Nonlinear Second-order Partial Differential Equations Research Articles

A technique for the calculation of evaporation from the soil surface based on moisture profiles

The problem of calculating the evaporation from the soil surface is formulated as an optimal control problem. The controlled process of the vertical water transfer in soil is described by a onedimensional nonlinear second-order parabolic partial differential equation. The objective function is the squared Euclidean distance between the calculated values of the soil moisture at various depths and certain prescribed values. To improve the efficiency of finding a numerical solution, the sensitivity of the soil moisture at various depths to the variations of evaporation is estimated by means of fast automatic differentiation. The analysis of these estimates made it possible to determine an effective thin subsurface soil layer in which the moisture is most sensitive to the variations of evaporation; it is in this soil layer where the objective function should be calculated.

Finite element procedure for one-dimensional viscoplastic problems of an isotropic slab

This study, based on a modified form of the Gurtin–Anand model (J Mech Phys Solids 53:1624–1649, 2005), investigates the effect of an internal microforce on the flow rule of viscoplastic bodies. A generalized microforce balance is obtained via the virtual power principle, and using the free-energy imbalance to suitably restrict the constitutive relations, a flow rule which accommodates two energetic length scales is obtained. This leads to a nonlinear second-order partial differential equation in the plastic strain. Finite element models for one-dimensional problems arising from a pure shear state of a viscoplastic slab are derived, while solutions for linear rate-dependent and nonlinear small strain-gradient rate-independent problems are presented.

Krylov implicit integration factor WENO method for SIR model with directed diffusion

SIR models with directed diffusions are important in describing the population movement. However, efficient numerical simulations of such systems of fully nonlinear second order partial differential equations (PDEs) are challenging. They are often mixed type PDEs with ill-posed or degenerate components. The solutions may develop singularities along with time evolution. Stiffness due to nonlinear diffusions in the system gives strict constraints in time step sizes for numerical methods. In this paper, we design efficient Krylov implicit integration factor (IIF) Weighted Essentially Non-Oscillatory (WENO) method to solve SIR models with directed diffusions. Numerical experiments are performed to show the good accuracy and stability of the method. Singularities in the solutions are resolved stably and sharply by the WENO approximations in the scheme. Unlike a usual implicit method for solving stiff nonlinear PDEs, the Krylov IIF WENO method avoids solving large coupled nonlinear algebraic systems at every time step. Large time step size computations are achieved for solving the fully nonlinear second-order PDEs, namely, the time step size is proportional to the spatial grid size as that for solving a pure hyperbolic PDE. Two biologically interesting cases are simulated by the developed scheme to study the finite-time blow-up time and location or discontinuity locations in the solution of the SIR model.

Study of the solvability of a boundary value problem for the system of nonlinear differential equations of the theory of shallow shells of the Timoshenko type

We study the solvability of a boundary value problem for a system of nonlinear second-order partial differential equations under given boundary conditions, which describes the equilibrium of elastic shallow shells with hinged edges in the framework of the Timoshenko shear model. The study method implies the reduction of the original system of equations to a single nonlinear differential equation whose solvability is proved with the use of the contraction mapping principle.

Smooth Surfaces of Constant Mean Curvature in Biology (Video Link)

Differential geometry is concerned with the calculus of smooth surfaces. The field rose to prominence through its incredible power to describe Einstein’s Theory of General Relativity, in which spacetime is considered to be a smooth four-dimensional manifold. More recently, differential geometry has been used to describe stress and strain for elastic bodies, in digital signals processing and in probability theory. Smooth surfaces appear in a variety of natural systems. This research focusses on the salvinia leaf, a floating fern whose skin has the unusual property that, when immersed in water, a stable, persistent air layer is retained on the surface of the leaf. This air-water interface is made possible by the phenomenon of surface tension and a forest-like structure on the surface of the leaf, which forms a ‘tent’ of air. This interface is a smooth surface that can be investigated using differential geometry, and has the particular property that it’s mean curvature is constant. The equation of capillary pressure, developed in the early nineteenth century by Thomas Young and Pierre-Simon Laplace, governs the system. However, the traditional formulation of the problem can be extremely difficult to solve. The project involved reformulating the Young-Laplace equation, a second-order nonlinear partial differential equation, in a coordinate independent fashion using differential geometry. Finite element analysis was then used to obtain numerical solutions for specific domain geometries. The ability to find domain geometries which form surfaces with specified properties will aid the design of effective bionic devices.

Shaping Single Offset Reflector Antennas Using Local Axis-Displaced Confocal Quadrics

This work investigates a novel numerical procedure for the solution of an exact formulation for the Geometrical Optics synthesis of a single reflector antenna by simultaneously imposing Snell’s Law and Conservation of Energy in a tube of rays, yielding a second-order nonlinear partial differential equation of Monge-Ampère type, which can be solved as a boundary value problem. The investigation explores the interpolating properties of confocal quadrics to locally represent the shaped reflector surface. It allows the partial derivatives involved in the formulation to be analytically expressed. To illustrate the method, two examples of offset single reflectors shaped to radiate a Gaussian power density within a superelliptical contoured beam are presented. The results are validated by Physical Optics analysis with equivalent edge currents.

Local Decomposition and Accessibility of Nonlinear Infinite-Dimensional Systems

This article deals with the local system decomposition of infinite-dimensional systems, which are described by second-order nonlinear partial differential equations. We show that if there exists a certain codistribution which is invariant under the generalized system vector field, a local triangular decomposition can be obtained. Furthermore, we draw connections to a different approach which is based on transformation groups. Throughout the article we apply differential geometric methods, highlighting the geometric picture behind the system description. The article is closed with a nonlinear example.

A PUT OPTION'S VALUE FOR A NONLINEAR BLACK-SCHOLES EQUATION

We study a nonlinear Black-Scholes partial differential equation for modelling illiquid markets with feedback effects. After reducing the equation into a second-order nonlinear partial differential equation, we find that the assumption of a traveling wave profile to the second-order equation reduces it further to ordinary differential equations. Solutions to all these transformed equations facilitate an analytic solution to the nonlinear Black-Scholes equation. Use of the put-call parity gave rise to the put option's current value. These solutions can be used for pricing a European call and put options respectively at t ≥ 0 and when c 6 r.

Designing illumination lenses and mirrors by the numerical solution of Monge–Ampère equations

We consider the inverse refractor and the inverse reflector problem. The task is to design a free-form lens or a free-form mirror that, when illuminated by a point light source, produces a given illumination pattern on a target. Both problems can be modeled by strongly nonlinear second-order partial differential equations of Monge-Ampère type. In [Math. Models Methods Appl. Sci.25, 803 (2015)MMMSEU0218-202510.1142/S0218202515500190], the authors have proposed a B-spline collocation method, which has been applied to the inverse reflector problem. Now this approach is extended to the inverse refractor problem. We explain in depth the collocation method and how to handle boundary conditions and constraints. The paper concludes with numerical results of refracting and reflecting optical surfaces and their verification via ray tracing.

Dynamic response of composite laminated beams under asynchronous/repeated low-velocity impacts of multiple masses

This paper presents a theoretical method for low-velocity impact analysis of composite laminated beams with arbitrary lay-ups and various boundary conditions subjected to asynchronous/repeated impacts of multiple masses. The analysis is based on the higher-order shear deformation beam theory and the modified Hertzian contact law is used for modeling the interaction between the laminated beam and the impactors. Non-linear second-order partial differential equations of the motion are derived using the Lagrangian equations of the second kind. According to the Ritz method, simple polynomials are assigned to the shape functions of the displacement field of the laminated beam in order to satisfy arbitrary boundary conditions. The validity of the formulation is confirmed by comparing the present solutions with results available in the literature. Numerical examples showed that the time of impact play an important role on contact forces, beam displacements, absorbed energies by the beam and normal and shear stresses by positive and negative superposition of induced waves.

On the existence of solutions of a nonlinear boundary value problem for the system of partial differential equations of the theory of Timoshenko type shallow shells with free edges

We study the solvability of a system of nonlinear second-order partial differential equations with given boundary conditions. The method is to reduce the original system to a single nonlinear differential equation, whose solvability is then proved with the use of the contraction mapping principle.

Worst-case portfolio optimization with proportional transaction costs

We study optimal asset allocation in a crash-threatened financial market with proportional transaction costs. The market is assumed to be either in a normal state, in which the risky asset follows a geometric Brownian motion, or in a crash state, in which the price of the risky asset can suddenly drop by a certain relative amount. We only assume the maximum number and the maximum relative size of the crashes to be given and do not make any assumptions about their distributions. For every investment strategy, we identify the worst-case scenario in the sense that the expected utility of terminal wealth is minimized. The objective is then to determine the investment strategy which yields the highest expected utility in its worst-case scenario. We solve the problem for utility functions with constant relative risk aversion using a stochastic control approach. We characterize the value function as the unique viscosity solution of a second-order nonlinear partial differential equation. The optimal strategies are characterized by time-dependent free boundaries which we compute numerically. The numerical examples suggest that it is not optimal to invest any wealth in the risky asset close to the investment horizon, while a long position in the risky asset is optimal if the remaining investment period is sufficiently large.

THEORETICAL FORMULATION OF A TIME-DOMAIN FINITE ELEMENT METHOD FOR NONLINEAR MAGNETIC PROBLEMS IN THREE DIMENSIONS (Invited Paper)

In this work, a numerical solution of nonlinear ferromagnetic problems is formulated using the three-dimensional time-domain finite element method (TDFEM) combined with the inverse JilesAtherton (J-A) vector hysteresis model. After a brief introduction of the J-A constitutive model, the second-order nonlinear partial differential equation (PDE) is constructed through the magnetic vector potential in the time domain, which is then discretized by employing the Newmark-β scheme, and solved by applying the Newton-Raphson method. Different Newton-Raphson schemes are constructed and compared. The capability of the proposed methods is demonstrated by several numerical examples including the simulation of the physical demagnetization process, the prediction of the magnetic remanence in the ferromagnetic material, and the generation of higher-order harmonics.

Adaptive FEM with Optimal Convergence Rates for a Certain Class of Nonsymmetric and Possibly Nonlinear Problems

We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain non-linear second-order partial differential equations. We allow continuous polynomials of arbitrary, but fixed polynomial order. The adaptivity is driven by the residual error estimator. We prove convergence even with optimal algebraic convergence rates. In particular, our analysis covers general linear second-order elliptic operators. Unlike prior works for linear non-symmetric operators, our analysis avoids the interior node property for the refinement, and the differential operator has to satisfy a G\r{a}rding inequality only. If the differential operator is uniformly elliptic, no additional assumption on the initial mesh is posed.

Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations

We propose an efficient Legendre-Gauss collocation algorithm forsecond-order nonlinear ordinary differential equations (ODEs). Wealso design a Legendre-Gauss-type collocation algorithm for time-dependent second-order nonlinear partialdifferential equations (PDEs), which can be implemented in asynchronous parallel fashion. Numerical results indicate the highaccuracy and effectiveness of the suggested algorithms.

Conjugate flow action functionals

We present a new general framework to construct an action functional for a non-potential field theory. The key idea relies on representing the governing equations relative to a diffeomorphic flow of curvilinear coordinates which is assumed to be functionally dependent on the solution field. Such flow, which will be called the conjugate flow, evolves in space and time similarly to a physical fluid flow of classical mechanics and it can be selected in order to symmetrize the Gâteaux derivative of the field equations with respect to suitable local bilinear forms. This is equivalent to requiring that the governing equations of the field theory can be derived from a principle of stationary action on a Lie group manifold. By using a general operator framework, we obtain the determining equations of such manifold and the corresponding conjugate flow action functional. In particular, we study scalar and vector field theories governed by second-order nonlinear partial differential equations. The identification of transformation groups leaving the conjugate flow action functional invariant could lead to the discovery of new conservation laws in fluid dynamics and other disciplines.

Shock Diffraction by Convex Cornered Wedges for the Nonlinear Wave System

We are concerned with rigorous mathematical analysis of shock diffraction by two-dimensional convex cornered wedges in compressible fluid flow, through the nonlinear wave system. This shock diffraction problem can be formulated as a boundary value problem for second-order nonlinear partial differential equations of mixed elliptic-hyperbolic type in an unbounded domain. It can be further reformulated as a free boundary problem for nonlinear degenerate elliptic equations of second order with a degenerate oblique derivative boundary condition. We establish a global theory of existence and optimal regularity for this shock diffraction problem. To achieve this, we develop several mathematical ideas and techniques, which are also useful for other related problems involving similar analytical difficulties.

Nonlinear dynamics of two degree of freedom systems with linear and nonlinear stiffnesses

In this study, a new analytical approach is developed to analyze the free nonlinear vibration of conservative two-degree-of-freedom (TDOF) systems. The mathematical models of these systems are governed by second-order nonlinear partial differential equations. Nonlinear differential equations were transferred into a single equation by using some intermediate variables. The single nonlinear differential equations are solved by using the first order of the Hamiltonian approach (HA). Different parameters, which have a significant impact on the response of the systems, are considered and discussed. Some comparisons are presented to verify the results between the Hamiltonian approach and the exact solution. The maximum relative error is less than 2.2124 % for large amplitudes of vibration. It has been established that the first iteration of the Hamiltonian approach achieves very accurate results, does not require any small perturbations, and can be used for a wide range of nonlinear problems.

Vectorial Total Variation-Based Regularization for Variational Image Registration

To use interdependence between the primary components of the deformation field for smooth and non-smooth registration problems, the channel-by-channel total variation- or standard vectorial total variation (SVTV)-based regularization has been extended to a more flexible and efficient technique, allowing high quality regularization procedures. Based on this method, this paper proposes a fast nonlinear multigrid (NMG) method for solving the underlying Euler-Lagrange system of two coupled second-order nonlinear partial differential equations. Numerical experiments using both synthetic and realistic images not only confirm that the recommended VTV-based regularization yields better registration qualities for a wide range of applications than those of the SVTV-based regularization, but also that the proposed NMG method is fast, accurate, and reliable in delivering visually-pleasing registration results.

Implicit Difference Methods for Differential Functional Parabolic Equations with Dirichlet's Condition

Classical solutions of nonlinear second-order partial differential functional equations of parabolic type with Dirichlet's condition are approximated in the paper by solutions of associated implicit difference functional equations. The functional dependence is of the Volterra type. Nonlinear estimates of the generalized Perron type for given functions are assumed. The convergence and stability results are proved with the use of discrete functional inequalities and the comparison technique. In particular, these theorems cover quasi-linear equations. However, such equations are also treated separately. The known results on similar difference methods can be obtained as particular cases of our simple result.