Visiting Researchers:

Prof. D. Natroshvili,

**Objectives**

1. To analyse the equivalence of
Localised Boundary-Domain Integral and Integro-Differential Equations, LBDI(D)Es, to the original boundary value problems for
partial differential equations with variable coefficients, existence and
uniqueness of LBDI(D)E solutions, and invertibility of their operators. |

**Summary of
project findings**

The project developed rigorous
mathematical backgrounds of an emerging new family of computational methods
for solution of partial differential equations (PDEs) of science and
engineering. The approach is based on reducing the original linear or
nonlinear boundary value problems for PDEs to localised boundary-domain
integral or integro-differential equations, which
after mesh-based or mesh-less discretisation lead to systems of algebraic
equations with sparse matrices. This is especially beneficial for problems
with variable coefficients, where no fundamental solution is available in an
analytical and/or cheaply calculated form, but the approach employs a widely
available localised parametrix instead. The following objectives
have been reached in the project. 1.
The LBDI(D)Es, of linear boundary value problems for elliptic scalar PDEs and PDE systems of the
second order with variable coefficients were analysed. This includes the proofs
of the LBDI(D)E equivalence to the original BVPs,
existence and uniqueness of LBDI(D)E solutions, and invertibility of their
operators. 2.
The spectral properties of the BDI(D)Es of the second kind were
investigated and iterative methods for their solution, using the information
about the spectral properties, were developed applied in numerical
calculations. 3.
The analysis was extended to
some nonlinear LBDI(D)Es. |

Further implementation of the results in effective and robust computer codes based on LBDI(D)Es to solve problems of heat transfer and stress analysis of structure elements made of "functionally graded" materials, variable-curvature inhomogeneous elastic shells, filtration through inhomogeneous rocks etc, will have a very definite impact in the area of numerical methods and computational mechanics both in the UK and internationally.

The project also paves the way to extend the LBDIE approach to non-elliptic PDEs of the second order, e.g. Maxwell, parabolic and hyperbolic PDE systems, as well as to higher order equations. The project results for some nonlinear LBDI(D)Es can be also essentially generalised.

**Potential use in non-academic contexts**

The project analytical results were implemented in numerical algorithms and experimental computer codes and the obtained results were 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.

Chkadua O., Mikhailov S.E., Natroshvili D., 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)*, accepted. PDF

Chkadua O., Mikhailov S.E., Natroshvili D., Analysis of direct segregated
boundary-domain integral equations for variable-coefficient mixed BVPs in
exterior domains, *Analysis and Applications**,*
Vol.11, 2013, DOI: 10.1142/S0219530513500061
(accepted, to appear). PDF

Mikhailov S.E. Solution regularity and co-normal
derivatives for elliptic systems with non-smooth coefficients on Lipschitz
domains. *J.
Math. Analysis and Appl**.,** *Vol. 400,*
*2013, 48-67. PDF

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

Mikhailov S.E., Mohamed N.A., Numerical solution and spectrum of
boundary-domain integral equation for the Neumann BVP with variable
coefficient, *International J. Computer Math.**,* Vol. 89, 2012, 1488-1503. PDF

Mikhailov
S.E. Traces, extensions and co-normal derivatives for elliptic
systems on Lipschitz domains. *J.
Math. Analysis and Appl**.,*
Vol. 378, 2011, 324-342. PDF

Chkadua O., Mikhailov S.E., Natroshvili D. Localized
direct segregated boundary-domain integral equations for variable-coefficient
transmission problems with interface crack, *Mem**. Differential Equations Math. Phys.* Vol. **52,** 2011,
17-64. PDF

Chkadua O., Mikhailov S.E., Natroshvili D. Analysis of segregated
boundary-domain integral equations for variable-coefficient problems with
cracks, *Numer**. Meth. for PDEs**, *Vol. 27, 2011, 121-140. PDF

Ayele T.G., Mikhailov
S.E., Analysis of Two-Operator Boundary-Domain Integral Equations for
Variable-Coefficient Mixed BVP,* **Eurasian Math. J. **, *Vol. 2, No 3, 20-41. PDF

Chkadua O., Mikhailov S.E., Natroshvili D. 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
(BirkhŠuser): Boston, ISBN 978 0
85316 2957, 2011, 91-108. PDF

Chkadua O., Mikhailov S.E., Natroshvili D., Analysis of segregated
boundary-domain integral equations for mixed variable-coefficient BVPs in
exterior domains, in: *Integral
Methods in Science and Engineering**:* *Computational and Analytic Aspects.* C. Constanda, P. Harris, eds., Springer (BirkhŠuser): Boston, ISBN 978 0 85316 2957, 2011,
109-128. PDF

Chkadua O., Mikhailov S.E.,
Natroshvili D., Localized boundary-domain integral equations for Dirichlet
problem for second order elliptic equations with matrix variable coefficients,
in: *Proceedings of the 8th UK Conference on Boundary Integral Methods, *(edited
by D. Lesnic), Leeds University Press, Leeds, UK , ISBN 978 0 85316 2957,
(2011), 119-126. PDF

Chkadua O., Mikhailov S.E., Natroshvili D., Localized boundary-domain
integral equations method for an interface crack problem, in: *Proceedings of
the 8th UK Conference on Boundary Integral Methods, *(edited by D. Lesnic), Leeds University Press, Leeds, UK , ISBN 978 0
85316 2957, (2011), 49-56. PDF

Chkadua O., Mikhailov S.E., Natroshvili D. Localized boundary-domain
integral equation formulation for mixed type problems, *Georgian Math.* *J.,* Vol.17, 2010, 469-494. PDF

Chkadua O., Mikhailov S.E., Natroshvili D. Analysis of
some localized boundary-domain integral equations, *J. Integral Equations and
Appl.* Vol.21, 2009, 405-445. PDF

Chkadua O.,
Mikhailov S.E., Natroshvili D. Analysis of direct boundary-domain integral
equations for a mixed BVP with variable coefficient, I: Equivalence and
Invertibility*, **J. Integral Equations and Appl.* Vol.21(4), 2009, 499-543. PDF

Chkadua O., Mikhailov S.E., Natroshvili D. 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. PDF

Mikhailov S.E. Analysis of united boundary-domain integro-differential and integral equations for a mixed BVP
with variable coefficient. *Math. Methods in Applied Sciences*, Vol. 29,
2006, 715-739. PDF

Mikhailov S.E. Localized direct
boundary-domain integro-differential formulations for
incremental elasto-plasticity of inhomogeneous body. *Engineering
Analysis with Boundary Elements*, Vol. 30, 2006, 218-226. PDF

Mikhailov S. E. Incremental Localized
Boundary-Domain Integro-Differential Equations of Elastic Damage Mechanics for
Inhomogeneous Body, In: *Advances in Meshless Methods* (Edited by J.
Sladek & V. Sladek), Tech Science Press, Forsyth, USA, ISBN: 0-9717880-2-2,
2006, 105-123. PDF

Mikhailov S.E. Will the boundary (-domain) integral
equation method survive? Preface to the special issue on non-traditional
boundary (-domain) integral equation methods.* J. Engineering Math*., Vol.
51, 2005, 197-198. PDF

Mikhailov S.E., Nakhova I.S. Mesh-based numerical
implementation of the localized boundary-domain integral equation method to a
variable-coefficient Neumann problem.* J. Engineering Math*., Vol. 51,
2005, 251-259. PDF

Mikhailov S.E. Localized direct boundary–domain integro–differential formulations for scalar
nonlinear boundary-value problems with variable coefficients.* J. Engineering
Math*. , Vol. 51, 2005, 283-302. PDF

Mikhailov S.E. Direct localized boundary-domain integro-differential formulations for physically nonlinear
elasticity of inhomogeneous body. *Engineering Analysis with Boundary
Elements*, Vol. 29, 2005, 1008–1015. PDF

Mikhailov S. E. Analysis of extended boundary-domain
integral and integro-differential equations of some
variable-coefficient BVP. In: *Advances in Boundary Integral Methods -
Proceedings of the 5th UK Conference on Boundary Integral Methods* (Edited
by Ke Chen), University of Liverpool Publ., UK, ISBN
0 906370 39 6, 2005, 106-125. PDF

Mikhailov S.E. Analysis of boundary-domain integral and integro-differential equations for a Dirichlet problem with
variable coefficient. In: *Integral Methods in Science and Engineering:
Theoretical and Practical Aspects* (Edited by C.Constanda,
Z.Nashed, D.Rolins),
Boston-Basel-Berlin: BirkhŠuser, ISBN 0-8176-4377-X, 2005, 161-176. PDF

Mikhailov S.E. Boundary-Domain
Integro-Differential Equation of Elastic Damage Mechanics Model of Stationary
Drilling. In: *Advances in Boundary Element Techniques VI *(Edited
by A.P.S.Selvadurai, C.L.Tan
& M.H.Aliabadi),* *EC Ltd., UK, ISBN
09547783-2-4, 2005, 107-114. PDF

Mikhailov S.E. Localized boundary-domain
integral formulations for problems with variable coefficients. *Engineering Analysis with Boundary Elements*, Vol. 26, 2002,
681-690. PDF

Mikhailov S.E. Finite-dimensional
perturbations of linear operators and some applications to boundary integral
equations, *Engineering Analysis with Boundary Elements*, 1999, Vol.23,
805-813. PDF