Scalar Partial Differential Equations Research Articles

Modeling viscous dissipation in MHD boundary layer flow: A two-stage in time and compact in space finite difference approach

It is suggested that an explicit-implicit third-order in-time numerical scheme be used to solve time-dependent partial differential equations (PDEs). A fourth-order compact scheme in space is adopted for discretizing space variables. The work creatively blends a third-order explicit-implicit numerical scheme in time with a fourth-order compact scheme in space to solve time-dependent PDEs. Notably, the proposed numerical scheme demonstrates stability for scalar PDEs and exhibits convergence for both linear and nonlinear PDE systems. In addition, a mathematical model is given for the heat transfer of magnetohydrodynamic (MHD) flow over flat and oscillatory sheets, and the effect of viscous dissipation is also considered. For various governing parameters, features of the flow and heat transfer characteristics are discussed and examined. Some results are verified using an exact solution. We use an explicit-implicit finite difference method guaranteed to be stable and convergent under all conditions to solve the coupled nonlinear partial differential equations that describe this problem. Physical aspects of the problem are discussed in light of the system’s embedded parameters, and numerical and graphical results are presented. To sum up, progress in our knowledge of such systems relies heavily on creating an accurate and efficient numerical scheme for modeling MHD boundary layer flows while considering the influence of viscous dissipation. This study aims to improve the current state of MHD flow simulations and provide a useful resource for engineers and scientists engaged in developing and improving MHD-based technologies.

Rigorous higher-order Poincaré optical vortex modes

The state of polarization of a general form of an optical vortex mode is represented by the vector ϵ^ m , which is associated with a vector light mode of order m>0. It is formed as a linear combination of two product terms involving the phase functions e±imϕ times the optical spin unit vectors σ∓. Any such state of polarization corresponds to a unique point (Θ P ,Φ P ) on the surface of the order m unit Poincaré sphere. However, albeit a key property, the general form of the vector potential in the Lorenz gauge A=ϵ^ m Ψ m , from which the fields are derived, including the longitudinal fields, has neither been considered nor has had its consequences been explored. Here, we show that the spatial dependence of Ψ m can be found by rigorously demanding that the product ϵ^ m Ψ m satisfies the vector paraxial equation. For a given order m this leads to a unique Ψ m , which has no azimuthal phase of the kind e i ℓ ϕ , and it is a solution of a scalar partial differential equation with ρ and z as the only variables. The theory is employed to evaluate the angular momentum for a general Poincaré mode of order m yielding the angular momentum for right- and left- circularly polarized, elliptically polarized, linearly polarized and radially and azimuthally polarized higher-order modes. We find that in applications involving Laguerre–Gaussian modes, only the modes of order m≥2 have non-zero angular momentum. All modes have zero angular momentum for points on the equatorial circle for which cos⁡Θ P =0.

A two-stage multi-step numerical scheme for mixed convective Williamson nanofluid flow over flat and oscillatory sheets

The novelty of this contribution is to propose an implicit numerical scheme for solving time-dependent boundary layer problems. The scheme is multi-step and consists of two stages. It is third-order accurate in time and constructed on three-time levels. For spatial discretization, a fourth-order compact scheme is adopted. The stability of the proposed scheme is analyzed for scalar linear partial differential equation (PDE) that shows its conditional stability. The convergence of the scheme is also provided for a system of time-dependent parabolic equations. Moreover, a mathematical model for heat and mass transfer of mixed convective Williamson nanofluid flow over flat and oscillatory sheets is modified with the characteristic of the Darcy–Forchheimer model. The results show that the temperature profile rises by developing thermophoresis and Brownian motion parameter values. Also, the proposed scheme is compared with an existing Crank–Nicolson method. It is found that the proposed scheme converges faster than the existing one for solving scalar linear PDE as well as the system of linear and nonlinear parabolic equations, which are dimensionless forms of governing equations of a flow phenomenon. The findings provided in this study can serve as a helpful guide for future investigations into fluid flow in closed-off industrial settings.

Design of Sturm global attractors 1: Meanders with three noses, and reversibility.

We systematically explore a simple class of global attractors, called Sturm due to nodal properties, for the semilinear scalar parabolic partial differential equation (PDE) ut=uxx+f(x,u,ux) on the unit interval 0<x<1, under Neumann boundary conditions. This models the interplay of reaction, advection, and diffusion. Our classification is based on the Sturm meanders, which arise from a shooting approach to the ordinary differential equation boundary value problem of equilibrium solutions ut=0. Specifically, we address meanders with only three "noses," each of which is innermost to a nested family of upper or lower meander arcs. The Chafee-Infante paradigm, with cubic nonlinearity f=f(u), features just two noses. Our results on the gradient-like global PDE dynamics include a precise description of the connection graphs. The edges denote PDE heteroclinic orbits v1⇝v2 between equilibrium vertices v1,v2 of adjacent Morse index. The global attractor turns out to be a ball of dimension d, given as the closure of the unstable manifold Wu(O) of the unique equilibrium with maximal Morse index d. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graph indicates time reversibility on the (d-1)-sphere boundary of the global attractor.

Partial Euler operators and the efficient inversion of Div

AbstractThe problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems have conservation laws of arbitrarily high order that must be found with the aid of computer algebra. Even low-order conservation laws of complex systems can be hard to find and invert. This paper describes a new, efficient approach to the inversion problem. Two main tools are developed: partial Euler operators and partial scalings. These lead to a line integral formula for the inversion of a total derivative and a procedure for inverting a given total divergence concisely.

Finite difference schemes for MHD mixed convective Darcy–forchheimer flow of Non-Newtonian fluid over oscillatory sheet: A computational study

This contribution proposes two third-order numerical schemes for solving time-dependent linear and non-linear partial differential equations (PDEs). For spatial discretization, a compact fourth-order scheme is deliberated. The stability of the proposed scheme is set for scalar partial differential equation, whereas its convergence is specified for a system of parabolic equations. The scheme is applied to linear scalar partial differential equation and non-linear systems of time-dependent partial differential equations. The non-linear system comprises a set of governing equations for the heat and mass transfer of magnetohydrodynamics (MHD) mixed convective Casson nanofluid flow across the oscillatory sheet with the Darcy–Forchheimer model, joule heating, viscous dissipation, and chemical reaction. It is noted that the concentration profile is escalated by mounting the thermophoresis parameter. Also, the proposed scheme converges faster than the existing Crank-Nicolson scheme. The findings that were provided in this study have the potential to serve as a helpful guide for investigations into fluid flow in closed-off industrial settings in the future.

A result of convergence for a mono-dimensional two-velocities lattice Boltzmann scheme

We consider a mono-dimensional two-velocities scheme used to approximate the solutions of a scalar hyperbolic conservative partial differential equation. We prove the convergence of the discrete solution towards the unique entropy solution by first estimating the supremum norm and the total variation of the discrete solution, and second by constructing a discrete kinetic entropy-entropy flux pair being given a continuous entropy-entropy flux pair of the hyperbolic system. We finally illustrate our results with numerical simulations of the advection equation and the Burgers equation.

Grassmannian Flows and Applications to Nonlinear Partial Differential Equations

We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto a coordinate chart of the underlying Grassmann manifold. Our method relies on extending this idea to the infinite dimensional case. The key is an integral equation analogous to the Marchenko equation in integrable systems, that represents the coodinate chart map. We show explicitly how to generate such solutions to scalar partial differential equations of arbitrary order with nonlocal quadratic nonlinearities using our approach. We provide numerical simulations that demonstrate the generation of solutions to Fisher--Kolmogorov--Petrovskii--Piskunov equations with nonlocal nonlinearities. We also indicate how the method might extend to more general classes of nonlinear partial differential systems.

Exact nonclassical symmetry solutions of Lotka–Volterra-type population systems

AbstractNew classes of conditionally integrable systems of nonlinear reaction–diffusion equations are introduced. They are obtained by extending a well-known nonclassical symmetry of a scalar partial differential equation to a vector equation. New exact solutions of nonlinear predator–prey systems with cross-diffusion are constructed. Infinite dimensional classes of exact solutions are made available for such nonlinear systems. Some of these solutions decay towards extinction and some oscillate or spiral around an interior fixed point. It is shown that the conditionally integrable systems are closely related to the standard diffusive Lotka–Volterra system, but they have additional features.

Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh–Nagumo equations

We investigate the stability of traveling-pulse solutions to the stochastic FitzHughNagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a scalar stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a system of a scalar stochastic partial differential equation (SPDE) coupled to a scalar ordinary differential equation (ODE). This approach has been recently employed by Kr\"uger and Stannat for scalar stochastic bistable reaction-diffusion equations such as the Nagumo equation. A main difference in our situation of an SPDE coupled to an ODE is that the linearization has essential spectrum parallel to the imaginary axis and thus only generates a strongly continuous semigroup. Furthermore, the linearization around the traveling wave is not self-adjoint anymore, so that fluctuations around the wave cannot be expected to be orthogonal in a corresponding inner product. We demonstrate that this problem can be overcome by making use of Riesz instead of orthogonal spectral projections as recently employed in a series of papers by Hamster and Hupkes in case of analytic semigroups. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs with linearizations only generating a strongly continuous semigroup. This provides a relevant generalization as these systems are prevalent in many applications.

Model-Based Optimal Adaptive Monitoring of Oil Spills

This article presents a novel model-based adaptive monitoring framework for the estimation of oil spills using mobile sensors. In the first of a four-stage process, simulation of a combined ocean, wind, and oil model provides a state trajectory over a finite time horizon, used in the second stage to solve an adjoint optimization problem for sensing locations. In the third stage, a reduced-order model is identified from the state trajectory, utilized alongside measurements to produce smoothed state estimates in the fourth stage, which update and re-initialize the first-stage simulation. In the second stage, sensors are directed to optimal sensing locations using the solution of a partial differential equation (PDE)-constrained optimization problem. This problem formulation represents a key contributory idea, utilizing the definition of spill uncertainty as a scalar PDE to be minimized subject to sensor, ocean, wind, and oil constraints. Spill uncertainty is a function of uncertainty in 1) the bespoke model of the ocean, wind, and oil spill; 2) the reduced order model identified from sensor data; and 3) the data assimilation method employed to estimate the states of the environment and spill. The uncertainty minimization is spatiotemporally weighted by a function of spill probability and information utility, prioritizing critical measurements. A numerical case study spanning a 2500-km <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> coastal area is presented, with four mobile sensors arriving 12 h after an oil leak. Compared to industry standard “ladder pathing,” the proposed method achieves an 80% reduction in oil distribution error and a 62% reduction in sensor distance traveled.

Extending FEniCS to work in higher dimensions using tensor product finite elements

We present a method to extend the finite element library FEniCS to solve problems with domains in dimensions above three by constructing tensor product finite elements. This methodology only requires that the high dimensional domain is structured as a Cartesian product of two lower dimensional subdomains. In this study we consider Dirichlet problems for scalar linear partial differential equations, though the methodology can be extended to non-linear problems. The utilization of tensor product finite elements allows us to construct a global system of linear algebraic equations that only relies on the finite element infrastructure of the lower dimensional subdomains contained in FEniCS. We demonstrate the effectiveness of our methodology in four distinctive test cases. The first test case is a Poisson equation posed in a four dimensional domain which is a Cartesian product of two unit squares solved using the classical Galerkin finite element method. The second test case is the wave equation in space–time, where the computational domain is a Cartesian product of a two dimensional space grid and a one dimensional time interval. In this second case we also employ the Galerkin method. The third test case is an advection dominated advection–diffusion equation where the global domain is a Cartesian product of two one dimensional intervals in which the streamline upwind Petrov–Galerkin method is applied to ensure discrete stability. The final test case uses the Galerkin approach to solve a Poisson problem on a Cartesian product of two intervals with a spatially varying, non-separable diffusivity term In all cases, a p=1 basis is used and optimal L2 convergence rates of order hp+1 of the errors are achieved with respect to h refinement.

Output‐feedback boundary adaptive fault‐tolerant control for scalar hyperbolic partial differential equation systems with actuator faults

SummaryThis article studies the output‐feedback adaptive fault‐tolerant boundary control problem for scalar hyperbolic partial differential equation systems with actuator faults. All the coefficients of the controlled plant are unknown, and two types of actuator faults, that is, multiplicative faults and additive faults, are considered simultaneously. For the state estimation problem, two filters are constructed, based on which an observer is obtained to estimate the system state. A parametric model is established for actuator faults, based on which the parameter updating laws of gradient type are then developed to identify actuator faults and to estimate the unknown system coefficients. With the observer and the parameter updating laws, an output‐feedback adaptive fault‐tolerant boundary control law is developed via infinite‐dimensional backstepping method. The boundness of all the signals involved in the control design is guaranteed and the convergence of system states is also confirmed. Finally, the simulation results are given to testify the effectiveness of the proposed control scheme.

Soliton based director deformation in a twist grain boundary liquid crystal

We investigate the director dynamics of a twist grain boundary liquid crystal under the one constant approximation for the different elastic constants representing the various deformation present in the liquid crystal medium. The free energy density is deduced to a higher-order vector nonlinear partial differential equation by balancing the torque experienced by the nematic molecules under a viscous field and the molecular field arises due to the presence of elastic constants. Upon employing the stereographic projection method we further reduced the vector nonlinear differential equation into a complex scalar nonlinear partial differential equation. We obtain a series of localized solutions for the complex scalar nonlinear partial differential equation through the standard tanh method.

The mean-field approximation for higher-dimensional Coulomb flows in the scaling-critical L ∞ space

In the mean-field scaling regime, a first-order system of particles with binary interactions naturally gives rise to a scalar partial differential equation (PDE), which, depending on the nature of the interaction, corresponds to either the Hamiltonian or gradient flow of the effective energy of the system for a large number of particles. The empirical measure of such systems is a weak solution to this limiting mean-field PDE, and one expects that as the number of particles tends to infinity, it converges along its lifespan in the weak-* sense to a more regular solution of the PDE, provided it does so initially. Much effort has been invested over the years in proving and quantifying this convergence under varying regularity assumptions. When the interaction potential is Coulomb, the mean-field PDE has a scaling invariance which leaves the L ∞ norm unchanged; i.e., L ∞ is a critical function space for the equation. Moreover, the L ∞ norm is either conserved or decreasing, and the equation is globally well-posed in this space, making it a natural choice for studying the dynamics. Building on our previous work (Rosenzweig 2022 Arch. Ration. Mech. Anal. 243 1361–431) for point vortices (i.e. d = 2), we prove quantitative convergence of the empirical measure to the L ∞ solution of the mean-field PDE for short times in dimensions d ⩾ 3. To the best of our knowledge, this is the first such work outside of the 2D case. Our proof is based on a combination of the modulated-energy method of Serfaty (2020 Duke Math. J. 169 2887–935) and a novel mollification argument first introduced by the author in Rosenzweig (2022 Arch. Ration. Mech. Anal. 243 1361–431). Compared to our prior work (Rosenzweig 2022 Arch. Ration. Mech. Anal. 243 1361–431), the new challenge is the non-logarithmic nature of the potential.

Leibniz-type rule of variable-order fractional derivative and application to build Lie symmetry framework

In this paper, we first study some properties of the variable-order fractional derivative defined by the Caputo fractional derivative and particularly present a Leibniz-type rule, which makes the variable-order fractional derivative to be expressed as an infinite sum of integer-order derivatives. Then we use such properties to build a Lie symmetry framework for a class of scalar variable-order fractional partial differential equations and show that such type of equations has an elegant symmetry structure, which facilitates us to explore the symmetry properties. By means of the Lie symmetry framework and the symmetry structure, we perform a Lie symmetry classification of the variable-order fractional diffusion equations and construct the corresponding symmetry reductions and certain exact solutions.

Future prevalence of type 2 diabetes-A comparative analysis of chronic disease projection methods.

Accurate projections of the future number of people with chronic diseases are necessary for effective resource allocation and health care planning in response to changes in disease burden. To introduce and compare different projection methods to estimate the number of people with diagnosed type 2 diabetes (T2D) in Germany in 2040. We compare three methods to project the number of males with T2D in Germany in 2040. Method 1) simply combines the sex- and age-specific prevalence of T2D in 2010 with future population distributions projected by the German Federal Statistical Office (FSO). Methods 2) and 3) additionally account for the incidence of T2D and mortality rates using partial differential equations (PDEs). Method 2) models the prevalence of T2D employing a scalar PDE which incorporates incidence and mortality rates. Subsequently, the estimated prevalence is applied to the population projection of the FSO. Method 3) uses a two-dimensional system of PDEs and estimates future case numbers directly while future mortality of people with and without T2D is modelled independently from the projection of the FSO. Method 1) projects 3.6 million male people with diagnosed T2D in Germany in 2040. Compared to 2.8 million males in 2010, this equals an increase by 29%. Methods 2) and 3) project 5.9 million (+104% compared to 2010) and 6.0 million (+116%) male T2D patients, respectively. The results of the three methods differ substantially. It appears that ignoring temporal trends in incidence and mortality may result in misleading projections of the future number of people with chronic diseases. Hence, it is essential to include these rates as is done by method 2) and 3).

Modified equation based mesh adaptation algorithm for evolutionary scalar partial differential equations

AbstractIt is well known that on uniform mesh classical higher order schemes for evolutionary problems yield an oscillatory approximation of the solution containing discontinuity or boundary layers. In this article, an entirely new approach for constructing locally adaptive mesh is given to compute nonoscillatory solution by representative “second” order schemes. This is done using modified equation analysis and a notion of data dependent stability of schemes to identify the solution regions for local mesh adaptation. The proposed algorithm is applied on scalar problems to compute the solution with discontinuity or boundary layer. Presented numerical results show underlying second order schemes approximate discontinuities and boundary layers without spurious oscillations.

Local defect-correction method based on multilevel discretization for Steklov eigenvalue problem

In this paper, we propose a local defect-correction method for solving the Steklov eigenvalue problem arising from the scalar second order positive definite partial differential equations based on the multilevel discretization. The objective is to avoid solving large-scale equations especially the large-scale Steklov eigenvalue problem whose computational cost increases exponentially. The proposed algorithm transforms the Steklov eigenvalue problem into a series of linear boundary value problems, which are defined in a multigrid space sequence, and a series of small-scale Steklov eigenvalue problems in a coarse correction space. Furthermore, we use the local defect-correction technique to divide the large-scale boundary value problems into small-scale subproblems. Through our proposed algorithm, we avoid solving large-scale Steklov eigenvalue problems. As a result, our proposed algorithm demonstrates significantly improved the solving efficiency. Additionally, we conduct numerical experiments and a rigorous theoretical analysis to verify the effectiveness of our proposed approach.

Frame-independent vector-cloud neural network for nonlocal constitutive modeling on arbitrary grids

Constitutive models are widely used for modeling complex systems in science and engineering, where first-principle-based, well-resolved simulations are often prohibitively expensive. For example, in fluid dynamics, constitutive models are required to describe nonlocal, unresolved physics such as turbulence and laminar–turbulent transition. However, traditional constitutive models based on partial differential equations (PDEs) often lack robustness and are too rigid to accommodate diverse calibration datasets. We propose a frame-independent, nonlocal constitutive model based on a vector-cloud neural network that can be learned with data. The model predicts the closure variable at a point based on the flow information in its neighborhood. Such nonlocal information is represented by a group of points, each having a feature vector attached to it, and thus the input is referred to as vector cloud. The cloud is mapped to the closure variable through a frame-independent neural network, invariant both to coordinate translation and rotation and to the ordering of points in the cloud. As such, the network can deal with any number of arbitrarily arranged grid points and thus is suitable for unstructured meshes in fluid simulations. The merits of the proposed network are demonstrated for scalar transport PDEs on a family of parameterized periodic hill geometries. The vector-cloud neural network is a promising tool not only as nonlocal constitutive models and but also as general surrogate models for PDEs on irregular domains.