# Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs

Lead Research Organisation:
Brunel University

Department Name: Mathematics

### Abstract

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.

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.

### Planned Impact

Although this is mainly mathematical analysis project, in a longer run its results can be implemented in effective and robust computer codes for solving problems of nonlinear heat transfer, nonlinearly elastic or elasto-plastic stress analysis of structure elements, including elastic shells under large deformations, nonlinear filtration through inhomogeneous rocks and Navier-Stokes problem of fluid flow. Through application in engineering and design, these computer codes will increase efficiency and reliability of engineering structures and machines, leading to the material and energy savings and increased competitiveness.

The PhD students in the PI research group will do some experimental prototype computer implementation, in parallel to the project, and its results will be informed to the prospective users through journal and conference publications and the project web-site, as well as through individual contacts with prospective users in computational mechanics. If the experimental numerical implementation proves to be successful, a commercial software can stem from it in 5-10 year period. This would then benefit the software developers and numerous users in mechanical, structural, civil, marine, and aerospace engineering including design.

The PhD students in the PI research group will do some experimental prototype computer implementation, in parallel to the project, and its results will be informed to the prospective users through journal and conference publications and the project web-site, as well as through individual contacts with prospective users in computational mechanics. If the experimental numerical implementation proves to be successful, a commercial software can stem from it in 5-10 year period. This would then benefit the software developers and numerous users in mechanical, structural, civil, marine, and aerospace engineering including design.

## People |
## ORCID iD |

Sergey E. Mikhailov (Principal Investigator) |

### Publications

Ayele T
(2017)

*Integral Methods in Science and Engineering, Volume 1*
Chkadua O
(2016)

*Localized boundary-domain singular integral equations of Dirichlet problem for self-adjoint second-order strongly elliptic PDE systems*in Mathematical Methods in the Applied Sciences
Dufera T
(2015)

*Integral Methods in Science and Engineering*
Gutt R
(2017)

*On the mixed problem for the semilinear Darcy-Forchheimer-Brinkman PDE system in Besov spaces on creased Lipschitz domains*in Mathematical Methods in the Applied Sciences
Kohr M
(2016)

*Integral potential method for a transmission problem with Lipschitz interface in $$\mathbb{R}^{3}$$ R 3 for the Stokes and Darcy-Forchheimer-Brinkman PDE systems*in Zeitschrift für angewandte Mathematik und Physik
Kohr M
(2016)

*Transmission Problems for the Navier-Stokes and Darcy-Forchheimer-Brinkman Systems in Lipschitz Domains on Compact Riemannian Manifolds*in Journal of Mathematical Fluid Mechanics
Lanza De Cristoforis M
(2018)

*Two-parameter anisotropic homogenization for a Dirichlet problem for the Poisson equation in an unbounded periodically perforated domain. A functional analytic approach*in Mathematische Nachrichten
Mikhailov S
(2015)

*Integral Methods in Science and Engineering*Description | 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. 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. Some preliminary findings are given below. (1) We considered the three-dimensional Dirichlet boundary value problem (BVP) for a second-order strongly elliptic self-adjoint system of partial differential equations in the divergence form with variable coef?cients and developed the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localised layer and volume potentials, we reduce the Dirichlet BVP to a system of localised boundary-domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localised boundary-domain integral equation system was studied. We establish that the obtained localised boundary-domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener-Hopf factorisation method, we investigated corresponding Fredholm properties and proved invertibility of the localised operator in appropriate Sobolev (Bessel potential) spaces. (2) A novel family of boundary-domain Integral equations for a mixed elliptic BVP with variable coefficient based on a novel parametrix was analysed. The mixed (Dirichlet-Neumann) boundary value problem for the steady-state Stokes system of PDEs for an incompressible viscous fluid with variable viscosity coefficient was reduced to a system of direct segregated Boundary-Domain Integral Equations (BDIEs). Mapping properties of the potential type integral operators appearing in these equations were presented in appropriate Sobolev spaces. Equivalence between the original BVP and the corresponding BDIE system has been proved. (3) We obtained existence and uniqueness results in weighted Sobolev spaces for transmission problems for the nonlinear Darcy-Forchheimer-Brinkman system and the linear Stokes system of partial differential equations in two complementary Lipschitz domains in R^3, one of them is a bounded Lipschitz domain O with connected boundary, and the other one is the exterior Lipschitz domain R^3 \ O. We exploited a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces. (4) The transmission type problems for the Navier-Stokes and Darcy-Forchheimer-Brinkman partial differential systems in two complementary Lipschitz domains on a compact Riemannian manifold of dimension 2 or 3 were studied. We exploited a layer potential method combined with a fixed point theorem in order to show existence and uniqueness results when the given data are suitably small in L_2-based Sobolev spaces. |

Exploitation Route | It is expected that the project results will be useful for mathematicians working in applied analysis and also mathematicians and engineers engaged in numerical solution of nonlinear BVPs of science and engineering, particularly in computational solid mechanics, fluid dynamics, diffusion, electro- and magnetodynamics. Further implementation of the results in effective and robust computer codes based on BDI(D)Es to solve nonlinear problems of heat transfer and stress analysis of structure elements, variable-curvature inhomogeneous elasto-plastic shells, filtration through inhomogeneous rocks, transonic flows, Navier-Stokes equations, etc, will have a very definite impact in the area of numerical methods and computational mechanics both in the UK and internationally. |

Sectors | Aerospace, Defence and Marine |

Description | Ethiopia |

Organisation | Addis Ababa University |

Department | Department of Mathematics |

Country | Ethiopia, Federal Democratic Republic of |

Sector | Academic/University |

PI Contribution | Formulation and analysis of Boundary-Domain Integral Equations |

Collaborator Contribution | Extension to analysis of two-operator Boundary-Domain Integral Equations |

Impact | Dufera T.T., Mikhailov S.E. (2015) Analysis of Boundary-Domain Integral Equations for Variable-Coefficient Dirichlet BVP in 2D, In: Integral Methods in Science and Engineering: Theoretical and Computational Advances. C. Constanda and A. Kirsh, eds., Springer (Birkhäuser): Boston, ISBN 978-3-319-16727-5, 163-175, DOI: 10.1007/978-3-319-16727-5_15. Ayele T.G., Mikhailov S.E., (2011) Analysis of Two-Operator Boundary-Domain Integral Equations for Variable-Coefficient Mixed BVP, Eurasian Math. J., Vol. 2, 2011, No 3, 20-41. |

Start Year | 2010 |

Description | Georgia |

Organisation | Georgian Technical University |

Country | Georgia |

Sector | Academic/University |

PI Contribution | Boundary-Domain Integral Equation formulation and analysis. |

Collaborator Contribution | Pseudo-Differential Equation technique for Boundary-Domain Integral Equations. |

Impact | Papers: Chkadua O., Mikhailov S.E., Natroshvili D. (2016) Localized boundary-domain singular integral equations of Dirichlet problem for self-adjoint second order strongly elliptic PDE systems, Math. Methods in Appl. Sci., DOI: 10.1002/mma.4100, 1-21. Chkadua O., Mikhailov S.E., Natroshvili D. (2013a) Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients, Integral Equations and Operator Theory (IEOT), Vol. 76, 2013, 509-547, DOI 10.1007/s00020-013-2054-4. Chkadua O., Mikhailov S.E., Natroshvili D. (20113b) Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed BVPs in exterior domains, Analysis and Applications, Vol.11, 2013, No 4, 1350006(1-33), DOI: 10.1142/S0219530513500061. Chkadua O., Mikhailov S.E., Natroshvili D. (2011a) Localized direct segregated boundary-domain integral equations for variable-coefficient transmission problems with interface crack, Mem. Differential Equations Math. Phys., Vol. 52, 17-64. Chkadua O., Mikhailov S.E., Natroshvili D. (2011b) Analysis of segregated boundary-domain integral equations for variable-coefficient problems with cracks, Numerical Meth. for PDEs, Vol. 27, 121-140. Chkadua O., Mikhailov S.E., Natroshvili D. (2011c) Analysis of some localized boundary-domain integral equations for transmission problems with variable coefficients, In: Integral Methods in Science and Engineering: Computational and Analytic Aspects. C. Constanda and P. Harris, eds., Springer (Birkhäuser): Boston, ISBN 978-0-8176-8237-8, 91-108. Chkadua O., Mikhailov S.E., Natroshvili D. (2010a) Localized boundary-domain integral equation formulation for mixed type problems, Georgian Math. J., Vol.17, 469-494. PDF Chkadua O., Mikhailov S.E., Natroshvili D. (2010b) Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, II: Solution regularity and asymptotics, J. Integral Equations and Appl. Vol.22, 2010, 19-37. |

Start Year | 2010 |

Description | Georgia |

Organisation | Tbilisi State Medical University |

Country | Georgia |

Sector | Academic/University |

PI Contribution | Boundary-Domain Integral Equation formulation and analysis. |

Collaborator Contribution | Pseudo-Differential Equation technique for Boundary-Domain Integral Equations. |

Impact | Papers: Chkadua O., Mikhailov S.E., Natroshvili D. (2016) Localized boundary-domain singular integral equations of Dirichlet problem for self-adjoint second order strongly elliptic PDE systems, Math. Methods in Appl. Sci., DOI: 10.1002/mma.4100, 1-21. Chkadua O., Mikhailov S.E., Natroshvili D. (2013a) Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients, Integral Equations and Operator Theory (IEOT), Vol. 76, 2013, 509-547, DOI 10.1007/s00020-013-2054-4. Chkadua O., Mikhailov S.E., Natroshvili D. (20113b) Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed BVPs in exterior domains, Analysis and Applications, Vol.11, 2013, No 4, 1350006(1-33), DOI: 10.1142/S0219530513500061. Chkadua O., Mikhailov S.E., Natroshvili D. (2011a) Localized direct segregated boundary-domain integral equations for variable-coefficient transmission problems with interface crack, Mem. Differential Equations Math. Phys., Vol. 52, 17-64. Chkadua O., Mikhailov S.E., Natroshvili D. (2011b) Analysis of segregated boundary-domain integral equations for variable-coefficient problems with cracks, Numerical Meth. for PDEs, Vol. 27, 121-140. Chkadua O., Mikhailov S.E., Natroshvili D. (2011c) Analysis of some localized boundary-domain integral equations for transmission problems with variable coefficients, In: Integral Methods in Science and Engineering: Computational and Analytic Aspects. C. Constanda and P. Harris, eds., Springer (Birkhäuser): Boston, ISBN 978-0-8176-8237-8, 91-108. Chkadua O., Mikhailov S.E., Natroshvili D. (2010a) Localized boundary-domain integral equation formulation for mixed type problems, Georgian Math. J., Vol.17, 469-494. PDF Chkadua O., Mikhailov S.E., Natroshvili D. (2010b) Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, II: Solution regularity and asymptotics, J. Integral Equations and Appl. Vol.22, 2010, 19-37. |

Start Year | 2010 |

Description | Saarbrucken |

Organisation | Saarland University |

Department | Institute of Applied Mathematics |

Country | Germany, Federal Republic of |

Sector | Academic/University |

PI Contribution | Boundary-Domain Integral Equation methodology. |

Collaborator Contribution | Numerical implementation of the Boundary-Domain Integral Equations in a computer code numerical solution of some boundary value problems. |

Impact | Journal paper: Grzhibovskis R., Mikhailov S., Rjasanow S. (2013) Numerics of boundary-domain integral and integro-differential equations for BVP with variable coefficient in 3D, Comput. Mech., Vol. 51, 495-503, DOI: 10.1007/s00466-012-0777-8. |

Start Year | 2012 |