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The authors analyze continuity equations with Stratonovich stochasticity, defined on a smooth closed Riemannian manifold M with metric h . The velocity field u is perturbed by Gaussian noise terms Ẇ 1 ( t ), …, Ẇ N ( t ) driven by smooth spatially dependent vector fields a 1 ( x ), …, a N ( x ) on M . The velocity u belongs to L t 1 W x 1,2 with div h u bounded in L t,x p for p > d + 2, where d is the dimension of M (they do not assume div h u ∈ L t,x ∞ ). For carefully chosen noise vector fields a i (and the number N of them), they show that the initial-value problem is well-posed in the class of weak L 2 solutions, although the problem can be ill-posed in the deterministic case because of concentration effects. The proof of this “regularization by noise” result is based on a L 2 estimate, which is obtained by a duality method, and a weak compactness argument.

In this paper, we examine the appearance of high-order terms in the continuity equation for an incompressible fluid obtained by L. Euler in 1752 from the linear Cauchy–Helmholtz equations. Solution of the inhomogeneous wave equation allows one calculate or estimate the intensity of vibrations and self-oscillations, which are sometimes considered spontaneous.

Impacts of solar flare vary at different parts of the lower ionosphere depending on it’s proximity to the direct exposure of incoming solar radiation. The quantitative analysis of this phenomena can be attributed to ‘solar zenith angle (χ(t))’ profile over ionosphere. We numerically solve the ‘electron continuity equation’ to obtain the lower ionospheric electron density profile (Ne(t)). The electron production rate (q(t)) is governed by the (i) X-ray profile (ϕ(t)) of the flare, (ii) χ(t)-values during the flare occurrence etc. For analyzing the X-ray profile during flares, we use the GOES-15 satellite observations. Since we’re working on electron continuity equation based simplified ionospheric model, we confined our analysis for comparatively stable mid-latitude ionosphere only. We choose three flares each from C, M and X-classes for Ne(t)-profile computation. We observe that temporal Ne(t)-profiles differ when computed for lower ionosphere over different discrete latitudes. Further, we compute the spatial Ne(t)-profile across mid-latitude at the time when ϕ(t)=ϕmax. Now we assume that, these flares repeat themselves every day of a year (DoY) at the same time of a day and we compute Ne(t)-profiles for each day. We found a seasonal effect on Ne(t)-profile due to solar flare. Further, we investigate the response time delay (Δt) of the lower ionosphere, which is the time difference between incidence of X-ray and the respective change in Ne(t)-profiles during solar flares. Strong seasonal effects on Ne(t)-profile and Δt are the unique results of this work.

This paper presents a groundbreaking numerical technique for solving nonlinear time fractional differential equations, combining the conformable continuity equation (CCE) with the Non-Polynomial Spline (NPS) interpolation to address complex mathematical challenges. By employing conformable descriptions of fractional derivatives within the CCE framework, our method ensures enhanced accuracy and robustness when dealing with fractional order equations. To validate our approach’s applicability and effectiveness, we conduct a comprehensive set of numerical examples and assess stability using the Fourier method. The proposed technique demonstrates unconditional stability within specific parameter ranges, ensuring reliable performance across diverse scenarios. The convergence order analysis reveals its efficiency in handling complex mathematical models. Graphical comparisons with analytical solutions substantiate the accuracy and efficacy of our approach, establishing it as a powerful tool for solving nonlinear time-fractional differential equations. We further demonstrate its broad applicability by testing it on the Burgers–Fisher equations and comparing it with existing approaches, highlighting its superiority in biology, ecology, physics, and other fields. Moreover, meticulous evaluations of accuracy and efficiency using (L2 and L∞) norm errors reinforce its robustness and suitability for real-world applications. In conclusion, this paper presents a novel numerical technique for nonlinear time fractional differential equations, with the CCE and NPS methods’ unique combination driving its effectiveness and broad applicability in computational mathematics, scientific research, and engineering endeavors.

This work explores explicit reduced-order modeling of linear and nonlinear joint interface effects on the overall dynamic response of free-free and cantilever lap-jointed beam structures. Joint interfaces are a significant source of uncertainties in assembled structures due to their inherent nonlinearities. These nonlinearities create amplitude-dependent frequency and damping characteristics in the system’s response. The dynamic variability of jointed structures and their wide range of use makes it necessary to model joint interface characteristics accurately and simply. Additionally, it is desired to define joint parameters that are intuitive and easy to tune to realistic systems. To accomplish this, the linear and nonlinear joint interface effects will be explicitly modeled with various springs and dampers at a localized point along the beam. The assembled beams are modeled with the Euler–Bernoulli beam theory, while the joint effects are considered by partitioning the beam and implementing the damping and stiffness coefficients into the continuity equations. The influence of the modeled joint interface effects on the system’s response, linear properties and nonlinear trends is explored and discussed. This work develops a localized joint model which is beneficial to qualitatively understand the joint interface effects on the system’s linear and nonlinear properties. However, a distributed joint model with area-dependent forces is needed to quantitatively simulate the interface kinematics and better model the characteristics of the assembled lap-joint beam structure.

Numerical models for calculating bottom morphology are increasingly popular because of their long-term forecasting capabilities as well as their ability to identify causes. However, the simulation of bottom morphology, in addition to natural factors, is also governed by economic activities, especially sand mining. In this paper, we propose the development of the numerical model in curvilinear coordinates combined with a sand mining component that simulates the bottom morphology of the Tien river segment in the Mekong Delta as a study area. The modeling theory relies on the Reynolds equation system, which is combined with the suspended sediment transport equation in a two-dimensional curvilinear coordinate system. The process of averaging is performed over the depth. The suspended sediment transport equation incorporates a source function that describes the upward or downward movement of particles. In the model for bottom morphology, a component for sand mining is included, which accounts for the rate at which sand is extracted from the riverbed. This component is integrated into the equation governing the continuity of the bed load. The findings demonstrate that the numerical model effectively captures the changes in the river’s bottom morphology resulting from sand mining. Specifically, when the sand mining component is employed, the model accurately represents the actual development of the bottom morphology in river segments where sand mining occurs. Furthermore, the downstream bed alterations are significantly influenced by sand mining activities. By incorporating the sand mining component into the model, it becomes a valuable tool for simulating the bottom morphology in river segments subjected to sand mining, thus aiding in sand mining planning and the management of disaster risks associated with bank erosion.

The paper addresses an optimal ensemble control problem for nonlocal continuity equations on the space of probability measures. We admit the general nonlinear cost functional, and an option to directly control the nonlocal terms of the driving vector field. For this problem, we design a descent method based on Pontryagin’s maximum principle (PMP). To this end, we derive a new form of PMP with a decoupled Hamiltonian system. Specifically, we extract the adjoint system of linear nonlocal balance laws on the space of signed measures and prove its well-posedness. As an implementation of the designed descent method, we propose an indirect deterministic numeric algorithm with backtracking. We prove the convergence of the algorithm and illustrate its modus operandi by treating a simple case involving a Kuramoto-type model of a population of interacting oscillators.

We consider the Cauchy problem for the continuity equation in space dimension d ≥ 2 . We construct a divergence-free velocity field uniformly bounded in all Sobolev spaces W 1 , p , for 1 ≤ p < ∞ , and a smooth compactly supported initial datum such that the unique solution to the continuity equation with this initial datum and advecting field does not belong to any Sobolev space of positive fractional order at any positive time. We also construct velocity fields in W r , p , with r > 1 , and solutions of the continuity equation with these velocities that exhibit some loss of regularity, as long as the Sobolev space W r , p does not embed in the space of Lipschitz functions. Our constructions are based on examples of optimal mixers from the companion paper Exponential self-similar mixing by incompressible flows (Alberti et al. in J Am Math Soc 32(2):445–490, 2019 ), and have been announced in Exponential self-similar mixing and loss of regularity for continuity equations (Alberti et al. in Comptes Rendus Math Acad Sci Paris 352(11):901–906, 2014 ).

In this study, a bivariable coupling model for river channel routing is presented. The proposed model is developed from the Priessmann 4-point implicit differential scheme with a weight coefficient of river flow continuity equation. It is based on the transformation of two different expression forms of river channel storage equation. Furthermore, we consider the impact of lateral inflow along the study river channel from another perspective. In this paper we deduct lateral inflow from the lower section instead of adding lateral inflow to the upper section. In order to be representative of geographical range, river channel characteristics, flood magnitude, hydraulic characteristics and time, the proposed model is tested in 38 river channels of 6 river systems in China by using observed data during flood season. The rationality of model structure and the validity of model simulation are examined comprehensively. Comparison between the proposed model and Muskingum model shows that the proposed model can improve the simulation accuracy. The results show that the simulation accuracy and stability of the bivariable coupling model is much better than that of the Muskingum model.

A mathematical framework and accompanying numerical algorithm exploiting the continuity equation for 4D reconstruction of spatiotemporal attenuation fields from multi-angle full-field transmission measurements is presented. The algorithm is geared towards rotation-free dynamic multi-beam X-ray tomography measurements, for which angular information is sparse but the temporal information is rich. 3D attenuation maps are recovered by propagating an initial discretized density volume in time according to the advection equations using the Finite Volumes method with a total variation diminishing monotonic upstream-centered scheme (TVDMUSCL). The benefits and limitations of the algorithm are explored using dynamic granular system phantoms modelled via discrete elements and projected by an analytical ray model independent from the numerical ray model used in the reconstruction scheme. Three phantom scenarios of increasing complexity are presented and it is found that projections from only a few (unknowns:equations > 10) angles can be sufficient for characterisation of the 3D attenuation field evolution in time. It is shown that the artificial velocity field produced by the algorithm sub-iteration, which is used to propagate the attenuation field, can to some extent approximate the true kinematics of the system. Furthermore, it is found that the selection of a temporal interpolation scheme for projection data can have a significant impact on error build up in the reconstructed attenuation field.