Nonlinear Boundary Problem Research Articles

Green’s Functions, Positive Solutions, and a Lyapunov Inequality for a Caputo Fractional-Derivative Boundary Value Problem

Abstract We consider a nonlinear boundary problem whose highest-order derivative is a Caputo derivative of order α with 1 < α < 2. Properties of its associated Green’s function are derived. These properties enable us to deduce sufficient conditions for the existence of a positive solution to the boundary value problem and to prove a Lyapunov inequality for the problem. Our results sharpen and extend earlier results of other authors.

A dam break driven by a moving source: a simple model for a powder snow avalanche

We study the two-dimensional, irrotational flow of an inviscid, incompressible fluid injected from a line source moving at constant speed along a horizontal boundary, into a second, immiscible, inviscid fluid of lower density. A semi-infinite, horizontal layer sustained by the moving source has previously been studied as a simple model for a powder snow avalanche, an example of an eruption current, Carroll et al. (Phys. Fluids, vol. 24, 2012, 066603). We show that with fluids of unequal densities, in a frame of reference moving with the source, no steady solution exists, and formulate an initial/boundary value problem that allows us to study the evolution of the flow. After considering the limit of small density difference, we study the fully nonlinear initial/boundary value problem and find that the flow at the head of the layer is effectively a dam break for the initial conditions that we have used. We study the dynamics of this in detail for small times using the method of matched asymptotic expansions. Finally, we solve the fully nonlinear free boundary problem numerically using an adaptive vortex blob method, after regularising the flow by modifying the initial interface to include a thin layer of the denser fluid that extends to infinity ahead of the source. We find that the disturbance of the interface in the linear theory develops into a dispersive shock in the fully nonlinear initial/boundary value problem, which then overturns. For sufficiently large Richardson number, overturning can also occur at the head of the layer.

Radially symmetric growth of necrotic tumors and connection with nonnecrotic tumors

In this paper we study a nonlinear free boundary problem for radially symmetric growth of tumors with necrosis. We show this problem is globally well-posed and find a threshold value σ∗>0, such that if and only if external nutrient supply σ̄≥σ∗, there exists a unique necrotic stationary solution. We prove it is globally asymptotically stable. For σ̄<σ∗, asymptotic behavior of transient solution has been studied. The connection and mutual transition between necrotic and nonnecrotic tumors are also given.

On Solving Nonlinear Moving Boundary Problems with Heterogeneity Using the Collocation Meshless Method

In this article, a solution to nonlinear moving boundary problems in heterogeneous geological media using the meshless method is proposed. The free surface flow is a moving boundary problem governed by Laplace equation but has nonlinear boundary conditions. We adopt the collocation Trefftz method (CTM) to approximate the solution using Trefftz base functions, satisfying the Laplace equation. An iterative scheme in conjunction with the CTM for finding the phreatic line with over–specified nonlinear moving boundary conditions is developed. To deal with flow in the layered heterogeneous soil, the domain decomposition method is used so that the hydraulic conductivity in each subdomain can be different. The method proposed in this study is verified by several numerical examples. The results indicate the advantages of the collocation meshless method such as high accuracy and that only the surface of the problem domain needs to be discretized. Moreover, it is advantageous for solving nonlinear moving boundary problems with heterogeneity with extreme contrasts in the permeability coefficient.

Nonlinear Functional Boundary Value Problems for Conformable Fractional Dynamic Equations on Time Scales

In this paper, we establish the existence of solutions for the conformable fractional dynamic equations on time scales, with nonlinear functional boundary value conditions. We first obtain the exact expression of the fractional Green’s function related to the linear conformable fractional dynamic problem and then we study the nonlinear functional boundary problems, by means of the upper and lower solutions method together with Schauder’s fixed-point theorem.

Troesch’s problem solved by Sinc methods

In this paper, we apply Sinc methods to compute the approximate solution of Troesch’s problem which is a highly sensitive, singular, and nonlinear boundary problem. The methods applied to Troesch’s equation use non-equidistant interpolation points generated by conformal maps, also called Sinc points. We demonstrate that Sinc methods are highly effective and compare our results with those computed by other methods. Sinc methods provide an efficient, accurate and reliable approximations of singular problems.

A fully nonlinear free boundary problem arising from optimal dividend and risk control model

Focusing on the problem arising from a stochastic model of risk control and dividend optimization techniques for a financial corporation, this work considers a parabolic variational inequality with gradient constraint \begin{document}$\min\Big\{v_t-\max\limits_{0\leq a\leq1}\Big(\frac{1}{2}\sigma^2a^2v_{xx}+\mu av_x\Big)+cv,\;v_x-1\Big\} = 0.$ \end{document} Suppose the company's performance index is the total discounted expected dividends, our objective is to choose a pair of control variables so as to maximize the company's performance index, which is the solution to the above variational inequality under certain initial-boundary conditions. The main effort is to analyse the properties of the solution and two free boundaries arising from the above variational inequality, which we call dividend boundary and reinsurance boundary.

Feed-forward Control Based on a Neural Network Technology for an Optimal Slewing of the Flexible Spacecraft

The problem of construction of optimal program slewing of spacecraft with attached flexible elements in the class of slewing around Eulerian axis is considered. The mathematical model is developed, allowing considering a finite number of natural modes of flexible vibrations. The problem of optimal reorientation of spacecraft with nontraditional objective of control, reducing to a minimum dynamic overload of flexible elements in a relative motion is formulated. The algorithm of numerical solving of the corresponding nonlinear boundary problem, realized in the form of a package of applied programs in FORTRAN is developed. The network whose training can be made during of preflight preparation is created in space of the slewing parameters. Simulation of fast calculation of the parameters of the optimal slewing program in flight is made on the basis of use of known radial basic functions.

Existence results for impulsive semilinear differential inclusions with nonlinear boundary conditions

In this paper, we discuss the nonlinear boundary problem for first-order impulsive semilinear differential inclusions. We establish existence results by using Martelli’s fixed point theorem with upper and lower solutions method. We find that by giving different definitions of lower and upper solutions we can get all existence results. We also present an example.

A new comparison principle and its application to nonlinear impulsive functional integro-differential equations

In this article, we first create a new comparison principle for a nonlinear impulsive boundary problem involving different deviating arguments. Then we employ the new result and iterative method to study the existence of the max-minimal solution of a second-order impulsive functional integro-differential equation. The results achieved in this paper are more general and complement many previously known results.

TFmix: A high-precision implementation of the finite-temperature Thomas–Fermi model for a mixture of atoms

In this work we present a TFmix code intended for numerical calculation of the thermal part of electronic thermodynamic properties of a mixture of elements by the finite-temperature Thomas–Fermi model. The code is based on analytical models for both first and second derivatives of Helmholtz thermodynamic potential. All numerical calculations are made within a controlled high accuracy: tests for thermodynamic consistency give at least 11 coinciding decimal digits. The code calculates thermodynamic functions on a regular grid of isotherms and isochores; at each grid point some extensive parameters and the number of free electrons are output both for the whole mixture and for each component. Other extensive or intensive thermodynamic properties, including pressure, entropy, isochoric and isobaric heat capacities, isothermal and adiabatic sound velocities can be easily calculated from the information available at each grid point. Several unit systems are provided for convenience. A cross-platform graphical user interface is developed to simplify the use of the code. Program summaryProgram Title: TFmix, version 1.0Program Files doi:http://dx.doi.org/10.17632/mc3vj77jfn.1Licensing provisions: GPLv3Programming language: C, PythonNature of problem: Any substance consists of elements so its equation of state contains a contribution of electronic gas. Thermodynamics of the electronic gas in a mixture of ions and electrons has been studied in many approaches. Thermodynamic properties of a uniform ideal electron gas can be calculated using the well-known analytical model of Fermi-gas. On the other hand, models of electron gas which take into account interaction effects are quite complicated and require sophisticated computational techniques. Even a simplified semiclassical Thomas–Fermi model is based upon the numerical solution of a non-linear boundary problem. Two main issues of the Thomas–Fermi model restrict its usage: uncontrolled accuracy of calculated thermodynamic functions (especially second derivatives of a thermodynamic potential), and unphysical behavior of the model at relatively low temperatures.Solution method: Each atom in the mixture is surrounded by a spherical cell. The radii of the cells are fitted to equalize the chemical potentials of all atoms. A guaranteed accuracy of first derivatives of the thermodynamic potential is provided by a transformation of integrals over the Thomas–Fermi potential to a system of differential equations. One of equations in the system is the Thomas–Fermi equation. Second derivatives of the thermodynamic potential are calculated similarly with the only difference that a corresponding derivative of the Thomas–Fermi equation is used in the system of differential equation. To avoid the unphysical behavior of the Thomas–Fermi model at low temperatures we extract a thermal contribution to thermodynamic properties which vanishes at zero temperature. To eliminate the error which appears from the subtraction of the cold part at low temperatures we use asymptotic expressions for thermodynamic functions and the Thomas–Fermi equation. The code calculates regular tables of thermodynamic functions on a grid of isotherms and isochores including second derivatives of a thermodynamic potential. This information is necessary for astrophysical applications, for continuum mechanics simulation of processes in plasma and for the creation of wide-range equations of state. A graphical user interface is provided with the code and allows to specify input parameters, to perform calculations and to plot the results.Additional comments including restrictions and unusual features: GSL library version 1.16 or 2.x is required for compilation; matplotlib Python library is required to run the graphical user interface.

Existence and uniqueness of variational solution to the Neumann problem for the pth sub-Laplacian associated to a system of Hörmander vector fields

We establish the existence and uniqueness of variational solution to the nonlinear Neumann boundary problem for the $p^{th}$-Sub-Laplacian associated to a system of H\"ormander vector fields

Numerical Simulation of the Stress-Strain State of Thin-Layer Rubber-Metal Vibration Absorber Elements Under Nonlinear Deformation

Stress-strain state components of thin-layer rubber-metal elements are investigated. The compression level of a thin rubber layer under the action of vertically applied stress is calculated. For simplified hypotheses the relationship between the rubber layer thickness reduction and its radius-thickness ratio was analytically derived. The problem was solved under linear-elastic deformation of a rubber layer vulcanized onto the metal plates. For numerical calculations, weak rubber compressibility was simulated with the moment finite element scheme for weakly compressible materials, involving the triple approximation of displacement fields, strain components, and a volume variation function. The numerical solution was obtained by the finite element method for different layer radii and thicknesses under geometrically nonlinear elastic and viscoelastic deformation of the rubber material. The geometrical nonlinearity is described by the nonlinear strain tensor. Viscoelastic rubber properties are simulated by the hereditary Boltzmann– Volterra theory with the Rabotnov relaxation kernel. Nonlinear boundary problems are solved by the modified Newton–Kantorovich method. The calculation is effected for the two cases of curing the rubber layer onto the metal elements of the structure. The first case assumes that the rubber layer is vulcanized onto the metal plates, in the second one, it can freely slide over their surface. For the first case, numerical results are compared with the analytical solution. The effect of geometrical nonlinearity and viscoelastic rubber properties on the rubber layer thickness reduction was examined.

Về sự hội tụ của một sơ đồ sai phân đối với bài toán biên phi tuyến loại Parabol

This paper deals with a finite difference scheme for a nonlinear boundary problem of parabolic type. Under the essential assumption about the regularity of the solution of the continuous problem a second order convergence rate O(r2 + h2) is established for approximate solution of the finite difference scheme.

Rapid identification of pre-buckling states: A case of cylindrical shell

The problem to identify pre-buckling states for thin-walled shell corresponds to the problem to identify pre-bifurcation solutions (the inverse bifurcation problem) for von Karman equations that govern the structure. Typical solution sequences similar to those of post-bifurcation solutions observed along the bifurcation paths of the nonlinear boundary problem for von Karman equations are extracted to serve as precursors of bifurcation (tools to solve the problem). The method allows one to divide all operations required to solve the problem under study into two non-equal parts. The most time-consuming part (to trace bifurcation paths and cluster the respective solution) is performed off-line, while the part of the algorithm that is carried out on-line (the identification algorithm) requires a relatively small number of arithmetic operations. This allows development of the efficient system of rapid identification of pre-buckling states.

Error identities for variational problems with obstacles

AbstractThe paper is devoted to analysis of a class of nonlinear free boundary problems that are usually solved by variational methods based on primal, dual or primal‐dual variational settings. We deduce and investigate special relations (error identities). They show that a certain nonlinear measure of the distance to the exact solution (specific for each problem) is equivalent to the respective duality gap, whose minimization is the keystone of all variational numerical methods. Therefore, the identity actually sets the measure that contains maximal quantitative information on the quality of a numerical solution available through these methods. The measure has quadratic terms generated by the linear part of the differential operator and nonlinear terms associated with the free boundary. We obtain fully computable two sided bounds of this measure and show that they provide efficient estimates of the distance between the minimizer and any function (approximation) from the corresponding energy space. Several computational examples show that for different minimization sequences the balance between the quadratic and the nonlinear terms of the overall error measure may be different and essential contribution of nonlinear terms may serve as an indicator that the free boundaries are approximated very roughly.

Infiltration of MHD liquid into a deformable porous material

Abstract We analyze the capillary rise dynamics for magnetohydrodynamics (MHD) fluid flow through deformable porous material in the presence of gravity effects. The modeling is performed using mixture theory approach and mathematical manipulation yields a nonlinear free boundary problem. Due to the capillary rise action, the pressure gradient in the liquid generates a stress gradient that results in the deformation of porous substrate. The capillary rise process for MHD fluid slows down as compared to Newtonian fluid case. Numerical solutions are obtained using a method of lines approach. The graphical results are presented for important physical parameters, and comparison is presented with Newtonian fluid case.

Three-Dimensional Steady Supersonic Euler Flow Past a Concave Cornered Wedge with Lower Pressure at the Downstream

In this paper, we study the stability of the three-dimensional jet created by a supersonic flow past a concave cornered wedge with the lower pressure at the downstream. The gas beyond the jet boundary is assumed to be static. It can be formulated as a nonlinear hyperbolic free boundary problem in a cornered domain with two characteristic free boundaries of different types: one is the rarefaction wave, while the other one is the contact discontinuity, which can be either a vortex sheet or an entropy wave. A more delicate argument is developed to establish the existence and stability of the square jet structure under the perturbation of the supersonic incoming flow and the pressure at the downstream. The methods and techniques developed here are also helpful for other problems involving similar difficulties.

A nonlinear free boundary problem with a self-driven Bernoulli condition

We study a Bernoulli type free boundary problem with two phasesJ[u]=∫Ω|∇u(x)|2dx+Φ(M−(u),M+(u)),u−u¯∈W01,2(Ω), where u¯∈W1,2(Ω) is a given boundary datum. Here, M1 and M2 are weighted volumes of {u⩽0}∩Ω and {u>0}∩Ω, respectively, and Φ is a nonnegative function of two real variables.We show that, for this problem, the Bernoulli constant, which determines the gradient jump condition across the free boundary, is of global type and it is indeed determined by the weighted volumes of the phases.In particular, the Bernoulli condition that we obtain can be seen as a pressure prescription in terms of the volume of the two phases of the minimizer itself (and therefore it depends on the minimizer itself and not only on the structural constants of the problem).Another property of this type of problems is that the minimizer in Ω is not necessarily a minimizer in a smaller subdomain, due to the nonlinear structure of the problem.Due to these features, this problem is highly unstable as opposed to the classical case studied by Alt, Caffarelli and Friedman. It also interpolates the classical case, in the sense that the blow-up limits of u are minimizers of the Alt–Caffarelli–Friedman functional. Namely, the energy of the problem somehow linearizes in the blow-up limit.As a special case, we can deal with the energy levels generated by the volume term Φ(0,r2)≃r2n−1n, which interpolates the Athanasopoulos–Caffarelli–Kenig–Salsa energy, thanks to the isoperimetric inequality.In particular, we develop a detailed optimal regularity theory for the minimizers and for their free boundaries.

Monotone iterates for systems of nonlinear integro-elliptic equations

This paper deals with numerically solving systems of nonlinear integro-elliptic equations. We give a monotone iterative method, based on the method of upper and lower solutions. The construction of the initial upper and lower solution is discussed, and numerical experiments are presented. References I. Boglaev, Numerical solving systems of nonlinear integro-parabolic equations of Volterra type. J. Integral Equ. Appl. , 28(3) :309–342, 2016. doi:10.1216/JIE-2016-28-3-1 C. V. Pao, Nonlinear Parabolic and Elliptic Equations . Plenum Press, New York, 1992. C. V. Pao, Monotone iterative methods for numerical solutions of nonlinear integro-elliptic boundary problems. Appl. Math. Comput. , 186 :1624–1642, 2007. doi:10.1016/j.amc.2006.08.074 A. Samarskii, The Theory of Difference Schemes . Marcel Dekker, New York, 2001. A. H. Stroud, Approximate Calculation of Multiple Integrals . Prentice–Hall, Englewood Cliffs, New Jersey, 1971.