Two operator Boundary-Domain Integral Equations for variable coefficient Mixed BVP in 2D

Ayele, Tsegaye G.; Bekele, Solomon

doi:10.22541/au.158074299.98130594

Cited by 6 publications

(17 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The mapping properties of the parametrix-based volume and surface potentials formulated in Appendix C1 are proved or immediately follow from the literature [3][4][5]11,13 since the spaces H s,t (Ω, A) and H s,t (Ω, B) coincide algebraically and topologically, and T + a = a b T + b . Theorem 19.…”

Section: Conflict Of Interestmentioning

confidence: 60%

“…Corresponding formulations for the mixed problem for u ∈ H 1,0 (Ω, Δ) with f ∈ L 2 (Ω) were introduced and analyzed in the literature. 4,5,11,12,16 Let us represent in (44), (63), and (64) the generalised conormal derivative and the trace of the function u as…”

Section: (I) the Homogeneous Neumann Problems (106) And (107) Admit Only Linearly Independent Solution U 0 = 1 In H 1 (ω) (Ii) The Non-homentioning

confidence: 99%

“…To reduce BVP (127)-( 129) to one or another two-operator BDIE system, we shall use Equation (44) in Ω, and restrictions of Equation( 63) or (64) to appropriate parts of the boundary. We shall always substitute Φ 0 + for + u and Ψ 0 + for T + a (̃, u), cf., the literature 5,11,12 where Φ 0 ∈ H 1 2 ( Ω) and Ψ 0 ∈ H − 1 2 ( Ω) are considered as known, while belongs tõ H − 1 2 ( Ω D ) and toH 1 2 ( Ω N ) due to the boundary conditions ( 128) and ( 129) and are to be found along with u ∈ H 1 (Ω). This will lead us to segregated BDIE systems with respect to the unknown triple…”

Section: Two-operator Boundary-domain Integral Equationsmentioning

confidence: 99%

“…This needed a nontrivial generalization of the third Green identity and its conormal derivative for such functions, which extends the approach implemented in previous studies [4][5][6]16,17 for the PDE right-hand from L 2 (Ω). In this paper, using the two-operator approach in extended settings, different from the one in other works 11,12 and using the results in the literature, 13,18 we derive generalization of the two-operator third Green identity and its conormal derivative and give a rigorous analysis of the two-operator BDIEs for Dirichlet, Neumann, and mixed (Dirichlet-Neumann) problems in the appropriate Sobolev-Slobodetski (Bessel-potential) spaces. This paper extends our publication.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Analysis of two‐operator boundary‐domain integral equations for variable coefficient BVPs with general data

Ayele

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

The Dirichlet, Neumann, and mixed boundary value problems for the linear second-order scalar elliptic differential equation with variable coefficient in a bounded three-dimensional domain are considered. The PDE right-hand side belongs to H −1 (Ω) orH −1 (Ω), when neither classical nor canonical conormal derivatives of solutions are well defined. Using the two-operator approach and appropriate parametrix (Levi function), each problem is reduced to different systems of boundary domain integral equations (BDIEs). Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/nonuniqueness, and invertibility of the BDIE operators are analyzed in appropriate Sobolev (Bessel potential) spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible, and appropriate finite-dimensional perturbations are constructed leading to invertibility of the perturbed operators.

show abstract

Section: Conflict Of Interestmentioning

confidence: 60%

Section: (I) the Homogeneous Neumann Problems (106) And (107) Admit Only Linearly Independent Solution U 0 = 1 In H 1 (ω) (Ii) The Non-homentioning

confidence: 99%

Section: Two-operator Boundary-domain Integral Equationsmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Analysis of two‐operator boundary‐domain integral equations for variable coefficient BVPs with general data

Ayele

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…are Fredholm operators with the same index, ind (T : The same approach can be used to extend, to the general PDE right hand sides, the BDIE systems for the mixed problems, unbounded domains, BDIEs of more general scalar PDEs and the systems of PDEs, as well as to the united and localised BDIEs, for which the analysis is now available for the right hand sides only from L 2 (Ω), with smooth coefficient and domain boundary, see [3]- [10], [29], [2], [13], [35], [36].…”

mentioning

confidence: 99%

Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains

Mikhailov

2018

Bound Value Probl

View full text Add to dashboard Cite

Segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable Hölder-continuous coefficients on Lipschitz domains are formulated. The PDE right-hand sides belong to the Sobolev (Bessel potential) space, when neither strong classical nor weak canonical co-normal derivatives are well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and the Fredholm property and invertibility of the BDIE operators are analysed in appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible; however, some finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators. MSC: 35J25; 31B10; 45K05; 45A05

show abstract

Two‐operator boundary‐domain integral equation systems for variable‐coefficient mixed Boundary Value Problem in 2D

Ayele

Bekele

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, the mixed (Dirichlet-Neumann) boundary value problem (BVP) for the linear second-order scalar elliptic differential equation with variable coefficients in a bounded two-dimensional domain is considered. The two-operator approach and appropriate parametrix (Levi function) are used to reduce this BVP to four systems of two-operator boundary-domain integral equations (BDIEs).Although the theory of two-operator BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the associated Sobolev spaces to ensure the invertibility of corresponding parametrix-based integral layer potentials and hence the unique solvability of BDIEs. The equivalence of the two-operator BDIE systems to the original BVP is shown. The invertibility of the associated operators is proved in the appropriate Sobolev spaces.

show abstract

Two operator Boundary-Domain Integral Equations for variable coefficient Mixed BVP in 2D

Cited by 6 publications

References 0 publications

Analysis of two‐operator boundary‐domain integral equations for variable coefficient BVPs with general data

Analysis of two‐operator boundary‐domain integral equations for variable coefficient BVPs with general data

Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains

Two‐operator boundary‐domain integral equation systems for variable‐coefficient mixed Boundary Value Problem in 2D

Contact Info

Product

Resources

About