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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.

The main motivation of this paper is to present  the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L²-convergence rate is established.

The authors establish a result concerning the L²-convergence rate of the solution of backward stochastic differential equation with jumps with respect to the Yosida approximation.

The authors carry out a convergence rate of Yosida approximation to the semi-linear backward stochastic differential equation in infinite dimension.

In this paper, the authors present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L²-convergence rate is established.

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.

The purpose of this paper is to investigate the free vibration response of a laminated honeycomb sandwich panels (LHSP) for aerospace applications. Higher order shear deformation theory (HSDT) was simplified for the dynamic analysis of LHSP. Furthermore, the effects of honeycomb parameters on the value of natural frequency (NF) of vibration were explored.

This paper applies HSDT to the analysis of composite LHSP to derive four vibration differential equations of motion and solve it to find the NF of vibration. Two analytical models (Nayak and Meunier models) were selected from literature for comparison of the NF of vibration. In addition, a numerical model was built by using ABAQUS and the results were compared. Furthermore, parametric studies were conducted to explore the effect of honeycomb parameters on the value of the NF of vibration.

The present model is successful in simplifying HSDT for the analysis of LHSP. The first five natural frequencies of vibration were calculated analytically and numerically. In the parametric study, increasing core height or young’s modulus or changing laminate layup will increase the value of NF of vibration. Furthermore, increasing plate constraint (using clamped edge boundary condition) will increase the value of NF of vibrations.

The current analysis is suitable for all-composite symmetric LHSP. However, for isotropic or non-symmetric materials, minor modifications might be adopted.

The application of simplified HSDT to the analysis of LHSP is one of the important values of this research. The other is the successful and complete dynamic analysis of all-composite LHSP.

The purpose of this paper is to study large deviations for the solution processes of a stochastic equation incorporated with the effects of nonlocal condition.

A weak convergence approach is adopted to establish the Laplace principle, which is same as the large deviation principle in a Polish space. The sufficient condition for any family of solutions to satisfy the Laplace principle formulated by Budhiraja and Dupuis is used in this work.

Freidlin–Wentzell type large deviation principle holds good for the solution processes of the stochastic functional integral equation with nonlocal condition.

The asymptotic exponential decay rate of the solution processes of the considered equation towards its deterministic counterpart can be estimated using the established results.

The purpose of this article is to treat the nonlinear optimal control problem in EV traction systems which are based on 5-phase induction motors. Five-phase permanent magnet synchronous motors and five-phase asynchronous induction motors (IMs) are among the types of multiphase motors one can consider for the traction system of electric vehicles (EVs). By distributing the required power in a large number of phases, the power load of each individual phase is reduced. The cumulative rates of power in multiphase machines can be raised without stressing the connected converters. Multiphase motors are also fault tolerant because such machines remain functional even if failures affect certain phases.

A novel nonlinear optimal control approach has been developed for five-phase IMs. The dynamic model of the five-phase IM undergoes approximate linearization using Taylor series expansion and the computation of the associated Jacobian matrices. The linearization takes place at each sampling instance. For the linearized model of the motor, an H-infinity feedback controller is designed. This controller achieves the solution of the optimal control problem under model uncertainty and disturbances.

To select the feedback gains of the nonlinear optimal (H-infinity) controller, an algebraic Riccati equation has to be solved repetitively at each time-step of the control method. The global stability properties of the control loop are demonstrated through Lyapunov analysis. Under moderate conditions, the global asymptotic stability properties of the control scheme are proven. The proposed nonlinear optimal control method achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs.

Comparing to other nonlinear control methods that one could have considered for five-phase IMs, the presented nonlinear optimal (H-infinity) control approach avoids complicated state-space model transformations, is of proven global stability and its use does not require the model of the motor to be brought into a specific state-space form. The nonlinear optimal control method has clear implementation stages and moderate computational effort.

In the transportation sector, there is progressive transition to EVs. The use of five-phase IMs in EVs exhibits specific advantages, by achieving a more balanced distribution of power in the multiple phases of the motor and by providing fault tolerance. The study’s nonlinear optimal control method for five-phase IMs enables high performance for such motors and their efficient use in the traction system of EVs.

Nonlinear optimal control for five-phase IMs supports the deployment of their use in EVs. Therefore, it contributes to the net-zero objective that aims at eliminating the emission of harmful exhaust gases coming from human activities. Most known manufacturers of vehicles have shifted to the production of all-electric cars. The study’s findings can optimize the traction system of EVs thus also contributing to the growth of the EV industry.

The proposed nonlinear optimal control method is novel comparing to past attempts for solving the optimal control problem for nonlinear dynamical systems. It uses a novel approach for selecting the linearization points and a new Riccati equation for computing the feedback gains of the controller. 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.

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.

This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm.

The authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations.

Approaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approach’s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software.

The convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement.

There is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively.

This work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution.

To the best of the authors’ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.

This paper aims at studying numerically the entropy generation of nanofluid flowing over an inclined sheet in the presence of external magnetic field, heat source/sink, chemical reaction along with slip boundary conditions imposed on an impermeable wall.

A suitable similarity transformation technique has been used to convert the coupled nonlinear partial differential equations to ordinary differential equations (ODEs). The ODEs are then solved simultaneously using the finite difference method implemented through an in-house computer program. The effects of different controlling parameters such as magnetic parameter, radiation parameter, Brownian motion parameter, thermophoresis parameter, chemical reaction parameter, Reynolds number, Brinkmann number, Prandtl number, velocity slip parameter, temperature slip parameter and the concentration slip parameter on the entropy generation and Bejan number have been discussed comprehensively through the relevant physical insights for the first time.

The relative strengths of the irreversibilities due to heat transfer, fluid friction and the mass diffusion arising due to the change in each of the controlling variables have been delineated both in the near-wall and far-away-wall regions, which may be helpful for a better understanding of the thermo-fluid dynamics of nanofluid in boundary layer flows. The numerical results obtained from the present study have also been validated with results published in open literature.

The effects of different controlling parameters such as magnetic parameter, radiation parameter, Brownian motion parameter, thermophoresis parameter, chemical reaction parameter, Reynolds number, Brinkmann number, Prandtl number, velocity slip parameter, temperature slip parameter and the concentration slip parameter on the entropy generation and Bejan number have been discussed comprehensively through the relevant physical insights for the first time.

This paper aims to find a closed-form expression for the frequency and amplitude of single-ended ring oscillators when transistors experience all regions.

In this paper, the analytical relationships presented for ring oscillator amplitude and frequency are approximately derived due to the nonlinear nature of this oscillator, taking into account the differential equation that governs the ring oscillator and its output waveform.

In the case where the transistors experience the cut-off region, the relationships presented so far have no connection between the frequency and the dimensions of the transistor, which is not valid in practice. The relationship is presented for the frequency, including the dimensions of the transistor. Also, a simple and approximately accurate relationship for the oscillator amplitude is provided in this case.

The validity of these relationships has been investigated by analyzing and simulating a single-ended oscillator in 0.18 µm technology.