number and local Sherwood number are computed and are graphically displayed.

It is observed that the squeeze number has a direct relationship with Sherwood number and an inverse relationship with skin friction as Biot number increases. With enhanced Biot numbers, the temperature value increases during both squeeze and non-squeeze moments, but the temperature values are higher for squeeze moments compared to the other case.

This research has potential applications in various large-scale enterprises that might benefit from increased productivity.

The results are useful to thermal science community.

Unique and valuable insights are provided by studying the impact of chemical reaction on double diffusive MHD squeezing copper–water nanofluid flow between parallel plates filled with porous medium. In addition, this research has potential applications in various large-scale enterprises that might benefit from increased productivity.

The purpose of this paper is to assess the feasibility of analytical models, specifically the radial basis function method, Akbari–Ganji method and Gaussian method, in conjunction with the finite element method. The aim is to examine the impact of processing parameters on temperature history.

Through analytical investigation and finite element simulation, this research examines the influence of processing parameters on temperature history. Simufact software with a thermomechanical approach was used for finite element simulation, while radial basis function, Akbari–Ganji and Gaussian methods were used for analytical modeling to solve the heat transfer differential equation.

The accuracy of both finite element and analytical methods was validated with about 90%. The findings revealed direct relationships between thermal conductivity (from 100 to 200), laser power (from 400 to 800 W), heat source depth (from 0.35 to 0.75) and power absorption coefficient (from 0.4 to 0.8). Increasing the values of these parameters led to higher temperature history. On the other hand, density (from 7,600 to 8,200), emission coefficient (from 0.5 to 0.7) and convective heat transfer (from 35 to 90) exhibited an inverse relationship with temperature history.

The application of analytical modeling, particularly the utilization of the Akbari–Ganji, radial basis functions and Gaussian methods, showcases an innovative approach to studying directed energy deposition. This analytical investigation offers an alternative to relying solely on experimental procedures, potentially saving time and resources in the optimization of DED processes.

The transient loads on the spherical hybrid sliding bearings (SHSBs) rotor system during the process of accelerating to stable speed are related to time, which exhibits a complex transient response of the rotor dynamics. The current study of the shaft center trajectory of the SHSBs rotor system is based on the assumption that the rotational speed is constant, which cannot truly reflect the trajectory of the rotor during operation. The purpose of this paper truly reflects the trajectory of the rotor and further investigates the stability of the rotor system during acceleration of SHSBs.

The model for accelerated rotor dynamics of SHSBs is established. The model is efficiently solved based on the fourth-order Runge–Kutta method and then to obtain the shaft center trajectory of the rotor during acceleration.

Results show that the bearing should choose larger angular acceleration in the acceleration process from startup to the working speed; rotor system is more stable. With the target rotational speed increasing, the changes in the shaft trajectory of the acceleration process are becoming more complex, resulting in more time required for the bearing stability. When considering the stability of the rotor system during acceleration, the rotor equations of motion provide a feasible solution for the simulation of bearing rotor system.

The study can simulate the running stability of the shaft system from startup to the working speed in this process, which provides theoretical guidance for the stability of the rotor system of the SHSBs in the acceleration process.

The purpose of this study is to investigate the impact of varying baffle height and spacing distance on heat transfer and cooling performance of electronic components in a baffled horizontal channel, using a Cu-H₂O nanofluid under mixed convection and laminar flow.

The mathematical model is two-dimensional and comprises a system of four governing equations, such as the conservation of continuity, momentum and energy. To obtain numerical solutions for these equations, the finite volume method was used for discretization. A validation process was performed by comparing this study’s results with those of previously published studies. The comparison revealed a close agreement. The numerical study was performed for a wide range of key parameters: The baffle height (0 ≤ h ≤ 0.7), the spacing distance between baffle and blocks (0.25 ≤ w ≤ 3), the Grashof and Reynolds numbers are kept equal to 10⁴ and 75, respectively, the channel aspect ratio is L/H = 10, and the volume fraction of Cu nanoparticles is fixed at φ = 5%.

The results of the study reveal a significant improvement in heat transfer in terms of total Nusselt number of the top and bottom hot components, which exhibited an improvement of 16.89% and 17.23% when the baffle height increases from h = 0 to h = 0.7. Additionally, the study found that reducing the distance between the baffle and the electronic components up to a certain limit can improve the heat transfer rate. Therefore, the optimal height of the baffle was found to be no lower than 0.6, and the recommended distance between the heaters and the baffle was 0.5.

This study provides valuable insights into the optimization of the design of baffled channels for improved heat transfer performance. The findings of study can be used to improve heat exchangers and cooling systems in various applications. The use of Cu-H₂O nanofluid under mixed convection and laminar flow conditions in channel with baffle and electronic components is also unique, making this study an original contribution to the field.

The purpose of this study is to obtain the optimal frequency for low-frequency transmission lines while minimizing losses and maintaining the voltage stability of low-frequency systems. This study also emphasizes a reduction in calculations based on mathematical approaches.

Telegrapher’s method has been used to reduce large calculations in low-frequency high-voltage alternating current (LF-HVac) lines. The static compensator (STATCOM) has been used to maintain voltage stability. For optimal frequency selection, a modified Jaya algorithm (MJAYA) for optimal load flow analysis was implemented.

The MJAYA algorithm performed better than other conventional algorithms and determined the optimum frequency selection while minimizing losses. Voltage stability was also achieved with the proposed optimal load flow (OLF), and statistical analysis showed that the proposed OLF reduces the frequency deviation and standard error of the LF-HVac lines.

The optimal frequency for LF-HVac lines has been achieved, Telegrapher’s method has been used in OLF, and STATCOM has been used in LF-HVac transmission lines.

The purpose of this study is to solve two unique but difficult partial differential equations: the foam drainage equation and the nonlinear time-fractional fisher’s equation. Through our methods, we aim to provide accurate solutions and gain a deeper understanding of the intricate behaviors exhibited by these systems.

In this study, we use a dual technique that combines the Aboodh residual power series method and the Aboodh transform iteration method, both of which are combined with the Caputo operator.

We develop exact and efficient solutions by merging these unique methodologies. Our results, presented through illustrative figures and data, demonstrate the efficacy and versatility of the Aboodh methods in tackling such complex mathematical models.

Owing to their fractional derivatives and nonlinear behavior, these equations are crucial in modeling complex processes and confront analytical complications in various scientific and engineering contexts.

Voltage source inverter-fed permanent magnet synchronous motors (VSI-PMSMs) are widely used in industrial actuation and mechatronic systems in water pumping stations, as well as in the traction of transportation systems (such as electric vehicles and electric trains or ships with electric propulsion). The dynamic model of VSI-PMSMs is multivariable and exhibits complicated nonlinear dynamics. The inverters’ currents, which are generated through a pulsewidth modulation process, are used to control the stator currents of the PMSM, which in turn control the rotational speed of this electric machine. So far, several nonlinear control schemes for VSI-PMSMs have been developed, having as primary objectives the precise tracking of setpoints by the system’s state variables and robustness to parametric changes or external perturbations. However, little has been done for the solution of the associated nonlinear optimal control problem. The purpose of this study/paper is to provide a novel nonlinear optimal control method for VSI-fed three-phase PMSMs.

The present article proposes a nonlinear optimal control approach for VSI-PMSMs. The nonlinear dynamic model of VSI-PMSMs undergoes approximate linearization around a temporary operating point, which is recomputed at each iteration of the control method. This temporary operating point is defined by the present value of the voltage source inverter-fed PMSM state vector and by the last sampled value of the motor’s control input vector. The linearization relies on Taylor series expansion and the calculation of the system’s Jacobian matrices. For the approximately linearized model of the voltage source inverter-fed PMSM, an H-infinity feedback controller is designed. For the computation of the controller’s feedback gains, an algebraic Riccati equation is iteratively solved at each time-step of the control method. The global asymptotic stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control for this system, the H-infinity Kalman filter is proposed as a state observer. The proposed control method achieves fast and accurate tracking of the reference setpoints of the VSI-fed PMSM under moderate variations of the control inputs.

The proposed H-infinity controller provides the solution to the optimal control problem for the VSI-PMSM system under model uncertainty and external perturbations. Actually, this controller represents a min–max differential game taking place between the control inputs, which try to minimize a cost function that contains a quadratic term of the state vector’s tracking error, the model uncertainty, and exogenous disturbance terms, which try to maximize this cost function. To select the feedback gains of the stabilizing feedback controller, an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. To analyze the stability properties of the control scheme, the Lyapunov method is used. It is proven that the VSI-PMSM loop has the H-infinity tracking performance property, which signifies robustness against model uncertainty and disturbances. Moreover, under moderate conditions, the global asymptotic stability properties of this control scheme are proven. The proposed control method achieves fast tracking of reference setpoints by the VSI-PMSM state variables, while keeping also moderate the variations of the control inputs. The latter property indicates that energy consumption by the VSI-PMSM control loop can be minimized.

The proposed nonlinear optimal control method for the VSI-PMSM system exhibits several advantages: Comparing to global linearization-based control methods, such as Lie algebra-based control or differential flatness theory-based control, the nonlinear optimal control scheme avoids complicated state variable transformations (diffeomorphisms). Besides, its control inputs are applied directly to the initial nonlinear model of the VSI-PMSM system, and thus inverse transformations and the related singularity problems are also avoided. Compared with backstepping control, the nonlinear optimal control scheme does not require the state-space description of the controlled system to be found in the triangular (backstepping integral) form. Compared with sliding-mode control, there is no need to define in an often intuitive manner the sliding surfaces of the controlled system. Finally, compared with local model-based control, the article’s nonlinear optimal control method avoids linearization around multiple operating points and does not need the solution of multiple Riccati equations or LMIs. As a result of this, the nonlinear optimal control method requires less computational effort.

Voltage source inverter-fed permanent magnet synchronous motors (VSI-PMSMs) are widely used in industrial actuation and mechatronic systems in water pumping stations, as well as in the traction of transportation systems (such as electric vehicles and electric trains or ships with electric propulsion), The solution of the associated nonlinear control problem enables reliable and precise functioning of VSI-fd PMSMs. This in turn has a positive impact in all related industrial applications and in tasks of electric traction and propulsion where VSI-fed PMSMs are used. It is particularly important for electric transportation systems and for the wide use of electric vehicles as expected by green policies which aim at deploying electromotion and at achieving the Net Zero objective.

Unlike past approaches, in the new nonlinear optimal control method, linearization is performed around a temporary operating point, which is defined by the present value of the system’s state vector and by the last sampled value of the control input vector and not at points that belong to the desirable trajectory (setpoints). Besides, the Riccati equation, which is used for computing the feedback gains of the controller, is new, as is the global stability proof for this control method. Comparing with nonlinear model predictive control, which is a popular approach for treating the optimal control problem in industry, the new nonlinear optimal (H-infinity) control scheme is of proven global stability, and the convergence of its iterative search for the optimum does not depend on initial conditions and trials with multiple sets of controller parameters. It is also noteworthy that the nonlinear optimal control method is applicable to a wider class of dynamical systems than approaches based on the solution of state-dependent Riccati equations (SDRE). The SDRE approaches can be applied only to dynamical systems that can be transformed to the linear parameter varying form. Besides, the nonlinear optimal control method performs better than nonlinear optimal control schemes which use approximation of the solution of the Hamilton–Jacobi–Bellman equation by Galerkin series expansions.