This paper proposes a linearly implicit, structure-preserving scheme for the generalized Zakharov system. For the construction of the numerical scheme, we first rewrite the system into an equivalent first-order formulation. We then construct a linearly implicit scheme based on the leap-frog method, together with a suitable starting procedure. We prove that the proposed scheme preserves both mass and energy in the discrete sense. Numerical experiments are carried out to validate the theoretical results.

In this paper, we are interested in developing a linearly implicit and structure-preserving scheme to solve the Schr & ouml;dinger-Helmholtz equation. The spatial discretization is done by finite element methods. The time-discretization is done by the modified Leap-frog scheme, with carefully choosing the approximation of the intermediate average variable. The fully-discrete is linearly implicit. And it is proved that the proposed scheme is mass-and energy-conserving unconditionally. Moreover, unconditional error estimates of the fully-discrete scheme can be obtained in both two-dimensional and three-dimensional cases, allowing for optimal error estimates without any time-step restrictions. The key to our proof lies in using error estimates in certain norms at different time levels, along with the Sobolev embedding and inverse inequality in high-dimensional spaces. Numerical experiments are presented to confirm the theoretical results.

This study provides an efficient, adaptive computational framework for Bayesian model selection when likelihood functions are difficult to handle. To address model selection challenges in scenarios where complex model likelihood functions are difficult to handle, this paper proposes an adaptive approximate Bayesian computation with probability-minimizing computation (RL(Softmax)-ABC-PMC) algorithm that integrates a reinforcement learning (RL) Softmax decision mechanism. This method embeds Softmax agents into the standard ABC-PMC framework, dynamically evaluating the historical performance of candidate models to adaptively focus computational resources on potentially optimal models, thereby significantly enhancing selection efficiency. A numerical experiment is provided to select the best model from three Lotka-Volterra (LV) competition models, including a discretized LV model, a randomized LV model, and a fractional-order LV model, where the algorithm effectively distinguishes subtle differences in their fitting ability and yields reliable parameter estimates. Additionally, we further validate the efficiency of the provided algorithm by selecting the best model from three S-shaped growth curve models. Finally, an empirical analysis using Chinese automotive market sales data further validates the practicality and effectiveness of the proposed method in real-world applications, demonstrating that the RL(Softmax)-ABC-PMC algorithm can automatically identify model structures with stronger explanatory power while maintaining high computational efficiency.

An improved single-component Metropolis-Hastings (ISC-MH) algorithm for estimating a kind of time fractional delay diffusion equation (FDDE) is provided. We introduce the symmetric proposal distribution for generating the candidate point of each parameter to obtain the ISC-MH algorithm. To estimate the unknown parameter of the FDDE, firstly, we construct the compact difference scheme to numerically discretize the FDDE. Then the ISC-MH algorithm is used to estimate the fractional order, the diffusion coefficient, and the time delay in the system of the FDDE simultaneously. Finally, some simulation tests of two examples are provided to validate the performance of the algorithms.

A novel algorithm is proposed to increase the effectiveness of model selection and parameter estimation for the fractional susceptible-infected-recovered model. It combines reinforcement learning (RL) and approximate Bayesian computation sequential Monte Carlo (ABC-SMC) instead of ABC to improve the process of model selection and parameter estimation, where RL is used for model selection and ABC-SMC is exploited for parameter estimation. Numerical simulations illustrate that the combined algorithm (RL-ABC-SMC) significantly outperforms the ABC-SMC algorithm in terms of model selection. Finally, we consider the application of the proposed methodology.

Delay partial differential equations (PDEs) are widely utilized in many fields, such as climate prediction and epidemiology. But observation data in real world is often noisy and discrete. And in order to expand the applications of delay PDEs, we consider numerical Gaussian processes to solve these models. In this paper, numerical Gaussian processes for predicting the latent solution of a type of delay PDEs with multi-delays are investigated, and various delay PDEs are studied, including problems governed by variable-order fractional order operators and nonlinear operators, so as to adapt to the needs of practical applications. Numerical Gaussian processes are very good at fitting latent solution of PDEs, when all observation data is noisy and discontinuous. And the methodology can clearly quantify the uncertainty of the predicted solution. For complex boundaries controlled by ODEs, we consider mixed boundary conditions of delay PDEs in this paper. And we also apply Runge-Kutta methods to enhance the prediction accuracy of these problems. Finally, we design seven numerical examples to investigate the efficiency of NGPs and how the noisy data influences the solution of our studied problems.

To satisfy the space-borne tasks of meteorologic observation, planetary researches, and interferometric imaging, a new trial to develop a high efficiency Ka-band extended interaction klystron (EIK) is put on the agenda. In the high frequency circuit design, it is paid more attention to realizing high efficiency as far as possible; meanwhile, the circuit is kept stable using high lossy bunching cavities. The beam voltage and current are 17 kV and 0.72 A, respectively. The simulated output power is over 3 kW in the bandwidth of >100 MHz, and the maximum efficiency is above 30%. The preliminary test is finished with the beam transmission of 97% and the output power of 2.4 kW.

Structure-preserving numerical methods have extremely important applications in long-time simulations of highly oscillatory Hamiltonian systems. In this paper, we utilize the relaxation technique and propose a family of relaxation implicit-explicit Runge-Kutta methods for solving highly oscillatory Hamiltonian systems. Contrary to the standard implicit-explicit Runge-Kutta methods, the structure-preserving properties of the proposed methods enable them to be applied to long-time simulations. Besides, the proposed methods are linearly implicit and arbitrarily high-order accurate, which can greatly improve the computational efficiency in simulations. Finally, several numerical experiments are performed to verify the theoretical results in the article.

This paper presents a transformed L1 finite difference method for the time-fractional molecular beam epitaxy (TFMBE) model without slope selection, taking the initial singularity of the solution into account. A novel discrete fractional Gronwall inequality is used to provide unconditionally optimum error estimates. The convergence results indicate that the method has an order of 2 in the spatial direction and an order of $ 2-\alpha $ 2-alpha in the temporal direction. Numerical experiments are carried out to corroborate our theoretical findings.

A linearized Galerkin finite element method is presented for numerically solving the semi-linear time-fractional parabolic problems, whose solutions always display a initial weak singularity. The transformed L1 scheme based on a change of variable is used to approximate Caputo derivatives and the finite element approximations to the spatial variables. By the temporal-spatial error splitting argument, unconditionally optimal error estimates of the proposed schemes are proved. Finally, several numerical experiments are given to demonstrate our theoretical results.

Numerical solution and parameter estimation for a type of fractional diffusion equation are considered. Firstly, the symmetrical compact difference scheme is applied to solve the forward problem of the fractional diffusion equation. The stability and convergence of the symmetrical difference scheme are presented. Then, the Bayesian method is considered to estimate the unknown fractional order α of the fractional diffusion equation model. To validate the efficiency of the symmetrical numerical scheme and the estimation method, some simulation tests are considered. The simulation results demonstrate the accuracy of the compact difference scheme and show that the proposed estimation algorithm can provide effective statistical characteristics of the parameter.

For vacuum electronic devices (VEDs), the use of a hollow electron beam (HEB) increases the beam-wave interaction efficiency and permits a prominent decrease of the cutoff voltage in the focusing-electrode controlled electron gun. To obtain an HEB electron optics system (EOS) with stable propagation, lower ripple, high compression, and good laminarity, the issues related to the HEBs transmission characteristics in a uniform magnetic field are deeply investigated in this article. The studies on the Brillouin flow, which are different from the conclusion of the previous literature, are presented. Through combining the theoretical analysis and the simulation, the distinct motion ways of each layer in the HEB are clearly exhibited in both cases of the immersed flow and the partially shielding flow. This work can provide a meaningful reference for the design of a hollow beam EOS with high transmission and compression.

This paper mainly considers the parameter estimation problem for several types of differential equations controlled by linear operators, which may be partial differential, integro-differential and fractional order operators. Under the idea of data-driven methods, the algorithms based on Gaussian processes are constructed to solve the inverse problem, where we encode the distribution information of the data into the kernels and construct an efficient data learning machine. We then estimate the unknown parameters of the partial differential Equations (PDEs), which include high-order partial differential equations, partial integro-differential equations, fractional partial differential equations and a system of partial differential equations. Finally, several numerical tests are provided. The results of the numerical experiments prove that the data-driven methods based on Gaussian processes not only estimate the parameters of the considered PDEs with high accuracy but also approximate the latent solutions and the inhomogeneous terms of the PDEs simultaneously.

Probabilistic machine learning and data-driven methods gradually show their high efficiency in solving the forward and inverse problems of partial differential equations (PDEs). This paper will focus on investigating the forward problem of solving time-dependent nonlinear delay PDEs with multi-delays based on multi-prior numerical Gaussian processes (MP-NGPs), which are constructed by us to solve complex PDEs that may involve fractional operators, multi-delays and different types of boundary conditions. We also quantify the uncertainty of the prediction solution by the posterior distribution of the predicted solution. The core of MP-NGPs is to discretize time firstly, then a Gaussian process regression based on multi-priors is considered at each time step to obtain the solution of the next time step, and this procedure is repeated until the last time step. Different types of boundary conditions are studied in this paper, which include Dirichlet, Neumann and mixed boundary conditions. Several numerical tests are provided to show that the methods considered in this paper work well in solving nonlinear time-dependent PDEs with delay, where delay partial differential equations, delay partial integro-differential equations and delay fractional partial differential equations are considered. Furthermore, in order to improve the accuracy of the algorithm, we construct Runge–Kutta methods under the frame of multi-prior numerical Gaussian processes. The results of the numerical experiments prove that the prediction accuracy of the algorithm is obviously improved when the Runge–Kutta methods are employed.

In this article, the design and experiment of a hollow electron beam (HEB) electron optics system (EOS) for Ka-band extended interaction klystrons (EIKs) are presented. To realize the fast switch of the current emission at a low cutoff voltage, an electron gun with two focus electrodes (FEs) is chosen. The emitting surface of cathode is a ringed spherical surface. The first focus electrode (FE1) is around the cathode, and the second focus electrode (FE2) is set in the central hole of the cathode. The current emitted from the cathode surface is about 2A, when the operating voltage is 25 kV. Three simulation codes, Egun, Superfish, and CST, are used for designing the electron gun and the permanent magnet focusing system (PMFS). Some sensitivity analyses with respect to the PMFS’s critical parameters are presented. The simulation shows the dc beam transmission is 100%. The electron beam, confined by 6000 Gauss uniform field, propagates through the drift tube of 37 mm with good laminarity and small ripple. The emission current can be suppressed when the voltage of the FEs is −3.4 kV. According to the results of simulation, the Ka-band EIK had been constructed and the hot-test experiments had been fulfilled in succession. The experimental results exhibit that the dc beam transmission is about 99% and the beam can sustain 98% transmission at high frequency (HF) operation. In addition, the output power is higher than 8 kW in the bandwidth of over 100 MHz.

In this paper, we present a multiple layer device for investigating the impact of electric field on the conductance switching of GeTe phase change material excluding the contribution from Joule heat. The device includes a dielectric layer with excellent current-blocking which can result in a large electric field generated in the amorphous GeTe film and almost no current. With the generated electric field far beyond its threshold value for the conductance switching, our experimental data indicate that the conductance switching has not happened in the GeTe film. This indicates that the ovonic threshold switching (OTS) could not be induced by the purely electric field in amorphous chalcogenide film. Meanwhile, a modified thermal-assist model based on the Poole–Frenkel (PF) mechanism has been proposed to verify the thermal assistance is indispensable in the OTS process. And the modified model is well applied on the GeTe devices with different scales, which further supports the current experimental conclusion. This contributes to the further study of the OTS mechanism and application of the phase-change memory (PCM).

The design of a W-band gyrotron traveling wave tube amplifier is presented. The device with a lossy ceramic interaction circuit operates in the fundamental harmonic TE01 circular electric mode. The 70-kV, 6-A electron beam is produced by a double anode magnetron injection gun with an average perpendicular-to-parallel velocity ratio of 1.2 and a parallel velocity spread of less than 3%. The beam-wave interaction is investigated by using a particle-in-cell code. The simulations predict that the amplifier can produce an output peak power of over 100 kW, 28% efficiency, 42 dB gain, and a 3 GHz bandwidth, respectively under the assumption of beam velocity spread 3%.

A difference scheme is constructed for a type of variable coefficient time fractional subdiffusion equation with multidelay. Stability and convergence results of the scheme are obtained, and theoretical results are proved by two numerical tests.