Adequate Closed Form Wave Solutions to the Generalized KdV Equation in Mathematical Physics

In this paper, we consider the generalized Korteweg-de-Vries (KdV) equations which are remarkable models of the water waves mechanics, the shallow water waves, the quantum mechanics, the ion acoustic waves in plasma, the electro-hydro-dynamical model for local electric field, signal processing waves through optical fibers, etc. We determine the useful and further general exact traveling wave solutions of the above mentioned NLDEs by applying the exp(−τ(ξ))-expansion method by aid of traveling wave transformations. Furthermore, we explain the physical significance of the obtained solutions of its definite values of the involved parameters with graphic representations in order to know the physical phenomena. Finally, we show that the exp(−τ(ξ))-expansion method is convenient, powerful, straightforward and provide more general solutions and can be helping to examine vast amount of travelling wave solutions to the other different kinds of NLDEs.

科威特Equate公司实施乙二醇扩能

科威特Equate石化公司于2015年12月以32亿美元从陶氏化学公司和石化工业公司（PIC）手中收购了乙二醇（EG）领域的生产商MEGlobal公司,并于2016年11月28日完成在科威特舒巴的聚乙烯（PE）脱瓶颈扩能项目。该项目产能从82.5万t/a提高到100万t/a。

Investigation of adequate closed form travelling wave solution to the space-time fractional non-linear evolution equations

This work aims to construct exact solutions for the space-time fractional(2+1)-dimensional dispersive longwave(DLW)equation and approximate long water wave equation(ALW)utilizing the twovariable(G′/G,1/G)-expansion method and the modified Riemann-Liouville fractional derivative.The recommended equations play a significant role to describe the travel of the shallow water wave.The fractional complex transform is used to convert fractional differential equations into ordinary differential equations.Several wave solutions have been successfully achieved using the proposed approach and the symbolic computer Maple package.The Maple package program was used to set up and validate all of the computations in this investigation.By choosing particular values of the embedded parameters,we pro-duce multiple periodic solutions,periodic wave solutions,single soliton solutions,kink wave solutions,and more forms of soliton solutions.The achieved solutions might be useful to comprehend nonlinear phenomena.It is worth noting that the implemented method for solving nonlinear fractional partial dif-ferential equations(NLFPDEs)is efficient,and simple to find further and new-fangled solutions in the arena of mathematical physics and coastal engineering.

Accurate simulations of pure-viscoacoustic wave propagation in tilted transversely isotropic media

Forward modeling of seismic wave propagation is crucial for the realization of reverse time migration(RTM) and full waveform inversion(FWI) in attenuating transversely isotropic media. To describe the attenuation and anisotropy properties of subsurface media, the pure-viscoacoustic anisotropic wave equations are established for wavefield simulations, because they can provide clear and stable wavefields. However, due to the use of several approximations in deriving the wave equation and the introduction of a fractional Laplacian approximation in solving the derived equation, the wavefields simulated by the previous pure-viscoacoustic tilted transversely isotropic(TTI) wave equations has low accuracy. To accurately simulate wavefields in media with velocity anisotropy and attenuation anisotropy, we first derive a new pure-viscoacoustic TTI wave equation from the exact complex-valued dispersion formula in viscoelastic vertical transversely isotropic(VTI) media. Then, we present the hybrid finite-difference and low-rank decomposition(HFDLRD) method to accurately solve our proposed pure-viscoacoustic TTI wave equation. Theoretical analysis and numerical examples suggest that our pure-viscoacoustic TTI wave equation has higher accuracy than previous pure-viscoacoustic TTI wave equations in describing q P-wave kinematic and attenuation characteristics. Additionally, the numerical experiment in a simple two-layer model shows that the HFDLRD technique outperforms the hybrid finite-difference and pseudo-spectral(HFDPS) method in terms of accuracy of wavefield modeling.

Surrogate modeling for unsaturated infiltration via the physics and equality-constrained artificial neural networks

Machine learning(ML)provides a new surrogate method for investigating groundwater flow dynamics in unsaturated soils.Traditional pure data-driven methods(e.g.deep neural network,DNN)can provide rapid predictions,but they do require sufficient on-site data for accurate training,and lack interpretability to the physical processes within the data.In this paper,we provide a physics and equalityconstrained artificial neural network(PECANN),to derive unsaturated infiltration solutions with a small amount of initial and boundary data.PECANN takes the physics-informed neural network(PINN)as a foundation,encodes the unsaturated infiltration physical laws(i.e.Richards equation,RE)into the loss function,and uses the augmented Lagrangian method to constrain the learning process of the solutions of RE by adding stronger penalty for the initial and boundary conditions.Four unsaturated infiltration cases are designed to test the training performance of PECANN,i.e.one-dimensional(1D)steady-state unsaturated infiltration,1D transient-state infiltration,two-dimensional(2D)transient-state infiltration,and 1D coupled unsaturated infiltration and deformation.The predicted results of PECANN are compared with the finite difference solutions or analytical solutions.The results indicate that PECANN can accurately capture the variations of pressure head during the unsaturated infiltration,and present higher precision and robustness than DNN and PINN.It is also revealed that PECANN can achieve the same accuracy as the finite difference method with fewer initial and boundary training data.Additionally,we investigate the effect of the hyperparameters of PECANN on solving RE problem.PECANN provides an effective tool for simulating unsaturated infiltration.

Saturation Estimation with Complex Electrical Conductivity for Hydrate-Bearing Clayey Sediments:An Experimental Study

Clays have considerable influence on the electrical properties of hydrate-bearing sediments.It is desirable to understand the electrical properties of hydrate-bearing clayey sediments and to build hydrate saturation(S_(h))models for reservoir evaluation and monitoring.The electrical properties of tetrahydrofuran-hydrate-bearing sediments with montmorillonite are characterized by complex conductivity at frequencies from 0.01 Hz to 1 kHz.The effects of clay and Sh on the complex conductivity were analyzed.A decrease and increase in electrical conductance result from the clay-swelling-induced blockage and ion migration in the electrical double layer(EDL),respectively.The quadrature conductivity increases with the clay content up to 10%because of the increased surface site density of counterions in EDL.Both the in-phase conductivity and quadrature conductivity decrease consistently with increasing Sh from 0.50 to 0.90.Three sets of models for Sh evaluation were developed.The model based on the Simandoux equation outperforms Archie’s formula,with a root-mean-square error(E_(RMS))of 1.8%and 3.9%,respectively,highlighting the clay effects on the in-phase conductivity.The fre-quency effect correlations based on in-phase and quadrature conductivities exhibit inferior performance(E_(RMS)=11.6%and 13.2%,re-spectively)due to the challenge of choosing an appropriate pair of frequencies and intrinsic uncertainties from two measurements.The second-order Cole-Cole formula can be used to fit the complex-conductivity spectra.One pair of inverted Cole-Cole parameters,i.e.,characteristic time and chargeability,is employed to predict S_(h) with an E_(RMS) of 5.05%and 9.05%,respectively.

Pre-reduction of WO_(3)-Co_(3)O_(4)by H_(2)-C_(2)H_4 in a fluidized bed

In order to avoid the formation ofηphase(W_(6)Co_(6)C or W_(3)Co_(3)C)that adversely affects the sintering process and its products in the preparation process of ultra-fine WC-Co powder,a technical route of prereduction of WO_(3)-Co_(3)O_(4)to WO_(2)-Co and then deep reduction carbonization to WC-Co powder has been proposed.This study mainly investigates the influence of gas partial pressure on the pre-reduction process of WO_(3)-Co_(3)O_(4)under a mixed atmosphere of H_(2)-C_(2)H_(4)-Ar at 600℃and establishes the kinetic equations of pre-reduction and carbon evolution.The results indicate that increasing the partial pressure of hydrogen is conducive to the rapid and complete conversion of WO_(3) to WO_(2).High carbon content can be generated by the deposition of C_(2)H_(4),and it hinders the diffusion of the reducing gas;WO_(3)still cannot be completely reduced to WO_(2)as the partial pressure of C_(2)H_(4) increases to 60%.For the carbon evolution of C_(2)H_(4),the carbon amount is positively related to the H_(2)partial pressure,but it shows the highest amount and evolution rate when the ethylene partial pressure is 20%.Based on the reduction rate curves of WO_(3) and carbon evolution rate curves of C_(2)H_(4),the rate equations of pre-reduction and carbon evolution of WO_(3)-Co_(3)O_(4)system at 600℃are established.The pre-reduction reaction belongs to the first-order reaction,and its equation is expressed as follows:r=-(dw_(WO_(3)))/dt=(9±0.15)×10^(-2)×P_(H_(2))^(0.44)P_(C_(2)H_(4))&(0.57)The carbon deposition rate equation of C_(2)H_(4) can be expressed as follows:r=-(dc_C)/dt=r_f-r_b≌7.35×10^(-2)×P_(C_(2)H_(4))^(0.31)

STABILITY OF THE RAREFACTION WAVE IN THE SINGULAR LIMIT OF A SHARP INTERFACE PROBLEM FOR THE COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM

This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space.For the perturbations with small energy but possibly large oscillations of rarefaction wave solutions near phase separation,and where the strength of the initial phase field could be arbitrarily large,we prove that the solution of the Cauchy problem exists for all time,and converges to the centered rarefaction wave solution of the corresponding standard two-phase Euler equation as the viscosity and the thickness of the interface tend to zero.The proof is mainly based on a scaling argument and a basic energy method.

Some Modified Equations of the Sine-Hilbert Type

Three modified sine-Hilbert(sH)-type equations, i.e., the modified sH equation, the modified damped sH equation, and the modified nonlinear dissipative system, are proposed, and their bilinear forms are provided.Based on these bilinear equations, some exact solutions to the three modified equations are derived.

Viscoacoustic generalized screen propagator in constant-Q model

When seismic waves propagate through the geological formation,there is a significant loss of energy and a decrease in imaging resolution,because of the viscoacoustic properties of subsurface medium.This profoundly impacts seismic wavefield propagation,imaging and interpretation.To accurately image the true structure of subsurface medium,the consensus among geophysicists is to no longer treat subsurface medium as ideal homogeneous medium,but rather to incorporate the viscoacoustic properties of subsurface medium.Based on the generalized screen propagator using conventional acoustic wave equation(acoustic GSP),our developed method introduces viscoacoustic compensation strategy,and derives a one-way wave generalized screen propagator based on time-fractional viscoacoustic wave equation(viscoacoustic GSP).In numerical experiments,we conducted tests on two-dimensional multi-layer model and the Marmousi model.When comparing with the acoustic GSP using the acoustic data,we found that the imaging results of the viscoacoustic GSP using the viscoacoustic data showed a significant attenuation compensation effect,and achieved imaging results for both algorithms were essentially consistent.However,the imaging results of acoustic GSP using viscoacoustic data showed significant attenuation effects,especially for deep subsurface imaging.This indicates that we have proposed an effective method to compensate the attenuated seismic wavefield.Our application on a set of real seismic data demonstrated that the imaging performance of our proposed method in local areas surpassed that of the conventional acoustic GSP.This suggests that our proposed method holds practical value and can more accurately image real subsurface structures while enhancing imaging resolution compared with the conventional acoustic GSP.Finally,with respect to computational efficiency,we gathered statistics on running time to compare our proposed method with conventional Q-RTM,and it is evident that our method exhibits higher computational efficiency.In summary,our proposed viscoacoustic GSP method takes into account the true properties of the medium,still achieves migration results comparable to conventional acoustic GSP.

Analytical solutions fractional order partial differential equations arising in fluid dynamics

This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectional propagation of long-wave in dispersive media and GSEs are used to model the interaction between one-dimensional high,and low-frequency waves.Classes of trigonometric and hyperbolic function solutions in fractional calculus are discussed.Graphical simulations of the numerical solutions are flaunted by MATLAB.

Travelling wave solutions of nonlinear conformable analytical approaches

The presented study deals with the investigation of nonlinear Bogoyavlenskii equations with conformable time-derivative which has great importance in plasma physics and non-inspectoral scattering problems.Travelling wave solutions of this nonlinear conformable model are constructed by utilizing two powerful analytical approaches,namely,the modified auxiliary equation method and the Sardar sub-equation method.Many novel soliton solutions are extracted using these methods.Furthermore,3D surface graphs,contour plots and parametric graphs are drawn to show dynamical behavior of some obtained solutions with the aid of symbolic software such as Mathematica.The constructed solutions will help to understand the dynamical framework of nonlinear Bogoyavlenskii equations in the related physical phenomena.

THE STABILITY OF BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION AROUND THE HYDROSTATIC BALANCE

This paper is devoted to understanding the stability of perturbations around the hydrostatic equilibrium of the Boussinesq system in order to gain insight into certain atmospheric and oceanographic phenomena.The Boussinesq system focused on here is anisotropic,and involves only horizontal dissipation and thermal damping.In the 2D case R^(2),due to the lack of vertical dissipation,the stability and large-time behavior problems have remained open in a Sobolev setting.For the spatial domain T×R,this paper solves the stability problem and gives the precise large-time behavior of the perturbation.By decomposing the velocity u and temperatureθinto the horizontal average(ū,θ)and the corresponding oscillation(ū,θ),we can derive the global stability in H~2 and the exponential decay of(ū,θ)to zero in H^(1).Moreover,we also obtain that(ū_(2),θ)decays exponentially to zero in H^(1),and thatū_(1)decays exponentially toū_(1)(∞)in H^(1)as well;this reflects a strongly stratified phenomenon of buoyancy-driven fluids.In addition,we establish the global stability in H^(3)for the 3D case R^(3).

INTERFACE BEHAVIOR AND DECAY RATES OF COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND A VACUUM

In this paper,we study the one-dimensional motion of viscous gas near a vacuum,with the gas connecting to a vacuum state with a jump in density.The interface behavior,the pointwise decay rates of the density function and the expanding rates of the interface are obtained with the viscosity coefficientμ(ρ)=ρ^(α)for any 0<α<1;this includes the timeweighted boundedness from below and above.The smoothness of the solution is discussed.Moreover,we construct a class of self-similar classical solutions which exhibit some interesting properties,such as optimal estimates.The present paper extends the results in[Luo T,Xin Z P,Yang T.SIAM J Math Anal,2000,31(6):1175-1191]to the jump boundary conditions case with density-dependent viscosity.

Finite element model updating for structural damage detection using transmissibility data

This paper presents a new finite element model updating method for estimating structural parameters and detecting structural damage location and severity based on the structural responses(output-only data).The method uses the sensitivity relation of transmissibility data through a least-squares algorithm and appropriate normalization of the extracted equations.The proposed transmissibility-based sensitivity equation produces a more significant number of equations than the sensitivity equations based on the frequency response function(FRF),which can estimate the structural parameters with higher accuracy.The abilities of the proposed method are assessed by using numerical data of a two-story two-bay frame model and a plate structure model.In evaluating different damage cases,the number,location,and stiffness reduction of the damaged elements and the severity of the simulated damage have been accurately identified.The reliability and stability of the presented method against measurement and modeling errors are examined using error-contaminated data.The parameter estimation results prove the method’s capabilities as an accurate model updating algorithm.

Theoretical study of particle and energy balance equations in locally bounded plasmas

In this study,new particle and energy balance equations have been developed to predict the electron temperature and density in locally bounded plasmas.Classical particle and energy balance equations assume that all plasma within a reactor is completely confined only by the reactor walls.However,in industrial plasma reactors for semiconductor manufacturing,the plasma is partially confined by internal reactor structures.We predict the effect of the open boundary area(A′_(L,eff))and ion escape velocity(u_(i))on electron temperature and density by developing new particle and energy balance equations.Theoretically,we found a low ion escape velocity(u_(i)/u_(B)≈0.2)and high open boundary area(A′_(L,eff)/A_(T,eff)≈0.6)to result in an approximately 38%increase in electron density and an 8%decrease in electron temperature compared to values in a fully bounded reactor.Additionally,we suggest that the velocity of ions passing through the open boundary should exceedω_(pi)λ_(De)under the condition E^(2)_(0)?(Φ/λ_(De))^(2).

Data-Driven Ai-and Bi-Soliton of the Cylindrical Korteweg-de Vries Equation via Prior-Information Physics-Informed Neural Networks

By the modifying loss function MSE and training area of physics-informed neural networks(PINNs),we propose a neural networks model,namely prior-information PINNs(PIPINNs).We demonstrate the advantages of PIPINNs by simulating Ai-and Bi-soliton solutions of the cylindrical Korteweg-de Vries(cKdV)equation.

Correction to:Feasibility study of the photonuclear reaction cross section of medical radioisotopes using a laser Compton scattering gamma source

Correction to:Nuclear Science and Techniques(2024)35:155 https://doi.org/10.1007/s41365-024-01481-7 In Eq.(2)of this article,the term'i'should be denoted as a subscript,the corrected equation should read N_(0)=σ(E)N_(t)QC_(kbeam)^(e^(-λt_(i)))∫_(0)^(t_(i))e^(λt)dt,(2)The original article has been corrected.

Prediction of ILI following the COVID-19 pandemic in China by using a partial differential equation

The COVID-19 outbreak has significantly disrupted the lives of individuals worldwide.Following the lifting of COVID-19 interventions,there is a heightened risk of future outbreaks from other circulating respiratory infections,such as influenza-like illness(ILI).Accurate prediction models for ILI cases are crucial in enabling governments to implement necessary measures and persuade individuals to adopt personal precautions against the disease.This paper aims to provide a forecasting model for ILI cases with actual cases.We propose a specific model utilizing the partial differential equation(PDE)that will be developed and validated using real-world data obtained from the Chinese National Influenza Center.Our model combines the effects of transboundary spread among regions in China mainland and human activities’impact on ILI transmission dynamics.The simulated results demonstrate that our model achieves excellent predictive performance.Additionally,relevant factors influencing the dissemination are further examined in our analysis.Furthermore,we investigate the effectiveness of travel restrictions on ILI cases.Results can be used to utilize to mitigate the spread of disease.

THE LIMIT CYCLE BIFURCATIONS OF A WHIRLING PENDULUM WITH PIECEWISE SMOOTH PERTURBATIONS

This paper deals with the problem of limit cycles for the whirling pendulum equation x=y,y=sin x(cosx-r)under piecewise smooth perturbations of polynomials of cos x,sin x and y of degree n with the switching line x=0.The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained using the Picard-Fuchs equations,which the generating functions of the associated first order Melnikov functions satisfy.Furthermore,the exact bound of a special case is given using the Chebyshev system.At the end,some numerical simulations are given to illustrate the existence of limit cycles.