Boundary Problem Research Articles

The Born approximation in the three-dimensional Calderón problem Ⅱ: Numerical reconstruction in the radial case

In this work we illustrate a number of properties of the Born approximation in the three-dimensional Calderón inverse conductivity problem by numerical experiments. The results are based on an explicit representation formula for the Born approximation recently introduced by the authors. We focus on the particular case of radial conductivities in the ball $ B_R \subset \mathbb{R}^3 $ of radius $ R $, in which the linearization of the Calderón problem is equivalent to a Hausdorff moment problem. We give numerical evidences that the Born approximation is well defined for $ L^{\infty} $ conductivities, and present a novel numerical algorithm to reconstruct a radial conductivity from the Born approximation under a suitable smallness assumption. We also show that the Born approximation has depth-dependent uniqueness and approximation capabilities depending on the distance (depth) to the boundary $ \partial B_R $. We then investigate how increasing the radius $ R $ affects the quality of the Born approximation, and the existence of a scattering limit as $ R\to \infty $. Similar properties are also illustrated in the inverse boundary problem for the Schrödinger operator $ -\Delta +q $, and strong recovery of singularity results are observed in this case.

Solving Neumann Boundary Problem with Kernel-Regularized Learning Approach

Solving Neumann Boundary Problem with Kernel-Regularized Learning Approach

Analysis of a free boundary problem for vascularized tumor growth with time delays and almost periodic nutrient supply

<abstract><p>In this research, we have proposed and investigated a time-delayed free boundary problem concerning tumor growth in the presence of almost periodic nutrient supply with angiogenesis. This study primarily focused on examining the impact of almost periodic nutrient supply, angiogenesis, and time delay on tumor growth dynamics. We analyzed the existence, uniqueness, and exponential stability of almost periodic solutions. Furthermore, we established conditions for the disappearance of almost periodic oscillations in tumors. The existence and uniqueness of almost periodic solutions were proven, while sufficient conditions for the exponential stability of the unique solution were established. Finally, computer simulations were employed to illustrate our results.</p></abstract>

A survey of results of St.Petersburg State University research school on nonlinear partial differential equations I

The article contains a review of the most important results obtained in the framework of the St.Petersburg State University research school on nonlinear PDEs (the O.A. Ladyzhenskaya - N.N. Uraltseva school). The main attention is paid to the works carried out at our university over the past 50 years. The first part of the review concerns the solvability and qualitative properties of solutions to boundary value problems for the second order scalar quasilinear elliptic and parabolic equations, as well as variational problems. The planned second part of the review will include sections on fully nonlinear equations and systems of equations and on free boundary problems.

Dynamics of a Leslie-Gower predator-prey model with advection and free boundaries

This paper investigates a Leslie-Gower predator-prey model with advection and double free boundaries. Firstly, we establish the global existence and uniqueness of the solution. Subsequently, we analyze the long-time behaviors of the solution and demonstrate a spreading-vanishing dichotomy. If the initial habitat size or the expanding capacity of the boundaries is large, the invasive predator will spread throughout the entire space; otherwise, both the prey and predator will eventually vanish. When spreading occurs, two subcases arise, depending on the interaction coefficient $ c $ and the environmental support $ \delta $. If $ c\delta < 1 $, the prey and predator coexist, which we refer to as the weakly hunting case; conversely, if $ c\delta \geq 1 $, the predator survives and spreads throughout the space, while the prey vanishes, which we term the strongly hunting case. In the event of spreading, we also provide an estimate of the spreading speed. Finally, we introduce a front-fixing implicit-explicit finite difference method for the free boundary problem. Numerical simulations validate our theoretical findings and uncover some intriguing new phenomena, prompting further investigation into this and general free boundary problems.

Well-posedness of a free boundary problem of early atherosclerotic plaque formation with reverse cholesterol transport

Well-posedness of a free boundary problem of early atherosclerotic plaque formation with reverse cholesterol transport

On a class of Lyapunov's inequality involving $ \lambda $-Hilfer Hadamard fractional derivative

<abstract><p>In this paper, we presented and proved a general Lyapunov's inequality for a class of fractional boundary problems (FBPs) involving a new fractional derivative, named $ \lambda $-Hilfer. We proved a criterion of existence which extended that of Lyapunov concerning the ordinary case. We used this criterion to solve the fractional differential equation (FDE) subject to the Dirichlet boundary conditions. In order to do so, we invoked some properties and essential results of $ \lambda $-Hilfer fractional boundary value problem (HFBVP). This result also retrieved all previous Lyapunov-type inequalities for different types of boundary conditions as mixed. The order that we considered here only focused on $ 1 < r\leq 2 $. General Hartman-Wintner-type inequalities were also investigated. We presented an example in order to provide an application of this result.</p></abstract>

Dynamical behavior of solutions of a free boundary problem

<abstract><p>This paper is concerned with the spreading properties for a reaction-diffusion equation with free boundary condition. We obtained a complete description of the long-time dynamical behavior of this problem. By introducing a parameter $ \sigma $ in the initial data, we revealed a threshold value $ \sigma^* $ such that spreading happens when $ \sigma > \sigma^* $ and vanishing happens when $ \sigma\leq \sigma^* $. There exists a unique $ L^* > 0 $ independent of the initial data such that $ \sigma^* = 0 $ if and only if the length of initial occupying interval is no smaller than $ 2L^* $. These theoretical results may have important implications for prediction and prevention of biological invasions.</p></abstract>

Stability analysis of an implicit fractional integro-differential equation via integral boundary conditions

The primary objective of this research study is to analyze a boundary problem involving Caputo fractional integro-differential equations. The focus is on a differential equation with a nonlinear right-hand side composed of two terms. The stability analysis of a fractional integro-differential equation is presented using Ulam's concept. Furthermore, this research study establishes the correlation between the stated problem and the Volterra integral equation. The investigation proceeds by utilizing the renowned Banach and Krasnoselskii's fixed point theorems to explore the existence and uniqueness of solutions for the problem. Additionally, to provide tangible evidence of the abstract findings, two illustrative examples are presented.

LANDAU LEGENDRE WAVELET GALERKIN METHOD APPLIED TO STUDY TWO-PHASE MOVING BOUNDARY PROBLEM OF HEAT TRANSFER IN FINITE REGION

In this paper, we developed a mathematical model of solidification where specific heat and thermal conductivity are temperature dependent. This model is a two-phase moving boundary problem (MBP) of heat transfer in finite region and represents as MBP of system of parabolic nonlinear second-order partial differential equations (PDEs). We developed a Landau Legendre wavelet Galerkin method for finding the solution of the problem. The MBP of a system of PDEs is transformed into a variable boundary value problem of nonlinear ordinary defferential equations (ODEs) by the use of dimensionless variables and the Landau transform. The problem is converted into a system of algebraic equations with the application of Legendre wavelet Galerkin method. In particular case, we compared present solution with Laplace transform solution and found approximately the same. The whole investigation has been done in dimensionless form. When the specific heat and thermal conductivity exponentially vary in temperatures, it is the effect of dimensionless parameters: thermal diffusivity (α<sub>12</sub>), ratio of thermal conductivity (<i>k</i><sub>12</sub>), dimensionless temperature (θ<sub><i> f</i></sub>), Fourier number (F<sub>0</sub>), Stefan number (Ste), and ratio of densities (ρ<sub>1</sub> / ρ<sub>2</sub>) are discussed in detail.

Linear stability for a free boundary problem modeling the growth of tumor cord with time delay.

This paper was concerned with a free boundary problem modeling the growth of tumor cord with a time delay in cell proliferation, in which the cell location was incorporated, the domain was bounded in $ \mathbb{R}^2 $, and its boundary included two disjoint closed curves, one fixed and the other moving and a priori unknown. A parameter $ \mu $ represents the aggressiveness of the tumor. We proved that there exists a unique radially symmetric stationary solution for sufficiently small time delay, and this stationary solution is linearly stable under the nonradially symmetric perturbations for any $ \mu > 0 $. Moreover, adding the time delay in the model leads to a larger stationary tumor. If the tumor aggressiveness parameter is bigger, the time delay has a greater effect on the size of the stationary tumor, but it has no effect on the stability of the stationary solution.

Symmetry-breaking longitude bifurcation for a free boundary problem modeling the growth of tumor cord in three dimensions

Symmetry-breaking longitude bifurcation for a free boundary problem modeling the growth of tumor cord in three dimensions

Analysis and simulations of a free boundary problem modelling phototrophic granular biofilms

Analysis and simulations of a free boundary problem modelling phototrophic granular biofilms

Symmetry-breaking combined latitude-longitude bifurcations for a free boundary problem modeling small plaques

Symmetry-breaking combined latitude-longitude bifurcations for a free boundary problem modeling small plaques

Coefficient inverse problem for anisotropic time-domain wave equation

In this paper, we investigate an inverse problem of determining coefficient via partial boundary data in anisotropic media. More precisely, we study a scheme for asymptotic reconstruction on coefficient inverse problem for anisotropic time-domain wave equation by a geometric control method. As an initial attempt, it seems natural to treat the case of two dimensional Euclidean space in analogy with many other well-known inverse boundary problems.

A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions Ⅲ : General case

A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions Ⅲ : General case

Research on the Personal Data Portability Based on the Perspective of Property Rights Theory in the Context of Enterprise Digital Transformation

This study examines the right to data portability under the industrial economics theory, and addresses the problems of ambiguous boundaries, inconsistent formats, and data sharing. Based on the relational transactions in China, this study analyses the right to data portability from three aspects: transaction boundaries, transaction frequency, and exclusive nature. From the viewpoint of transaction uncertainty, it is necessary to clarify the content of transaction data and define the property rights boundary. From the view of transaction frequency, more detailed regulation of data processing mode is needed to reduce transaction costs in the future. The data processing mode needs to find a balance between security and rapidness. The data processing mode should imitate distributed data transfer protocol or decentralized recommendation system. From the perspective of asset exclusivity, data sharing should be classified for different users. The relationship between user privacy and data sharing should be properly balanced to prevent a “one-size-fits-all” data management model. This study applies property rights theory to address the issue of data portability, focuses on the economic benefits brought by data portability, and expands the perspective of the data portability’s nature. Based on relational transactions in China, this study enriches the research of personal data protection in developing countries. In addition, the results provide a theoretical basis for government management, help define the boundary of digital assets, promote the digital transformation of cities, and facilitate sustainable economic development.

On a boundary problem for the fourth order equation with the third derivative with respect to time

In this paper, we consider a boundary value problem in a rectangular domain for a fourth-order homogeneous partial differential equation containing the third derivative with respect to time. The uniqueness of the solution of the stated problem is proved by the method of energy integrals. Using the method of separation of variables, the solution of the considered problem is sought as a multiplication of two functions X (x) and Y (y). To determine X (x),we obtain a fourth-order ordinary differential equation with four boundary conditions at the segment boundary [0,p], and for a Y (y) – third-order ordinary differential equation with three boundary conditions at the boundary of the segment [0,q]. Imposing conditions on the given functions, we prove the existence theorem for a regular solution of the problem. The solution of the problem is constructed in the form of an infinite series, and the possibility of term-by-term differentiation of the series with respect to all variables is substantiated. When substantiating the uniform convergence, it is shown that the “small denominator” is different from zero.

HF radio channel modeling by a waveguide approach

We present a modified method of HF radio channel modeling based on a waveguide approach. The waveguide approach represents the electromagnetic field of radiation inside the Earth—ionosphere waveguide as an eigenfunction series of a radial boundary problem with impedance conditions on the Earth surface and radiation conditions at infinity. The transfer function of the radio channel is represented as a series of products of angular-operator Green functions, excitation coefficients, and coefficients for receiving individual normal modes. We have obtained a solution of the boundary value problem of determining the eigenfunctions and eigenvalues of the radial operator. The solution can be applied to the frequency range below the ionospheric F-layer critical frequency. We examine algorithms for calculating distance-frequency, frequency-angular, and amplitude characteristics of signals by analyzing and numerically summarizing the series in terms of strongly damped normal modes.

Pengukuran Dan Pemetaan Fotogrametris Menggunakan Unmaned Aerial Vehicle (UAV) Di Kota Samarinda Kelurahan Sungai Pinang Dalam

Soil plays an important role for all living things, where soil is a source of livelihood. It acts as a planting medium, a water reservoir, and a place for us to do activities. All of these activities are carried out on the land (Natasha Hutabarat et al., n.d.). In the community itself, land ownership is always a problem that is often encountered, whether it is the problem of unfixed land boundaries, land tenure, and land grabbing. The National Land Agency (BPN) as the institution that organizes land registration has a Complete Systematic Land Registration (PTSL) program that makes it easier for the community to register land so as to avoid land conflicts. In order to accelerate land registration, PTSL in Samarinda City this time uses media or tools to take measurements using the photogrammetric method, namely the Unmanned Aerial Vehicle (UAV) to obtain aerial images or the latest images. In its implementation, the community is asked to install boundary signs that have been delivered or socialized by BPN employees. So that the boundaries of the land plot can be seen from the UAV and visible from aerial images