Second-order Partial Differential Equations Research Articles

A [formula omitted]-integrability condition of surfaces with singular parametrizations in isogeometric analysis

Singularities of a surface’s geometric mapping (or parametrization) are often unavoidable, especially when a complex surface is considered. In isogeometric analysis, singularities impact the regularity of test functions. When a second-order partial differential equation, such as a Laplace–Beltrami equation, is solved, the test functions should satisfy the H1-regularity, which may be destroyed by singularities. In this paper, we consider the H1-regularity of test functions on a surface by parametrization with isolated singularities. A H1-integrability condition is presented, and we apply this condition to discuss the H1-regularity of test functions by two common singular parametrizations of surfaces such as spheres and D-patches.

The inversion formula for the Volterra integral equation with the Humbert function in the nuclear and its applications to the boundary value problems

Many problems of applied mathematics are reduced to the solution of integral equations with special functions in kernels, therefore the inversion formulas for such equations play an important role in solving boundary value problems for second-order partial differential equations. In this paper, we introduce one degenerate hypergeometric function of two variables through which the solution of the Volterra integral equation of the first kind studing here is expressed. The inversion formula found is applied to finding some relations between the desired solution and its derivative of one boundary value problem for a hyperbolic equation with two degeneration lines of the second kind and with a spectral parameter. Bibliogr.16.Keywords: Volterra integral equation of the first kind, inversion formula, Humbert function, degenerate hypergeometric function of two variables, degenerate hyperbolic equation of the second kind, spectral parameter.

Some properties of zeros of solutions of second-order linear partial differential equations of elliptic type

Some properties of zeros of solutions of second-order linear partial differential equations of elliptic type

An Introduction to Partial Differential Equations

This book is an introduction to methods for solving partial differential equations (PDEs). After the introduction of the main four PDEs that could be considered the cornerstone of Applied Mathematics, the reader is introduced to a variety of PDEs that come from a variety of fields in the Natural Sciences and Engineering and is a springboard into this wonderful subject. The chapters include the following topics: First-order PDEs, Second-order PDEs, Fourier Series, Separation of Variables, and the Fourier Transform. The reader is guided through these chapters where techniques for solving first- and second-order PDEs are introduced. Each chapter ends with a series of exercises illustrating the material presented in each chapter. The book can be used as a textbook for any introductory course in PDEs typically found in both science and engineering programs and has been used at the University of Central Arkansas for over ten years.

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.

Backstepping boundary observer based-control for hyperbolic PDE in rotary drilling system

It is well known that torsional vibrations in oil well system affect the drilling directions and may be inherent for drilling systems. The drill pipe model is described by second order hyperbolic Partial Differential Equation (PDE) with mixed boundary conditions in which a sliding velocity is considered at the top end. In this paper, we consider the problem of boundary observer design for one-dimensional PDE with the usually neglected damping term. The main purpose is the construction of a control law which stabilizes the damped wave PDE, using only boundary measurements. From the Lyapunov theory, we show an exponentially vibration stability of the partially equipped oil well drilling system. The observer-based control law is found using the backstepping approach for second-order hyperbolic PDE. The numerical simulations confirm the effectiveness of the proposed PDE observer based controller.

Friction Factors in Fully Developed MHD Laminar Flows for Oblique Magnetic Fields and High Hartmann Numbers in Rectangular Channels

Self-cooled liquid metal blanket of a thermonuclear fusion reactor has the most critical issue of strong MHD effects which influence the thermal efficiency of the reactor. Special attention needs to be paid to the MHD friction factors for flow in rectangular channels at low as well as at high Hartmann numbers. The effect on MHD friction factors of a magnetic field oblique to the channel walls for various channel wall conductivities and aspect ratios at different Hartmann numbers is investigated numerically. Even for relatively small channel wall conductivity at high Hartmann numbers, velocity jets are developed near the channel walls. Due to these jets, MHD friction factors increase by the orders of magnitude which can be observed in the numerical results. The finite-difference method has been used to discretize the coupled second-order linear partial differential equations of MHD by using both uniform and nonuniform grids. Nonuniform discretization of the mesh is more efficient than the uniform mesh due to steep velocity gradients near the walls. For parallel plates, an analytical solution for MHD friction factor has been developed, and then compared with the numerical solutions. It is found that an oblique magnetic field has a significant effect on MHD friction factors for different channel wall conductivities and aspect ratios at various Hartmann numbers.

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.

Resolución de la Ecuación Diferencial Parcial Hiperbólica de segundo orden con término fuente mediante el Método de D’Alembert-Green

In the present work, we study a non-homogeneous second-order partial hyperbolic differential equation, its canonical form, its resolution using D’Alembert’s formula and Green’s theorem. Only mixed initial conditions that are not homogeneous are required to solve this problem. There are several physical problems that lead to this type of mathematical model, so this technique of resolution contributes to the knowledge of finding explicit solutions of problems such as two-dimensional wave type. Within the results the explicit solution of three cases is generated: regarding the homogeneity and non-homogeneity of the initial conditions and the term source, from the point of view of analytical solution for continuous functions.

Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines

We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations in arbitrary space dimension [Formula: see text]. We employ hierarchical B-splines of arbitrary degree and different order of smoothness. We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (respectively, the sum of energy error plus data oscillations) with optimal algebraic rates. Numerical experiments underpin the theoretical findings.

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.

Nonlinear stability in three-layer channel flows

The nonlinear stability of viscous, immiscible multilayer flows in plane channels driven both by a pressure gradient and gravity is studied. Three fluid phases are present with two interfaces. Weakly nonlinear models of coupled evolution equations for the interfacial positions are derived and studied for inertialess, stably stratified flows in channels at small inclination angles. Interfacial tension is demoted and high-wavenumber stabilisation enters due to density stratification through second-order dissipation terms rather than the fourth-order ones found for strong interfacial tension. An asymptotic analysis is carried out to demonstrate how these models arise. The governing equations are $2\times 2$ systems of second-order semi-linear parabolic partial differential equations (PDEs) that can exhibit inertialess instabilities due to interaction between the interfaces. Mathematically this takes place due to a transition of the nonlinear flux function from hyperbolic to elliptic behaviour. The concept of hyperbolic invariant regions, found in nonlinear parabolic systems, is used to analyse this inertialess mechanism and to derive a transition criterion to predict the large-time nonlinear state of the system. The criterion is shown to predict nonlinear stability or instability of flows that are stable initially, i.e. the initial nonlinear fluxes are hyperbolic. Stability requires the hyperbolicity to persist at large times, whereas instability sets in when ellipticity is encountered as the system evolves. In the former case the solution decays asymptotically to its uniform base state, while in the latter case nonlinear travelling waves can emerge that could not be predicted by a linear stability analysis. The nonlinear analysis predicts threshold initial disturbances above which instability emerges.

Feedback design of differential equations of reconstruction for second–order distributed parameter systems

AbstractThe paper aims at studying a class of second-order partial differential equations subject to uncertainty involving unknown inputs for which no probabilistic information is available. Developing an approach of feedback control with a model, we derive an efficient reconstruction procedure and thereby design differential equations of reconstruction. A characteristic feature of the obtained equations is that their inputs formed by the feedback control principle constructively approximate unknown inputs of the given second-order distributed parameter system.

Displacement potentials for functionally graded piezoelectric solids

• Solutions of Functionally Graded Transversely Isotropic Piezoelectric Media studied. • Displacement potential functions for the media are introduced. • The system of governing differential equations has been reduced and decoupled. • Limiting cases of the problem are discussed in detail. Two new displacement potential functions are introduced for the general solution of a three-dimensional piezoelasticity problem for functionally graded transversely isotropic piezoelectric solids. The material properties vary continuously along the axis of symmetry of the medium. The four coupled equilibrium equations in terms of displacements and electric potential are reduced to two decoupled sixth- and second-order linear partial differential equations for the potential functions. The obtained results are verified with two limiting cases: (i) a functionally graded transversely isotropic medium, and (ii) a homogeneous transversely isotropic piezoelectric solid. The simplified relations corresponding to the special case of similar variation of material properties are also given. Furthermore, the special cases of axisymmetric problems, exponentially graded piezoelectric media and transversely isotropic piezoelectric media with power law variation are discussed in detail.

A Note about the Black-Scholes Option Pricing Model under Time-Varying Conditions

By solving the second-order parabolic partial differential equation, the BS option pricing formula can be obtained; The BS pricing model under time-varying conditions is hard to get an explicit solution; We found that there is a same mistake in Zhige Ren’s paper and Xiaochen Han’s paper, and trying to put forward an idea to solve a sort of second-order parabolic partial differential equations.

Chaotic semigroups from second order partial differential equations

We give general conditions on given parameters to ensure Devaney and distributional chaos for the solution C0-semigroup corresponding to a class of second-order partial differential equations. We also provide a critical parameter that led us to distinguish between stability and chaos for these semigroups. In the case of chaos, we prove that the C0-semigroup admits a strongly mixing measure with full support. We also give concrete examples of partial differential equations, such as the telegraph equation, whose solutions satisfy these properties.

John\u2019s Equation-based Consistency Condition and Corrupted Projection Restoration in Circular Trajectory Cone Beam CT

In transmitted X-ray tomography imaging, the acquired projections may be corrupted for various reasons, such as defective detector cells and beam-stop array scatter correction problems. In this study, we derive a consistency condition for cone-beam projections and propose a method to restore lost data in corrupted projections. In particular, the relationship of the geometry parameters in circular trajectory cone-beam computed tomography (CBCT) is utilized to convert an ultra-hyperbolic partial differential equation (PDE) into a second-order PDE. The second-order PDE is then transformed into a first-order ordinary differential equation in the frequency domain. The left side of the equation for the newly derived consistency condition is the projection derivative of the current and adjacent views, whereas the right side is the projection derivative of the geometry parameters. A projection restoration method is established based on the newly derived equation to restore corrupted data in projections in circular trajectory CBCT. The proposed method is tested in beam-stop array scatter correction, metal artifact reduction, and abnormal pixel correction cases to evaluate the performance of the consistency condition and corrupted projection restoration method. Qualitative and quantitative results demonstrate that the present method has considerable potential in restoring lost data in corrupted projections.

Spectral Shift of Cylindrical Heat Equations by Full-State Boundary Feedback

This contribution introduces a full-state boundary feedback for cylindrical heat conductors. The desired decay properties are imposed to the spectrum of linear, second-order parabolic partial differential equations (PDEs). The feedback-scheme exploits periodicity in angular direction for cylindrical coordinates to reduce the 3-dimensional (3-D) system to a set of 2-D subsystems by means of a Fourier transformation. A backstepping-based controller for shifting the spectra for each of the decoupled systems on 2-D domains is presented. It can be shown, that any decay rate of the overall tracking error can be achieved by superposing solely a finite number of these controllers. The validity of the approach is shown by simulations.

Second‐order PDE‐based image restoration algorithm using directional diffusion

Variational and partial differential equation (PDE)-based algorithms have been widely applied in image restoration. However, it may produce undesirable staircase effect or blur image edges. To avoid these problems, a second-order PDE model based on directional diffusion has been proposed for image restoration. This model can just diffuse along the edge's tangential direction of the original image. Thus it can preserve the edges, avoid the staircase effect in the restored image. Visual and quantitative results demonstrate that the proposed second-order PDE model is superior to other models in preserving edges and avoiding staircase effect for image restoration.

A regularity theory for quasi-linear Stochastic PDE[formula omitted] in weighted Sobolev spaces

We study the second-order quasi-linear stochastic partial differential equations (SPDEs) defined on C1-domains. The coefficients are random functions depending on t,x and the unknown solutions. We prove the uniqueness and existence of solutions in appropriate Sobolev spaces, and in addition, we obtain Lp and Hölder estimates of both the solution and its gradient.