Form Of Partial Differential Equations Research Articles

On hierarchical competition through reduction of individual growth

We consider a population organised hierarchically with respect to size in such a way that the growth rate of each individual depends only on the presence of larger individuals. As a concrete example one might think of a forest, in which the incidence of light on a tree (and hence how fast it grows) is affected by shading by taller trees. The classic formulation of a model for such a size-structured population employs a first order quasi-linear partial differential equation equipped with a non-local boundary condition. However, the model can also be formulated as a delay equation, more specifically a scalar renewal equation, for the population birth rate. After discussing the well-posedness of the delay formulation, we analyse how many stationary birth rates the equation can have in terms of the functional parameters of the model. In particular we show that, under reasonable and rather general assumptions, only one stationary birth rate can exist besides the trivial one (associated to the state in which there are no individuals and the population birth rate is zero). We give conditions for this non-trivial stationary birth rate to exist and analyse its stability using the principle of linearised stability for delay equations. Finally, we relate the results to the alternative, partial differential equation formulation of the model.

Disentangling Stochastic PDE Dynamics for Unsupervised Video Prediction.

Unsupervised video prediction aims to predict future outcomes based on the observed video frames, thus removing the need for supervisory annotations. This research task has been argued as a key component of intelligent decision-making systems, as it presents the potential capacities of modeling the underlying patterns of videos. Essentially, the challenge of video prediction is to effectively model the complex spatiotemporal and often uncertain dynamics of high-dimensional video data. In this context, an appealing way of modeling spatiotemporal dynamics is to explore prior physical knowledge, such as partial differential equations (PDEs). In this article, considering real-world video data as a partly observed stochastic environment, we introduce a new stochastic PDE predictor (SPDE-predictor), which models the spatiotemporal dynamics by approximating a generalized form of PDEs while dealing with the stochasticity. A second contribution is that we disentangle the high-dimensional video prediction into low-level dimensional factors of variations: time-varying stochastic PDE dynamics and time-invariant content factors. Extensive experiments on four various video datasets show that SPDE video prediction model (SPDE-VP) outperforms both deterministic and stochastic state-of-the-art methods. Ablation studies highlight our superiority driven by both PDE dynamics modeling and disentangled representation learning and their relevance in long-term video prediction.

Theoretical Study of the Effect of Permeability Tensor upon Drainage of Soils

the influence of permeability tensor upon drainage of anisotropic soils under ponded water and steady recharge (rainfall) is theoretically investigated. Tensorial permeability has led to the formulation of mixed type partial differential equations. Since there is no analytical solution to this problem, the formulation is therefore solved numerically by the method of finite elements. The finite element formulation is implemented into a computer model which can be applied to any problem of seepage under steady state conditions. Two different example problems representing two different flow conditions under full anisotropy have been studied. Results of the model for the isotropic case were checked against exact mathematical solutions derived analytically for isotropic soils and found to be accurate which indicates that using this model for anisotropic soils is safe and sound

Bifurcations and Early-Warning Signs for SPDEs with Spatial Heterogeneity

AbstractBistability is a key property of many systems arising in the nonlinear sciences. For example, it appears in many partial differential equations (PDEs). For scalar bistable reaction-diffusions PDEs, the bistable case even has taken on different names within communities such as Allee, Allen-Cahn, Chafee-Infante, Nagumo, Ginzburg-Landau,$$\Phi ^4$$Φ4, Schlögl, Stommel, just to name a few structurally similar bistable model names. One key mechanism, how bistability arises under parameter variation is a pitchfork bifurcation. In particular, taking the pitchfork bifurcation normal form for reaction-diffusion PDEs is yet another variant within the family of PDEs mentioned above. More generally, the study of this PDE class considering steady states and stability, related to bifurcations due to a parameter is well-understood for the deterministic case. For the stochastic PDE (SPDE) case, the situation is less well-understood and has been studied recently. In this paper we generalize and unify several recent results for SPDE bifurcations. Our generalisation is motivated directly by applications as we introduce in the equation a spatially heterogeneous term and relax the assumptions on the covariance operator that defines the noise. For this spatially heterogeneous SPDE, we prove a finite-time Lyapunov exponent bifurcation result. Furthermore, we extend the theory of early warning signs in our context and we explain, the role of universal exponents between covariance operator warning signs and the lack of finite-time Lyapunov uniformity. Our results are accompanied and cross-validated by numerical simulations.

Effects of radiation and heat generation for non‐Newtonian fluid flow over slendering stretching sheet: Numerically

AbstractA numerical analysis of MHD 3D second grade fluid over slendering stretching sheet is presented in this study. In addition, heat generation and thermal radiation impacts are included to inspect the heat transfer mechanism. Using the boundary layer approximation, the problem is stated mathematically in the form of PDEs (partial differential equations), which are altered to nonlinear ODEs (ordinary differential equations) by invoking appropriate similarity variables. Solutions of these nonlinear equations are computed by MATLAB bvp4c function. Graphs for involving parameters are plotted to view their impacts on velocity and temperature. Impression of numerous parameters on Skin friction and Nusselt number are also depicted in tables and the results are discussed in detail. The computed values of skin friction revealed constant behavior and Nusselt number depicted increasing performance due to boosting values of Prandtl number Pr. The computed values of skin friction revealed constant behavior and Nusselt number depicted reducing performance due to boosting values of heat generation parameter Q. The computed values of skin friction revealed constant behavior and Nusselt number depicted reducing performance due to boosting values of radiation parameter Rd.

Agrawal nanofluid flow towards a stagnation point past a moving disk with smoluchowski temperature and Maxwell velocity slip boundary conditions: The case of Buongiorno's model

AbstractNanofluid is a novel heat transfer fluid with the ability to significantly improve the heat transfer efficiency of regular fluids. Many efforts have been made to study its viscosity and thermal conductivity, both of which are crucial thermophysical properties. This work reports the significance of nanofluid along with thermophoretic and Brownian motion phenomena on the Agrawal flow towards a stagnation point past a permeable moving disk. The impacts of the Maxwell velocity slip and Smoluchowski temperature conditions are also incorporated. The leading equations in form of PDEs (Partial differential equations) are simplified into ODEs (Ordinary differential equations) by employing similarity theory. Then these simplified equations are solved numerically by utilizing a bvp4c solver (fourth‐order boundary value problem).The phenomenal effects of the relevant embedded distinguished parameters on the shear stress, rate of mass, and heat transfer are demonstrated through different graphs and tables. Graphical results indicate that double solutions are detected for stretching/shrinking parameter in a certain range. The heat transport and the wall drag force ascend with the suction aspect, while the mass transfer rate drops with suction in both solutions.

A one-shot overlapping Schwarz method for component-based model reduction: application to nonlinear elasticity

We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations (PDEs) based on overlapping subdomains. Our approach reads as a constrained optimization statement that penalizes the jump at the components’ interfaces subject to the approximate satisfaction of the PDE in each local subdomain. Furthermore, the approach relies on the decomposition of the local states into a port component – associated with the solution on interior boundaries – and a bubble component that vanishes at ports: since the bubble components are uniquely determined by the solution value at the corresponding port, we can recast the constrained optimization statement into an unconstrained statement, which reads as a nonlinear least-squares problem and can be solved using the Gauss–Newton method. We present thorough numerical investigations for a two-dimensional neo-Hookean nonlinear mechanics problem to validate our method; we further discuss the well-posedness of the mathematical formulation and the a priori error analysis for linear coercive problems.

On spectrally flowed local vertex operators in AdS$_3$

We provide a novel local definition for spectrally flowed vertex operators in the SL(2,\mathbb{R})SL(2,ℝ)-WZW model, generalising the proposal of Maldacena and Ooguri in [arXiv:hep-th/0111180] for the singly-flowed case to all \omega>1ω>1. This allows us to establish the precise connection between the computation of correlators using the so-called spectral flow operator, and the methods introduced recently by Dei and Eberhardt in [arXiv:2105.12130] based on local Ward identities. We show that the auxiliary variable yy used in the latter paper arises naturally from a point-splitting procedure in the space-time coordinate. The recursion relations satisfied by spectrally flowed correlators, which take the form of partial differential equations in yy-space, then correspond to null-state conditions for generalised spectral flowed operators. We highlight the role of certain SL(2,\mathbb{R})SL(2,ℝ) discrete module isomorphisms in this context, and prove the validity of the conjecture put forward in [arXiv:2105.12130] for yy-space structure constants of three-point functions with arbitrary spectral flow charges.

Mastering the Cahn–Hilliard equation and Camassa–Holm equation with cell-average-based neural network method

In this paper, we develop cell-average-based neural network (CANN) method to approximate solutions of nonlinear Cahn–Hilliard equation and Camassa–Holm equation. The CANN method is motivated by the finite volume scheme and evolved from the integral or weak formulation of partial differential equations. The major idea of cell-average-based neural network method is to explore a neural network to approximate the solution average difference or evolution between two neighboring time steps. Unlike traditional numerical methods, CANN method is not limited by the CFL restriction and can adapt large time steps for solution evolution, which is a significant advantage that classical numerical methods do not have. Once well trained, this method can be implemented as an fixed explicit finite volume scheme and applied to certain groups of initial value conditions for Cahn–Hilliard equation and Camassa–Holm equation without retraining the neural network. Furthermore, the CANN method also performs very well in handling data with corruption or low-quality data generated by Gaussian white noise. Numerical examples are presented to demonstrate effectiveness, accuracy and capability of the proposed method.

Non-Parametric Belief Propagation Solver for Stochastic Systems of Linear Equations

The striking growth of powerful computing resources allows time-efficient solution of computationally demanding problems. In particular, advances in high-performance computing have made stochastic approaches to real-world applications more practical. The belief propagation (BP) algorithm is a probabilistic method typically used in information theory and artificial intelligence. This article exploits the probabilistic message passing attribute of BP for solving stochastic linear systems that naturally arise from finite element formulation of stochastic partial differential equations (PDEs), establishing an explicit connection between the two fields for the first time. The accuracy of the algorithm is validated by comparison to the well-known Monte Carlo method.

A Hybrid SIE-PDE Formulation Without Boundary Condition Requirement for Transverse Magnetic Electromagnetic Analysis

A hybrid surface integral equation partial differential equation (SIE-PDE) formulation without the boundary condition requirement is proposed to solve the transverse magnetic (TM) electromagnetic problems. In the proposed formulation, the computational domain is decomposed into two overlapping domains: the SIE and PDE domains. In the SIE domain, complex structures with piecewise homogeneous media, e.g., highly conductive media, are included. An equivalent model for those structures is constructed by replacing them with the background medium and introducing a surface equivalent electric current density on an enclosed boundary to represent their electromagnetic effects. The remaining computational domain and homogeneous background medium replaced domain consist of the PDE domain, in which inhomogeneous or non-isotropic media are included. Through combining the surface equivalent electric current density and the inhomogeneous Helmholtz equation, a hybrid SIE-PDE formulation is derived. It requires no boundary conditions, and is mathematically equivalent to the original physical model. Through careful construction of basis functions to expand electric fields and the equivalent current density, the discretized formulation is made compatible with the SIE and PDE domain interface. The accuracy and efficiency are validated through two numerical examples. Results show that the proposed SIE-PDE formulation can obtain accurate results, and significant performance improvements in terms of CPU time and memory consumption compared with the FEM are achieved.

New self-force method via elliptic partial differential equations for Kerr inspiral models

We present a new method designed to avoid numerical challenges that have impeded calculation of the Lorenz gauge self-force acting on a compact object inspiraling into a Kerr black hole. This type of calculation is valuable in creating waveform templates for extreme mass-ratio inspirals, which are an important source of gravitational waves for the upcoming Laser Interferometer Space Antenna mission. Prior hyperbolic partial differential equation (PDE) formulations encountered numerical instabilities involving unchecked growth in time; our new method is based on elliptic PDEs, which do not exhibit instabilities of that kind. For proof of concept, we calculate the self-force acting on a scalar charge in a circular orbit around a Kerr black hole. We anticipate this method will subsequently facilitate calculation of first-order Lorenz gauge Kerr metric perturbations and self-force, which could serve as a foundation for second-order Kerr self-force investigations.

Data-Free Solution of Electromagnetic PDEs Using Neural Networks and Extension to Transfer Learning

In this article, we present a data-free method to solve differential equation using artificial neural networks (ANN). This method exploits the universal function approximation nature of a neural network to mimic a specified partial differential equation (PDE) and provide its solution. Specifically, use of simple feed-forward artificial neural network (FF-ANN) is shown to demonstrate the solution of 1-D second-order differential equations without use of any prior data. The article also demonstrates that the similarity in two PDEs is akin to similarity in the optimized weight matrices attained after the solution using FF-ANN. This property is then utilized for a transferred learning to enable faster convergence of a new PDE, based on the prior solution of a similar PDE. The concept is shown while considering a general form of second-order PDE, and considering specific cases of scalar in-homogeneous wave equation form and Poisson Equation form. Error convergence below 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−6</sup> is shown and the transferred learning process shows typical time acceleration by a factor of 1.5–3 for the considered equations.

A Compact Numerical Scheme for the Heat Transfer of Mixed Convection Flow in Quantum Calculus

This contribution aims to propose a compact numerical scheme to solve partial differential equations (PDEs) with q-spatial derivative terms. The numerical scheme is based on the q-Taylor series approach, and an operator is proposed, which is useful to discretize second-order spatial q-derivative terms. The compact numerical scheme is constructed using the proposed operator, which gives fourth-order accuracy for second-order q-derivative terms. For time discretization, Crank–Nicolson, and Runge–Kutta methods are applied. The stability for the scalar case and convergence conditions for the system of equations are provided. The mathematical model for the heat transfer of boundary layer flow under the effects of non-linear mixed convection is given in form of PDEs. The governing equations are transformed into dimensionless PDEs using suitable transformations. The velocity and temperature profiles with variations of mixed convection parameters and the Prandtl number are drawn graphically. From considered numerical experiments, it is pointed out that the proposed scheme in space and Crank–Nicolson in time is more effective than that in which discretization for the time derivative term is performed by applying the Runge–Kutta scheme. A comparison with existing schemes is carried out as part of the research. For future fluid-flow investigations in an enclosed industrial environment, the results presented in this study may serve as a useful guide.

A note on a modified Hilbert transform

The Hilbert transform is a useful tool in the mathematical analysis of time-dependent partial differential equations in order to prove coercivity estimates in anisotropic Sobolev spaces in case of a bounded spatial domain Ω, but an infinite time interval . Instead, a modified Hilbert transform can be used if we consider a finite time interval . In this note we prove that the classical and the modified Hilbert transformations differ by a compact perturbation, when a suitable extension of a function defined on a bounded time interval onto is used. This result is important when we deal with space–time variational formulations of time-dependent partial differential equations, and for the implementation of related space–time finite and boundary element methods for the numerical solution of parabolic and hyperbolic equations with the heat and wave equations as model problems, respectively.

A novel method to impose boundary conditions for higher-order partial differential equations

Partial differential equations (PDEs) with spatial derivatives of order higher than two are receiving increasing attention, partially due to the current popularity of the phase-field method. In the finite element community, the growth of Isogeometric Analysis and other discretization schemes that employ approximation spaces with high-order continuity has fostered the use of variational formulations that avoid the use of auxiliary unknowns representing derivatives of the solution. However, one of the caveats of this approach is the accurate and efficient imposition of boundary conditions, especially on complex geometries. This paper proposes a new method to impose boundary conditions naturally in the weak form of higher-order PDEs. Our method is based on a specially-designed weak form in which the boundary conditions to be imposed are weighted by derivatives of the weight functions. This requires multiple integrations by parts which are allowed due to the smoothness of the basis functions. The Cahn–Hilliard equation and the isothermal Navier–Stokes–Norteweg equations are used as examples to demonstrate the proposed method. We show that if the solutions of the PDEs are smooth enough, the proposed variational equations and the original PDEs are equivalent. We discretize the variational equations using Isogeometric Analysis. Convergence results of the proposed method agree with the best approximation errors of the basis functions. Numerical examples illustrate the applicability of the approach to mapped geometries with non-conformal grids.

Bioconvection Cross Diffusion Effects on MHD Flow of Nanofluids over Three Different Geometries with Melting

Currently, nanofluid is a hot area of interest for researchers. The nanofluid with bioconvection phenomenon attracted the researchers owing to its numerous applications in the field of nanotechnology, microbiology, nuclear science, heat storage devices, biosensors, biotechnology, hydrogen bomb, engine of motors, cancer treatment, the atomic reactor, cooling of devices, and in many more. This article presents the bioconvection cross-diffusion effects on the magnetohydrodynamic flow of nanofluids on three different geometries (cone, wedge, and plate) with mixed convection. The temperature-dependent thermal conductivity, thermal diffusivity, and Arrhenius activation energy applications are considered on the fluid flow with melting phenomenon. The flow is analyzed under thermal and solutal Robin’s conditions. The problem is formulated in the mathematical formulation of partial differential equations (PDEs). The similarity transformations are applied to diminish the governing nonlinear coupled boundary value problems into higher-order non-linear ordinary differential equations (ODEs). The resulting expressions/equation numerically tackled utilizing the famous bvp4c package by MATLAB for various interesting parameters. The results were physically and numerically calculated through graphics and tables for the velocity field, energy distribution, nanoparticles concentration, and microorganisms profile for numerous parameters. From the obtained results, we discern that the transfer of heat and mass coefficient is high over a plate and cone in the flow, respectively. The velocity profile is reduced via a larger magnetic parameter. Temperaturedependent thermal conductivity enhances the thermal field. Larger thermophoresis enhanced the concentration of nanoparticles. The microorganisms’ Biot number boosts the microorganism’s profile.

Symmetry considerations for differential equation formulations from classical and fractional Lagrangians

The utility of Noether’s classical theorem on differential equations extended to a generalized nonclassical theorem is the focus of this paper. After addressing a couple of standard related Partial Differential Equation (P.D.E.) formulations from classical Lagrangians, it culminates into a non-classical formulation of the diffusion equation in one spatial dimension from a fractional Lagrangian. Comparisons and contrasts between techniques for the classical and fractional formulations, as done here, facilitate the basic computational methods required for building analytical results. A noteworthy interface between Distribution theory, Trace theory and Lie symmetry theory is a key point of interest in this study.

Liouville partial-differential-equation methods for computing 2D complex multivalued eikonals in attenuating media

We have developed a Liouville partial-differential-equation (PDE)-based method for computing complex-valued eikonals in real phase space in the multivalued sense in attenuating media with frequency-independent qualify factors, where the new method computes the real and imaginary parts of the complex-valued eikonal in two steps by solving Liouville equations in real phase space. Because the earth is composed of attenuating materials, seismic waves usually attenuate so that seismic data processing calls for properly treating the resulting energy losses and phase distortions of wave propagation. In the regime of high-frequency asymptotics, the complex-valued eikonal is one essential ingredient for describing wave propagation in attenuating media because this unique quantity summarizes two wave properties into one function: Its real part describes the wave kinematics and its imaginary part captures the effects of phase dispersion and amplitude attenuation. Because some popular ordinary-differential-equation (ODE)-based ray-tracing methods for computing complex-valued eikonals in real space distribute the eikonal function irregularly in real space, we are motivated to develop PDE-based Eulerian methods for computing such complex-valued eikonals in real space on regular meshes. Therefore, we solved novel paraxial Liouville PDEs in real phase space so that we can compute the real and imaginary parts of the complex-valued eikonal in the multivalued sense on regular meshes. We call the resulting method the Liouville PDE method for complex-valued multivalued eikonals in attenuating media; moreover, this new method provides a unified framework for Eulerianizing several popular approximate real-space ray-tracing methods for complex-valued eikonals, such as viscoacoustic ray tracing, real viscoelastic ray tracing, and real elastic ray tracing. In addition, we also provide Liouville PDE formulations for computing multivalued ray amplitudes in a weakly viscoacoustic medium. Numerical examples, including a synthetic gas-cloud model, demonstrate that our methods yield highly accurate complex-valued eikonals in the multivalued sense.

Mathematical Modeling of Lymph Node Drainage Function by Neural Network

The lymph node (LN) represents a key structural component of the lymphatic system network responsible for the fluid balance in tissues and the immune system functioning. Playing an important role in providing the immune defense of the host organism, LNs can also contribute to the progression of pathological processes, e.g., the spreading of cancer cells. To gain a deeper understanding of the transport function of LNs, experimental approaches are used. Mathematical modeling of the fluid transport through the LN represents a complementary tool for studying the LN functioning under broadly varying physiological conditions. We developed an artificial neural network (NN) model to describe the lymph node drainage function. The NN model predicts the flow characteristics through the LN, including the exchange with the blood vascular systems in relation to the boundary and lymphodynamic conditions, such as the afferent lymph flow, Darcy’s law constants and Starling’s equation parameters. The model is formulated as a feedforward NN with one hidden layer. The NN complements the computational physics-based model of a stationary fluid flow through the LN and the fluid transport across the blood vessel system of the LN. The physical model is specified as a system of boundary integral equations (IEs) equivalent to the original partial differential equations (PDEs; Darcy’s Law and Starling’s equation) formulations. The IE model has been used to generate the training dataset for identifying the NN model architecture and parameters. The computation of the output LN drainage function characteristics (the fluid flow parameters and the exchange with blood) with the trained NN model required about 1000-fold less central processing unit (CPU) time than computationally tracing the flow characteristics of interest with the physics-based IE model. The use of the presented computational models will allow for a more realistic description and prediction of the immune cell circulation, cytokine distribution and drug pharmacokinetics in humans under various health and disease states as well as assisting in the development of artificial LN-on-a-chip technologies.