EPSRC Reference: 
EP/M013545/1 
Title: 
Mathematical Analysis of BoundaryDomain Integral Equations for Nonlinear PDEs 
Principal Investigator: 
Mikhailov, Professor SE 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
Brunel University London 
Scheme: 
Standard Research 
Starts: 
16 May 2015 
Ends: 
15 May 2018 
Value (£): 
180,968

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
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 BoundaryDomain Integral or IntegroDifferential Equations, BDI(D)Es, which after meshbased or meshless 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 thermoconductivity coefficients depending on the point temperature and coordinate, materials with damageinduced inhomogeneity, elastoplastic 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 boundaryvalue 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 boundarydomain integrodifferential 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 BoundaryDomain IntegroDifferential 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 meshbased or meshless 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 integrodifferential operators. Such analysis is available in the applicants publications for the global and localised BDIEs in the linear case, and for some global indirect nonlinear 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 fixedpoint 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.

Key Findings 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Potential use in nonacademic contexts 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Impacts 
Description 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk 
Summary 

Date Materialised 


Sectors submitted by the Researcher 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Project URL: 

Further Information: 

Organisation Website: 
http://www.brunel.ac.uk 