Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs

非线性偏微分方程边界域积分方程的数学分析

• 批准号: EP/M013545/1
• 负责人: Sergey Mikhailov
• 金额: $ 23.06万
• 依托单位: Brunel University London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2015
• 资助国家: 英国
• 起止时间: 2015 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FM013545%2F1
• 关键词: Mathematical; Analysis; Boundary; Domain; Integral

The proposal is aimed at developing rigorous mathematical backgrounds of an emerging new family of computational methods for solution of nonlinear Partial Differential Equations (PDEs). The approach is based on reducing the original nonlinear boundary value problems for PDEs to global or localised Boundary-Domain Integral or Integro-Differential Equations, BDI(D)Es, which after mesh-based or mesh-less discretisation lead to nonlinear systems of algebraic equations. In case of localised BDI(D)Es, the matrices of corresponding algebraic equations will be sparse. Nonlinear PDEs arise naturally in mathematical modelling of nonlinear physical processes, e.g. of nonlinear heat transfer in materials with the thermo-conductivity coefficients depending on the point temperature and coordinate, materials with damage-induced inhomogeneity, elasto-plastic materials, nonlinear equation of stationary potential compressible flow, nonlinear flows trough porous media, nonlinear electromagnetics and other areas of physics and engineering. The main ingredient for reducing a boundary-value problem for a linear PDE to a boundary integral equation is a fundamental solution to the original PDE. However, it is generally not available in an analytical and/or cheaply calculated form for linear PDEs with variable coefficients and for nonlinear PDEs. Developing ideas of Levi and Hilbert, one can use in this case a parametrix (Levi function) either to the original nonlinear PDE or to another, linear, PDE as a substitute for the fundamental solution. Parametrix is usually much wider available than fundamental solution and correctly describes the main part of the fundamental solution although does not have to satisfy the original PDE. This generally reduces the nonlinear boundary value problem not to a boundary integral equation but to a global nonlinear boundary-domain integro-differential equation. A discretisation of a global nonlinear BDIDE system leads to a system of nonlinear algebraic equations of the similar size as in the finite element method (FEM), however the matrix of the system is not sparse. The Localised Boundary-Domain Integro-Differential Equations, LBDIDEs, for nonlinear problems, emerged recently addressing this deficiency and making them competitive with the FEM for such problems. The LBDIDE method employs specially constructed localised parametrices to reduce nonlinear BVPs with variable coefficients to LBDIDEs. After employing a locally supported mesh-based or mesh-less discretisation, this leads to sparse systems of nonlinear algebraic equations efficient for computations. However implementation of this idea requires a deeper analytical insight into properties of the corresponding nonlinear integral and integro-differential operators. Such analysis is available in the applicants publications for the global and localised BDIEs in the linear case, and for some global indirect non-linear BDIEs. The project is intended to make a leap from these results to the analysis of much more general nonlinear global and localised BDIDEs. Further development of the project concerns the iterative algorithms to solve the global or localised nonlinear BDIDEs, particularly based on the fixed-point theorems. It is also expected that the project analytical results will be implemented in numerical algorithms and computer codes developed under the PI supervision by PhD students.

该提案旨在为求解非线性偏微分方程（PDEs）的新计算方法提供严格的数学背景。该方法基于将pde的原始非线性边值问题简化为全局或局部边界域积分或积分微分方程（BDI(D)Es），这些方程在基于网格或无网格的离散化之后导致代数方程的非线性系统。对于局部化的BDI(D)Es，相应的代数方程的矩阵将是稀疏的。非线性偏微分方程自然地出现在非线性物理过程的数学建模中，例如：依赖于点温度和坐标的导热系数的材料的非线性传热、具有损伤诱导的非均匀性的材料、弹塑性材料、定势可压缩流动的非线性方程、多孔介质的非线性流动、非线性电磁学以及其他物理和工程领域。将线性偏微分方程的边值问题化为边界积分方程的关键是对原偏微分方程的基本解。然而，对于变系数的线性偏微分方程和非线性偏微分方程，通常不能以解析和/或廉价的计算形式得到。发展列维和希尔伯特的思想，在这种情况下，我们可以使用参数矩阵（列维函数）来替代原始的非线性偏微分方程或另一个线性偏微分方程作为基本解的替代品。参数化通常比基本解广泛得多，虽然不需要满足原PDE，但它正确地描述了基本解的主要部分。这通常将非线性边值问题简化为全局非线性边域积分微分方程而不是边界积分方程。对全局非线性BDIDE系统进行离散化，得到与有限元法（FEM）相似大小的非线性代数方程组，但该系统的矩阵不是稀疏的。求解非线性问题的局部边界域积分微分方程（LBDIDEs）最近出现，解决了这一缺陷，并使其与FEM在此类问题上具有竞争力。LBDIDE方法采用特殊构造的局部参数将变系数非线性bvp降为LBDIDE。在采用局部支持的基于网格或无网格的离散化之后，这导致了计算效率高的非线性代数方程的稀疏系统。然而，实现这一思想需要对相应的非线性积分和积分微分算子的性质有更深的分析洞察力。这种分析可以在申请人的出版物中获得，用于线性情况下的全局和局部bdie，以及一些全局间接非线性bdie。该项目旨在实现从这些结果到更一般的非线性全局和局部BDIDEs分析的飞跃。该项目的进一步发展涉及求解全局或局部非线性BDIDEs的迭代算法，特别是基于不动点定理。预计项目分析结果将在PI指导下由博士生开发的数值算法和计算机代码中实现。

项目成果

Mapping properties of weakly singular periodic volume potentials in Roumieu classes

Roumieu 类中弱奇异周期体积势的映射特性

• DOI: 10.1216/jie.2020.32.129
• 发表时间: 2020
• 期刊: Journal of Integral Equations and Applications
• 影响因子: 0.8
• 作者: Dalla Riva M

Localized boundary-domain singular integral equations of Dirichlet problem for self-adjoint second-order strongly elliptic PDE systems

自伴二阶强椭圆偏微分方程组狄利克雷问题的局域边界域奇异积分方程

• DOI: 10.1002/mma.4100
• 发表时间: 2016
• 期刊: Mathematical Methods in the Applied Sciences
• 影响因子: 2.9
• 作者: Chkadua O

Developing a well-received pre-matriculation program: the evolution of MedFIT.

制定广受好评的预科课程：MedFIT 的演变。

• DOI: 10.1007/978-3-319-11970-0_12
• 发表时间: 2022
• 期刊: Discover education
• 作者: Allen A

Singular localised boundary-domain integral equations of acoustic scattering by inhomogeneous anisotropic obstacle

非均匀各向异性障碍物声散射的奇异局域边域积分方程

• DOI: 10.1002/mma.5268
• 发表时间: 2018
• 期刊: Mathematical Methods in the Applied Sciences
• 影响因子: 2.9
• 作者: Chkadua O

Integral Methods in Science and Engineering, Volume 1

科学与工程中的积分方法，第 1 卷

• DOI: 10.1007/978-3-319-59384-5_3
• 发表时间: 2017
• 作者: Ayele T