Nonlinear Second-order Partial Differential Equations Research Articles

Reconstruction of unknown storativity and transmissivity functions in 2D groundwater equations

ABSTRACT The paper deals with the identification from some pumping tests of unknown storativity and transmissivity functions defining a confined aquifer. We introduce an appropriate change of variables that transforms the groundwater equation into a diffusion-reaction one wherein the diffusion term is defined by the fraction transmissivity/storativity and the reaction term yields the right hand side of a second-order nonlinear elliptic partial differential equation satisfied by the unknown storativity function. Using time records of the drawdown taken at some measuring wells within the monitored aquifer, we establish identifiability results on the involved unknown variables. We develop an identification approach that starts by determining the two auxiliary diffusion and reaction variables. Afterwards, the developed approach proposes local and global procedures to reconstruct the unknown storativity function. That leads to deduce the sought transmissivity function from the product of the two identified storativity and diffusion functions. Some numerical experiments on a variant of groundwater equations are presented.

GO Synthesis of Offset Dual Reflector Antennas Using Local Axis-Displaced Confocal Quadrics

Abstract This work investigates an alternative numerical scheme for the solution of an exact formulation based on Geometrical Optics (GO) principles to synthesize offset dual reflector antennas. The technique is suited to solve a second-order nonlinear partial differential equation of the Monge-Ampere type as a boundary value problem. An iterative algorithm based on Newton's method was developed, using axis-displaced confocal quadrics to locally represent the subreflector surface, thus enabling an analytical description of the partial derivatives within the formulation. Such approach reduces discretization errors, as exact expressions for the mapping function and its derivatives are analytically determined. To check the robustness of the methodology, an offset dual-reflector Gregorian antenna was shaped to provide a Gaussian aperture field distribution with uniform phase within a superelliptical contour. The shaped surfaces were further interpolated by quintic pseudo-splines and analyzed by Physical Optics (PO) with equivalent edge currents to validate the synthesis procedure at 11,725 GHz.

Shape transformation using the modified Allen–Cahn equation

In this paper, we propose a phase-field model for the shape transformation and its simple numerical method. The proposed phase-field model is based on the Allen–Cahn equation, which is a second-order nonlinear parabolic partial differential equation and originally arises from modeling phase separation in alloys. The numerical scheme is a hybrid method using the operator splitting method. To validate the proposed phase-field model for the shape transformation, we perform two- and three-dimensional shape metamorphosis. The proposed phase-field model is also compared with the linear interpolation. The computational results demonstrate that the proposed mathematical model satisfactorily and naturally simulate the shape transformation.

Discontinuous Galerkin approximation for excitatory-inhibitory networks with delay and refractory periods

In this study, we consider the network of noisy leaky integrate-and-fire (NNLIF) model, which governs by a second-order nonlinear time-dependent partial differential equation (PDE). This equation uses the probability density approach to describe the behavior of neurons with refractory states and the transmission delays. A numerical approximation based on the discontinuous Galerkin (DG) method is used for the spatial discretization with the analysis of stability. The strong stability-preserving explicit Runge–Kutta (SSPERK) method is performed for the temporal discretization. Finally, some test examples and numerical simulations are given to examine the behavior of the solution. The execution of the constructed scheme is measured by the quantitative comparison with the existing finite difference technique, namely weighted essentially nonoscillatory (WENO) scheme.

Output-Feedback PDE Control of Traffic Flow on Cascaded Freeway Segments

We develop in this paper a boundary output feedback control law for an underactuated network of traffic flow on two connected roads; one incoming and one outgoing road connected by a junction. The macroscopic traffic dynamics on each road segment are governed by Aw-Rascle-Zhang (ARZ) model, consisting of second-order nonlinear partial differential equations (PDEs) of traffic density and velocity. The control objective is to stabilize the traffic network system on both roads around a chosen reference system. Using a ramp metering located at the outlet of the outgoing road, we actuate the traffic flux leaving this considered domain. Boundary measurements of traffic flux and velocity are taken at the junction connecting the two road segments. A delay-robust full state feedback control law and a boundary observer are designed for this under-actuated network of two systems interconnected through their boundaries. Each system consists of two hetero-directional linear first-order hyperbolic PDEs. The exponential convergence to the reference system is achieved.

Steady compressible subsonic impinging flows with non-zero vorticity

A well-posedness analysis of steady state of inviscid compressible subsonic impinging flows with non-zero vorticity is developed. A nonlinear nonhomogeneous second-order partial differential equation for the solution of the flow stream function is derived in terms of the inlet flow specifying the horizontal velocity. We first construct a subsonic solution to the impinging flow problem with special horizontal velocity and specific effluent fluxes in the outlets. Moreover, we showed that such a subsonic impinging flow with the same asymptotic behavior in the upstream and downstreams is unique. Furthermore, the existence, uniqueness and regularity of the interface separating the two fluids with different outlets are obtained. This result extends the recent result on the compressible subsonic irrotational impinging flows in [5]. Finally, as a byproduct, we obtain the existence of subsonic-sonic impinging flows.

Approximate analytic solutions for a chemical tubular reactor model using the Adomian decomposition method

The purpose of this paper is to apply the Adomian decomposition method to acquire the approximate analytical solutions for a chemical tubular reactor model composed of a system of strongly nonlinear second-order partial differential equations with standard boundary conditions. By properly assigning the dimensionless parameters in the model, the objective description of the solutions of temperature and process component concentration distributions can be obtained.

Method of Integral Equations for Studying the Solvability of Boundary Value Problems for the System of Nonlinear Differential Equations of the Theory of Timoshenko Type Shallow Inhomogeneous Shells

The solvability of the boundary value problem for a system of second-order nonlinear partial differential equations with given boundary conditions which describes the equilibrium of elastic inhomogeneous shallow shells with free edges in the framework of the Timoshenko shear model is considered. The boundary value problem is reduced to a single nonlinear equation whose solvability is established by using the contraction mapping principle.

О математической модели неизотермического ползущего течения жидкости через заданную область

Рассматривается математическая модель, описывающая стационарное ползущее течение неравномерно нагретой несжимаемой жидкости через ограниченную трехмерную область с локально-липшицевой границей. Выбранная модель представляет собой систему нелинейных дифференциальных уравнений в частных производных второго порядка со смешанными краевыми условиями: на участке протекания заданы давление, температура и касательная составляющая поля скоростей, а на твердых стенках сосуда используется условие прилипания и краевое условие типа Робена для температуры. Для данной краевой задачи вводится понятие слабого решения (пара «скорость-температура»), которое определяется как решение некоторой системы интегральных тождеств. Основной результат работы - теорема о существовании слабых решений в подпространстве декартова произведения двух соболевских пространств. Для доказательства этой теоремы дается операторная трактовка краевой задачи, выводятся априорные оценки решений и применяется теорема Лерэ-Шаудера о неподвижной точке вполне непрерывного отображения. Установлены энергетические равенства, которым удовлетворяют слабые решения.

An Efficient Method for Inverse Modeling of Soil Evaporation: Rationale and Numerical Example

An initial–boundary value problem for the one-dimensional nonlinear parabolic second-order partial differential equation that describes water movement in soil is considered. The inverse problem is to restore a boundary condition (evaporation) based on the comparison of modeled values of soil moisture at various points with some prescribed values. A criterion of the proximity of the selected evaporation to its true value is the mean-square deviation of the prescribed values of soil moisture at various points from the simulated soil moisture values corresponding to the selected evaporation. A numerical solution of a discretized optimization problem is investigated. The cases of accurate and inaccurate data are considered. The results of the numerical experiments are analyzed.

Traffic congestion control for Aw–Rascle–Zhang model

This paper develops boundary feedback control laws to reduce stop-and-go oscillations in congested traffic. The macroscopic traffic dynamics are governed by Aw–Rascle–Zhang (ARZ) model, consisting of second-order nonlinear partial differential equations (PDEs). A criterion to distinguish free and congested regimes for the ARZ traffic model leads to the study of hetero-directional hyperbolic PDE model of congested traffic regime. To stabilize the oscillations of traffic density and speed in a freeway segment, a boundary input through ramp metering is considered. We discuss the stabilization problem for freeway segments respectively, upstream and downstream of the ramp. For the more challenging upstream control problem, our full-state feedback control law employs a backstepping transformation. Both collocated and anti-collocated boundary observers are designed. The exponential stability in L2 sense and finite time convergence to equilibrium are achieved and validated with simulation. In the absence of relaxation time and boundary parameters’ knowledge, we propose adaptive output feedback control design. Control is applied at outlet and the measurement is taken from inlet of the freeway segment. We use the backstepping method to obtain an observer canonical form in which unknown parameters multiply with measured output. A parametric model based on this form is derived and gradient-based parameter estimators are designed. An explicit state observer involving the delayed values of the input and the output is introduced for state estimation. Using the parameter and state estimates, we develop an adaptive output feedback control law which achieves convergence to the steady regulation in the L2 sense.

Comparison study on the different dynamics between the Allen–Cahn and the Cahn–Hilliard equations

We perform a comparison study on the different dynamics between the Allen–Cahn (AC) and the Cahn–Hilliard (CH) equations. The AC equation describes the evolution of a non-conserved order field during anti-phase domain coarsening. The CH equation describes the process of phase separation of a conserved order field. The AC and the CH equations are second-order and fourth-order nonlinear parabolic partial differential equations, respectively. Linear stability analysis shows that growing and decaying modes for both the equations are the same. While the growth rates are monotonically decreasing with respect to the modes for the AC equation, the growth rates for the CH equation are increasing and then decreasing with respect to the modes. We perform various numerical tests using the Fourier spectral method to highlight the different evolutionary dynamics between the AC and the CH equations.

Proxy Pseudo3D model: the optimum of speed and accuracy in hydraulic fracturing simulation

The paper describes an efficient approach to simulating hydraulic fracturing in Pseudo3D setting for reservoirs with a low permeable matrix. By reformulating the problem, the authors succeeded in reducing it to two ordinary first-order differential equations instead of a nonlinear second-order partial differential equation, which fundamentally affects the computational complexity. Also, this approach allows a more correct computation of a fracture front and fracturing fluids interface versus a traditional Cell-Based Pseudo3D model. Nowadays the main limitation of the developed software module is the absence of matrix leak-off.

A novel approach for solving nonlinear flow equations: The next step towards an accurate assessment of shale gas resources

As ultra-tight porous media that include organic contents, shale gas resources are technically known as complex systems having various mechanisms that impact storage and flow. The slippage, Knudsen diffusion, the process of desorption, an adsorbed layer that affects apparent permeability, and solute gas in kerogen are recognized to be the most important ones. However, simultaneous effects of multi-mechanism flow and storage, and influences of scattered organic contents on shale gas flow behaviour are not well-understood yet.According to the mass conservation law, a basic mathematical model has been developed to investigate, step-by-step, the effects of different changes that are introduced, and examine whether patterns of how kerogen is distributed affect the production plateaus. The discretization of the second-order nonlinear Partial Differential Equation (PDE) that is evolved results in a certain number of nonlinear simultaneous algebraic equations, which are conventionally solved with the application of Newton’s method. To overcome the inherent difficulties of the initial guess, the derivations, and the inversion of the Jacobian matrix, a new application of Particle Swarm Optimization (PSO) as a nonlinear solver was applied to extract the anticipated pressure profile for each step in time outside the bounds of the reference equations.The results show that not only can the PSO effectively meet the required criteria, but also it performed faster than conventional techniques, especially in cases with a larger number of grids that encompass more phenomena. It was further revealed that the insertion of a multi-mechanism apparent permeability model in which the pore radius is also a pressure-dependent parameter causes the lower rate of production. A higher level of production has been recorded after including storage terms of adsorption and solute gas in kerogens. Although different patterns of kerogen distribution have finally overlapped, the different taken trend of each production profile underlines the impact of kerogen distribution as an important parameter within the procedure of history matching.

Nonlinear implicit Green’s functions for numerical approximation of partial differential equations: Generalized Burgers’ equation and nonlinear wave equation with damping

A representation formula for second-order nonhomogeneous nonlinear ordinary differential equations (ODEs) has been recently constructed by M. Frasca using its Green’s function, i.e. the solution of the corresponding nonlinear differential equation with a Dirac delta function instead of its nonhomogeneity. It has been shown that the first-order term–the convolution of the nonlinear Green’s function and the right-hand side, analogous to the Green’s representation formula for linear equations — provides a numerically efficient solution of the original equation, while the higher order terms add complementary corrections. The cases of square and sine nonlinearities have been studied by Frasca. Some new cases of explicit determination of nonlinear Green’s function have been studied previously by us. Here, we gather nonlinear equations and their explicitly determined Green’s functions from existing references, as well as investigate new nonlinearities leading to implicit determination of nonlinear Green’s function. Some transformations allowing to reduce second-order nonlinear partial differential equations (PDEs) to nonlinear ODEs are considered, meaning that Frasca’s method can be applied to second-order PDEs as well. We perform a numerical error analysis for a generalized Burgers’ equation and a nonlinear wave equation with a damping term in comparison with the method of lines.

Stochastic and partial differential equations on non-smooth time-dependent domains

In this article, we consider non-smooth time-dependent domains whose boundary is W1,p in time and single-valued, smoothly varying directions of reflection at the boundary. In this setting, we first prove existence and uniqueness of strong solutions to stochastic differential equations with oblique reflection. Secondly, we prove, using the theory of viscosity solutions, a comparison principle for fully nonlinear second-order parabolic partial differential equations with oblique derivative boundary conditions. As a consequence, we obtain uniqueness, and, by barrier construction and Perron’s method, we also conclude existence of viscosity solutions. Our results generalize two articles by Dupuis and Ishii to time-dependent domains.

Simulation of Drying Stresses in Eucalyptus nitens Wood

The objective of this work was to simulate the stresses produced during the drying of Eucalyptus nitens wood due to variations in the moisture content. The methodology involved experimental determination and simulation of drying stresses caused by the development of internal moisture content gradients. Modeling of the moisture transport was based on the concept of an effective diffusion coefficient. The mathematical model for stress-strain, and for moisture diffusion into the wood, was constituted by a system of second-order nonlinear partial differential equations with variable coefficients, which were numerically integrated by the control volume based on the finite element method (CVFEM). For validation purposes, tests were realized for evaluating deformations, stress drying, and moisture gradients that were produced during the drying of Eucalyptus nitens. The results showed satisfactory agreement between the experimental and simulated values, indicating an effective simulation.

G˚arding Cones and Bellman Equations in the Theory of Hessian Operators and Equations

In this work, we continue investigation of algebraic properties of G˚arding cones in the space of symmetric matrices. Based on this theory, we propose a new approach to study of fully nonlinear differential operators and second-order partial differential equations. We prove new-type comparison theorems for evolution Hessian operators and establish a relation between Hessian and Bellman equations.

Difference methods for parabolic equations with Robin condition

Classical solutions of nonlinear second-order partial differential functional equations of parabolic type with the Robin condition are approximated in the paper by solutions of associated boundedness-preserving implicit difference functional equations. It is proved that the discrete solutions uniquely exist, they are uniformly bounded with respect to meshes and the numerical method is convergent and stable. We also find the error estimate and its asymptotic behavior. The properties of some auxiliary nonlinear discrete recurrent equations are showed. The proofs are based on the comparison technique and the Banach fixed-point theorem.

On multilevel RBF collocation to solve nonlinear PDEs arising from endogenous stochastic volatility models

The main aim of this paper is the analytical and numerical study of a time-dependent second-order nonlinear partial differential equation (PDE) arising from the endogenous stochastic volatility model, introduced in [Bensoussan, A., Crouhy, M. and Galai, D., Stochastic equity volatility related to the leverage effect (I): equity volatility behavior. Applied Mathematical Finance, 1, 63–85, 1994]. As the first step, we derive a consistent set of initial and boundary conditions to complement the PDE, when the firm is financed by equity and debt. In the sequel, we propose a Newton-based iteration scheme for nonlinear parabolic PDEs which is an extension of a method for solving elliptic partial differential equations introduced in [Fasshauer, G. E., Newton iteration with multiquadrics for the solution of nonlinear PDEs. Computers and Mathematics with Applications, 43, 423–438, 2002]. The scheme is based on multilevel collocation using radial basis functions (RBFs) to solve the resulting locally linearized elliptic PDEs obtained at each level of the Newton iteration. We show the effectiveness of the resulting framework by solving a prototypical example from the field and compare the results with those obtained from three different techniques: (1) a finite difference discretization; (2) a naive RBF collocation and (3) a benchmark approximation, introduced for the first time in this paper. The numerical results confirm the robustness, higher convergence rate and good stability properties of the proposed scheme compared to other alternatives. We also comment on some possible research directions in this field.