Medium Equation Research Articles

Two-dimensional transient contrasting structure evolution in an inhomogeneous media with the advection

The inner transition layer evolution for two-dimensional quasi-linear initial-boundary value problem for the reaction-advection-diffusion equation in an inhomogeneous media with a small parameter within the high order derivatives is considered. Within the framework of the main (zero order) sum of the asymptotic series, the position of the inner transition layer is described by the Hamilton–Jacobi equation. The potential is calculated as an integrated density function of the reaction sources within the limits of the equilibrium levels. The front line of the transition layer evolves in the same way as the constant-eikonal line (the wavefront line in the other words) for the problem of wave propagation in an inhomogeneous medium in short-wave (geometro-optical) asymptotics. The sum of the asymptotic series of zero order and first order is found, the existence gap of a smooth front line, the time of destruction of the contrasting structure are calculated.

Direct numerical simulation of compressible turbulent channel flows over porous boundaries

The impact of porous boundaries on skin friction and heat transfer in compressible turbulent channel flows is explored through direct numerical simulations. Utilizing the volume-averaged Navier–Stokes equations for the porous media, we examine flows at Mach numbers Mb=0.3 and Mb=1.5, with friction Reynolds numbers ranging from 176 to 220, across three permeability levels: 0 (impermeable), 1.6×10−5, and 1.0×10−4. Our findings reveal a dual impact of porous boundaries on skin friction: a direct reduction due to averaged slip velocity at the interface and an indirect increase from amplified Reynolds shear stress due to enhanced wall-normal permeability. This permeability weakens the wall-blocking effect, leading to intensified wall-normal velocity fluctuations and consequently higher Reynolds stress. Additionally, the Mach number significantly influences flow structures, particularly at higher permeabilities, where it disrupts near-wall streaks at Mb=0.3 but has a diminished effect at Mb=1.5. Thermal analysis confirms the validity of the generalized Reynolds analogy for the mean temperature in these flows, with the turbulent Prandtl number approximating 1, in line with the strong Reynolds analogy. Moreover, a strong spatial correlation between heat flux and wall shear stress fluctuations underscores the intertwined dynamics of momentum and thermal transport at the porous–fluid interface.

Deriving an unsaturated hydraulic conductivity function based on ergun equation

Computational Fluid Dynamics (CFD) is a widely-used numerical technique for simulating flow phenomena, particularly unsaturated flow, which involves two-phase flow. This type of flow can be effectively modeled using the Volume of Fluid (VOF) method. The key to accurately depicting unsaturated flow is the unsaturated hydraulic function (K-function), which connects flow velocity to the hydraulic gradient. However, employing the VOF method alongside the commonly used Ergun equation for modeling porous media can result in an unrealistic K-function. This, in turn, often leads to unrepresentative models of unsaturated flow. This study focuses on determining the equivalent particle diameter, an essential factor in the Ergun equation, and uses this to derive an improved K-function. The modified K-function exhibits promising results, closely matching experimental data and outperforming other empirical methods. This improved function could enhance the description and simulation of unsaturated flows.

Computational analysis of power-law fluids for convective heat transfer in permeable enclosures using Darcy effects

Natural convection is a complex environmental phenomenon that typically occurs in engineering settings in porous structures. Shear thinning or shear thickening fluids are characteristics of power-law fluids, which are non-Newtonian in nature and find wide-ranging uses in various industrial processes. Non-Newtonian fluid flow in porous media is a difficult problem with important consequences for energy systems and heat transfer. In this paper, convective heat transmission in permeable enclosures will be thoroughly examined. The main goal is to comprehend the intricate interaction between the buoyancy-induced convection intensity, the porosity of the casing, and the fluid’s power-law rheology as indicated by the Rayleigh number. The objective is to comprehend the underlying mechanisms and identify the ideal conditions for improving heat transfer processes.The problem’s governing equations for a scientific investigation are predicated on the concepts of heat transport and fluid dynamics. The fluid flow and thermal behavior are represented using the energy equation, the Boussinesq approximation, and the Navier–Stokes equations. The continuity equation in a porous media represents the conservation of mass. Finite Element Analysis is the numerical method that is suggested for this challenging topic since it enables a comprehensive examination of the situation. The results of the investigation support several important conclusions. The power-law index directly impacts heat transmission patterns. A higher Rayleigh number indicates increased buoyancy-induced convection, which increases the heat transfer rates inside the shell. The porosity of the medium significantly affects temperature gradients and flow distribution, and it is most noticeable when permeability is present. The findings show how, in the context of porous media, these parameters have complicated relationships with one another.

The time‐asymptotic expansion for the compressible Euler equations with damping

AbstractIn 1992, Hsiao and Liu first showed that the solution to the compressible Euler equations with damping time‐asymptotically converges to the diffusion wave of the porous media equation. Geng et al. proposed a time‐asymptotic expansion around the diffusion wave , which is a better asymptotic profile than . In this paper, we rigorously justify the time‐asymptotic expansion by the approximate Green function method and the energy estimates. Moreover, the large time behavior of the solution to compressible Euler equations with damping is accurately characterized by the time‐asymptotic expansion.

Input Regularization for Integer Optimal Control in BV with Applications to Control of Poroelastic and Poroviscoelastic Systems

We revisit a class of integer optimal control problems for which a trust-region method has been proposed and analyzed in arXiv:2106.13453v3 [math.OC]. While the algorithm proposed in arXiv:2106.13453v3 [math.OC] successfully solves the class of optimization problems under consideration, its convergence analysis requires restrictive regularity assumptions. There are many examples of integer optimal control problems involving partial differential equations where these regularity assumptions are not satisfied. In this article we provide a way to bypass the restrictive regularity assumptions by introducing an additional partial regularization of the control inputs by means of mollification and proving a $\Gamma$-convergence-type result when the support parameter of the mollification is driven to zero. We highlight the applicability of this theory in the case of fluid flows through deformable porous media equations that arise in biomechanics. We show that the regularity assumptions are violated in the case of poro-visco-elastic systems, and thus one needs to use the regularization of the control input introduced in this article. Associated numerical results show that while the homotopy can help to find better objective values and points of lower instationarity, the practical performance of the algorithm without the input regularization may be on par with the homotopy.

Overcome Medical Algorithm Aversion: Conditional Joint Effect of Direct and Indirect Information

The just-ended COVID-19 pandemic and the looming global aging remind us that we need to be prepared for the shortage of doctors, which might become an urgent medical crisis in the future. Medical AI could relieve this urgency and sometimes perform better than human doctors; however, people are reluctant to trust medical AI because of algorithm aversion. Although several factors that can minimize algorithm aversion have been identified, they are not effective enough to promote medical AI as people’s first choice. Therefore, inspired by the direct and indirect information model of trust and media equation hypothesis, this research explored a new method to minimize aversion to medical AI by highlighting its social attributes. In 3 between-subject studies, a medical AI system’s direct information (i.e., transparency and quantitation of the decision-making process (DMP)) and indirect information (i.e., social proof) were manipulated. Study 1 (N = 193) and 2 (N = 429) showed that transparency of DMP and social proof increased trust in AI, but did not affect trust in human doctors. Social proof jointly affected trust in AI with non-quantitative DMP but not quantitative DMP. Study 3 (N = 184) further revealed the joint effect of the transparent non-quantitative DMP and near-perfect social proof, which could minimize algorithm aversion. These results extended the direct-indirect information model in interpersonal trust, revealed conditional media equation in human-AI trust, and offered practical implications for medical AI interface design.

Analyzing optical solitons in the generalized unstable NLSE in dispersive media

In this research, we utilize innovative methods to successfully address the challenges posed by the generalized unstable nonlinear Schrödinger equation (UNLSE) in dispersive media, yielding a rich array of solutions. Our investigation delves into a wide range of adapted exponential rational function (MERF) techniques, in addition to other unique and valuable approaches. By employing these methodologies in various scenarios, we unveil an extensive variety of wave solutions for the UNLSE. Furthermore, by satisfying specific parameter conditions, we identify novel optical solutions, encompassing both bright and dark solitons within the framework of this equation.

The time-domain Cartesian multipole expansion of electromagnetic fields

Time-domain solutions of Maxwell’s equations in homogeneous and isotropic media are paramount to studying transient or broadband phenomena. However, analytical solutions are generally unavailable for practical applications, while numerical solutions are computationally intensive and require significant memory. Semi-analytical solutions (e.g., series expansion), such as those provided by the current theoretical framework of the multipole expansion, can be discouraging for practical case studies. This paper shows how sophisticated mathematical tools standard in modern physics can be leveraged to find semi-analytical solutions for arbitrary localized time-varying current distributions thanks to the novel time-domain Cartesian multipole expansion. We present the theory, apply it to a concrete application involving the imaging of an intricate current distribution, verify our results with an existing analytical approach, and compare the proposed method to a finite-difference time-domain numerical simulation. Thanks to the concept of current “pixels” introduced in this paper, we derive time-domain semi-analytical solutions of Maxwell’s equations for arbitrary planar geometries.

Strichartz estimates for Maxwell equations in media: the partially anisotropic case

We prove Strichartz estimates for solutions to Maxwell equations in three dimensions with rough permittivities, which have less than three different eigenvalues. To this end, Maxwell equations are conjugated to half-wave equations in phase space. We use the Strichartz estimates in a known combination with energy estimates to show the new well-posedness results for quasilinear Maxwell equations.

Numerical homogenization of the gas filtration problem

The non-stationary gas flow equation in homogeneous and heterogeneous media is considered. Areas of this type are often encountered when considering gas production processes from a gas-bearing reservoir. The heterogeneous structure of the reservoir can strongly affect the extraction processes. For the numerical solution of the problem under consideration, we construct an approximation of the equations on a coarse grid using the numerical averaging method. The method allows us to solve the problem using less computational power and in a shorter time. A numerical comparison of the results of solving the model problem is carried out in a two-dimensional domain for the cases of linear and nonlinear variants of the equations, as well as in a homogeneous and heterogeneous medium. The finite element solution on a fine mesh was taken as a reference solution. The computational realization was performed using the FEniCS library. The construction of the geometric computational domain was performed in the program Gmsh. Paraview and the Matplotlib library in Python were used for data visualization.

A Score-Based Deterministic Diffusion Algorithm with Smooth Scores for General Distributions

Score matching based diffusion has shown to achieve the state of art results in generation modeling. In the original score matching based diffusion algorithm, the forward equation is a differential equation for which the probability density equation evolves according to a linear partial differential equation, the Fokker-Planck equation. A drawback of this approach is that one needs the data distribution to have a Lipschitz logarithmic gradient. This excludes a large class of data distributions that have a compact support. We present a deterministic diffusion process for which the vector fields are always Lipschitz and hence the score does not explode for probability measures with compact support. This deterministic diffusion process can be seen as a regularization of the porous media equation equation, which enables one to guarantee long term convergence of the forward process to the noise distribution. Though the porous media equation is itself not always guaranteed to have a Lipschitz vector field, it can be used to understand the closeness of the output of the algorithm to the data distribution as a function of the the time horizon and score matching error. This analysis enables us to show that the algorithm has better dependence on the score matching error than approaches based on stochastic diffusions. Using numerical experiments we verify our theoretical results on example one and two dimensional data distributions which are compactly supported. Additionally, we validate the approach on a modified MNIST data set for which the distribution is concentrated on a compact set. In each of the experiments, the approach using deterministic diffusion performs better that the diffusion algorithm with stochastic forward process, when considering the FID scores of the generated samples.

Fenomena Phubbing: Peran Teknologi Komunikasi Dalam Perubahan Interaksi Sosial

The rapid advances in communication technology have an impact on our daily lives, careers, and modes of communication. IoT (Internet of Things), e-learning, AI (Artificial Intelligence), cloud computing, mobile apps, virtual reality (VR), and augmented reality (AR) communication are some of the key aspects in the development of communication technology. These trends explain how advancements in communication technology could impact daily interactions and lifestyle choices. Even while these developments create opportunities for efficiency and connectivity, it's essential to understand and deal with the societal problems and consequences that these developments also bring with it. Phubbing has become a thing as a consequence of technology use gone wrong. "Phubbing" is the practice of someone spending excessive amounts of time, especially during social interactions, attached to an electronic gadget, such as a smartphone. The goal of this research is to determine how phubbing affects social relationships and how advancements in technology contributed to this trend. Media Dependency Theory and Media Equation Theory analysis are used to support this research, which uses a qualitative literature review approach from multiple relevant journals. The findings of this study demonstrate that the phenomenon of phubbing and its effects on social interaction are the outcome of media dependency caused by technological advancements.

Travelling modulating pulse solutions with small tails for a nonlinear wave equation in periodic media

Travelling modulating pulse solutions consist of a small amplitude pulse-like envelope moving with a constant speed and modulating a harmonic carrier wave. Such solutions can be approximated by solitons of an effective nonlinear Schrödinger equation arising as the envelope equation. We are interested in a rigorous existence proof of such solutions for a nonlinear wave equation with spatially periodic coefficients. Such solutions are quasi-periodic in a reference frame co-moving with the envelope. We use spatial dynamics, invariant manifolds, and near-identity transformations to construct such solutions on large domains in time and space. Although the spectrum of the linearised equations in the spatial dynamics formulation contains infinitely many eigenvalues on the imaginary axis or in the worst case the complete imaginary axis, a small denominator problem is avoided when the solutions are localised on a finite spatial domain with small tails in far fields.

Seismic amplitude inversion based on a new PP-wave reflection coefficient approximation equation for vertical transversely isotropic media

We develop a method for the simultaneous amplitude-variation-with-offset or -angle inversion of anisotropic parameters for transversely isotropic media with vertical axis of symmetry (VTI media). First, we introduce a nonlinear PP-wave reflection coefficient approximation equation in terms of only P- and S-wave impedances for isotropic elastic media. Then, by replacing the isotropic part of Rüger’s equation with this equation, we obtain a new PP-wave reflection coefficient approximation equation called the ASI Rüger equation for VTI media. To invert the parameters for VTI media based on the ASI Rüger equation, we adopt the Bayesian generalized linear inversion (GLI) method, a combination of GLI and Bayesian linear inversion, in which the noise and model perturbation are assumed to conform to the zero-mean Gaussian distribution. Compared with Rüger’s equation, the ASI Rüger equation decreases the trade-off between the parameters and reduces the ill-posedness of the inverse problem. The synthetic and field data tests demonstrate the feasibility of our method for inverting VTI media parameters (the vertical P-wave impedance, the vertical S-wave impedance, and Thomsen’s parameters [Formula: see text] and [Formula: see text]).

Improved numerical solution of anisotropic poroelastic wave equation in microseismicity: Graphic process unit acceleration and moment tensor implementation

AbstractThe accuracy and computational efficiency of full waveform forward modelling in poroelastic media are crucial for microseismic monitoring. It enables intuitive, precise and efficient simulation of subsurface responses, thereby improving the reliability of moment tensor inversion and seismic source mechanism interpretation. Additionally, it reflects the role of fluid effects in waveform evolution. In this paper, based on the Biot mechanism, we derived the first‐order velocity–stress equation of poroelastic media and discretized the model using a rotated staggered grid. The rotated staggered grid can well adapt to anisotropic media with high contrast parameters. We provide the finite difference formula based on graphic process unit–acceleration and moment tensor and also provide the graphic process unit workflow for forward modelling of anisotropic poroelastic media. First, two models with different grid sizes were run based on single graphic process unit, 1‐thread Central Processing Unit (CPU) and 16‐thread CPU. The results show that the speedup factors are approximately 14.3 and 3.7, respectively, compared with the running time of 1‐thread CPU and 16‐thread CPU. Then, we compare and evaluate the response of three basic source mechanisms (isotropic expansion, double couple and compensated linear vector dipole) in the model. The comparison of analytical and numerical results verifies the effectiveness of the method. Furthermore, wavefield snapshots of two typical anisotropic media (vertical transversely isotropic and horizontal transversely isotropic) are analysed to correspond to different moment tensor sources. The results showed that the source mechanism does not change the isotropic and anisotropic plane and the wave travel time, but it does change the polarization amplitude of the velocity component. The attenuation of slow qP‐wave increases along with the increase of the value of viscosity. The effect of permeability on wavefield appears with the opposite effect of viscosity. Finally, the seismic waveform differences between multi‐layer elastic media and poroelastic media are compared and analysed. The results showed that the seismic wavefield and waveform of poroelastic media are more complex, the propagation speed of seismic waves is faster, but the attenuation of seismic waves is stronger.

Stochastic Generalized Porous Media Equations Over $$\sigma $$-finite Measure Spaces with Non-continuous Diffusivity Function

Stochastic Generalized Porous Media Equations Over $$\sigma $$-finite Measure Spaces with Non-continuous Diffusivity Function

On a critical time-harmonic Maxwell equation in nonlocal media

In this paper, we study the existence of solutions for a critical time–harmonic Maxwell equation in nonlocal media \[ \begin{cases} \nabla\times(\nabla\times u)+\lambda u=\left(I_{\alpha}\ast|u|^{2^{{\ast}}_{\alpha}}\right)|u|^{2^{{\ast}}_{\alpha}-2}u & \mathrm{in}\ \Omega,\\ \nu\times u=0 & \mathrm{on}\ \partial\Omega, \end{cases} \] where $\Omega \subset \mathbb {R}^{3}$ is a bounded domain, either convex or with $\mathcal {C}^{1,1}$ boundary, $\nu$ is the exterior normal, $\lambda <0$ is a real parameter, $2^{\ast }_{\alpha }=3+\alpha$ with $0<\alpha <3$ is the upper critical exponent due to the Hardy–Littlewood–Sobolev inequality. By introducing some suitable Coulomb spaces involving curl operator $W^{\alpha,2^{\ast }_{\alpha }}_{0}(\mathrm {curl};\Omega )$ , we are able to obtain the ground state solutions of the curl–curl equation via the method of constraining Nehari–Pankov manifold. Correspondingly, some sharp constants of the Sobolev-like inequalities with curl operator are obtained by a nonlocal version of the concentration–compactness principle.

Efficient reverse time migration method in tilted transversely isotropic media based on a pure pseudoacoustic wave equation

In general, velocity anisotropy in shale media has been widely observed in laboratory and field work, which means that disregarding this characteristic can lead to inaccurate imaging locations when data are imaged with reverse time migration (RTM). Wavefields simulated with the conventional coupled pseudoacoustic wave equation may introduce S-wave noise and this equation is only valid in transversely isotropic media ([Formula: see text]). Certain decoupled qP-wave equations require the use of the pseudospectral method, which makes them computationally inefficient. To address these issues, we develop a new pure qP acoustic wave equation based on the acoustic assumption, which can be solved more efficiently using the finite-difference (FD) method. This equation can also be used in the forward-modeling process of RTM in tilted transversely isotropic (TTI) media. First, we perform a Taylor expansion of the root term in the pure qP-wave dispersion relation. This leads to an anisotropic dispersion relation that is decomposed into an elliptical anisotropic background factor and a circular correction factor. Second, we obtain the pure qP-wave equation in TTI media without a pseudodifferential operator. The new equation can be efficiently solved using FD methods and can be applied to RTM in TTI media with strong anisotropy. Our method indicates greater tolerance to numerical errors and is better suited for strong anisotropy, as compared with previously published methods. Numerical examples indicate the high kinematic and phase accuracy of our pure qP-wave equation along with its stability in TTI media characterized by ([Formula: see text]). By using a sag model and an overthrust TTI model, we determine the efficiency and accuracy of our TTI RTM.

Efficient least-squares reverse time migration in TTI media using a finite-difference solvable pure qP-wave equation

Abstract The anisotropic characteristics of underground media play a crucial role in seismic wave propagation and should be accounted for during seismic imaging. The least-squares reverse time migration (LSRTM) method achieves ideal imaging results by suppressing migration artifacts and balancing amplitude. Therefore, pure quasi-P (P-qP) wave equations in tilted transversely isotropic (TTI) media have been widely used to implement LSRTM, owing to their ability to produce stable and noise-free wavefields. However, solving the anisotropic P-qP-wave equations typically necessitates the use of spectral-based methods, making them computationally inefficient, especially in 3D applications. In this study, we first develop a P-qP-wave equation in TTI media that can be efficiently computed through the finite-difference (FD) approach. Numerical tests show that, in comparison to the previous TTI P-qP-wave equation, the newly derived FD solvable TTI P-qP-wave equation yields reasonably accurate and highly efficient wavefield simulations. Then, building on our newly derived wave equation, we derive its adjoint migration and demigration operator to implement TTI LSRTM. Two synthetic examples suggest that the newly presented P-qP-wave TTI LSRTM approach effectively correct for the anisotropy effects, providing high-quality imaging results. Additionally, our approach has superior computational efficiency over the conventional P-qP-wave TTI LSRTM technique based on a hybrid FD pseudo-spectral (HFDPS) solver.