On the existence of solution of the boundary-domain integral equation system derived from the 2D Dirichlet problem for the diffusion equation with variable coefficient using a novel parametrix

Fresneda-Portillo, Carlos; Woldemicheal, Zenebe W.

doi:10.1080/17476933.2019.1687457

Cited by 10 publications

(8 citation statements)

References 22 publications

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“…A discussion on fundamental solution existence theorems, algorithms for constructing fundamental solutions, and parametrices is available in Pomp. 5 Classical examples of derivation of boundary-domain integral equations (BDIEs) are for the diffusion equation 6 with variable coefficient in bounded domains in R 3 ; same problem applying a different parametrix 7 ; the Dirichlet problem 8 in R 2 and the mixed problem for the compressible Stokes system, 2 which is a good example of the application of the BDIE method to a PDE system.…”

Section: Introductionmentioning

confidence: 99%

On the existence and uniqueness of solution of boundary‐domain integral equations for the Dirichlet problem for the nonhomogeneous heat transfer equation defined on a 2D unbounded domain

Woldemicheal

Fresneda-Portillo

2020

Math Methods in App Sciences

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A system of boundary-domain integral equations (BDIEs) is obtained from the Dirichlet problem for the diffusion equation in nonhomogeneous media defined on an exterior two-dimensional domain. We use a parametrix different from the one employed in Dufera and Mikhailov (2019). The system of BDIEs is formulated in terms of parametrix-based surface and volume potentials whose mapping properties are analyzed in weighted Sobolev spaces. The system of BDIEs is shown to be equivalent to the original boundary value problem and uniquely solvable in appropriate weighted Sobolev spaces suitable for unbounded domains.

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Section: Introductionmentioning

confidence: 99%

On the existence and uniqueness of solution of boundary‐domain integral equations for the Dirichlet problem for the nonhomogeneous heat transfer equation defined on a 2D unbounded domain

Woldemicheal

Fresneda-Portillo

2020

Math Methods in App Sciences

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“…To overcome this obstacle, one can construct a parametrix using the known fundamental solution. A discussion on fundamental solution existence theorems, algorithms for constructing fundamental solutions and parametrices is available in [25]; for classical examples of derivation of Boundary Domain Integral Equations refer to [7] for the diffusion equation with variable coefficient in bounded domains in R 3 ; [26] for the same problem applying a different parametrix; [27] for the Dirichlet problem in R 2 and [23] for the mixed problem for the compressible Stokes system, as an example of derivation of BDIEs from a PDE system.…”

Section: Introductionmentioning

confidence: 99%

Boundary-domain integral equations for the diffusion equation in inhomogeneous media based on a new family of parametrices

Fresneda-Portillo

2019

Complex Variables and Elliptic Equations

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A system of Boundary-Domain Integral Equations is derived from the mixed (Dirichlet-Neumann) boundary value problem for the diffusion equation in inhomogeneous media defined on an unbounded domain. This paper extends the work introduced in [26] to unbounded domains. Mapping properties of parametrix-based potentials on weighted Sobolev spaces are analysed. Equivalence between the original boundary value problem and the system of BDIEs is shown. Uniqueness of solution of the BDIEs is proved using Fredholm Alternative and compactness arguments adapted to weigthed Sobolev spaces.

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“…To overcome this obstacle, one can construct a parametrix using the known fundamental solution. A discussion on fundamental solution existence theorems, algorithms for constructing fundamental solutions and parametrices is available in [24]; for classical examples of derivation of Boundary Domain Integral Equations refer to [7] for the diffusion equation with variable coefficient in bounded domains in R 3 ; [25] for the same problem applying a different parametrix; [26] for the Dirichlet problem in R 2 and [22] for the mixed problem for the compressible Stokes system, as an example of derivation of BDIEs from a PDE system.…”

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confidence: 99%

A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains

Fresneda-Portillo¹

2020

Communications on Pure &Amp; Applied Analysis

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A system of Boundary-Domain Integral Equations is derived from the mixed (Dirichlet-Neumann) boundary value problem for the diffusion equation in inhomogeneous media defined on an unbounded domain. This paper extends the work introduced in [25] to unbounded domains. Mapping properties of parametrix-based potentials on weighted Sobolev spaces are analysed. Equivalence between the original boundary value problem and the system of BDIEs is shown. Uniqueness of solution of the BDIEs is proved using Fredholm Alternative and compactness arguments adapted to weigthed Sobolev spaces.

show abstract

On the existence of solution of the boundary-domain integral equation system derived from the 2D Dirichlet problem for the diffusion equation with variable coefficient using a novel parametrix

Cited by 10 publications

References 22 publications

On the existence and uniqueness of solution of boundary‐domain integral equations for the Dirichlet problem for the nonhomogeneous heat transfer equation defined on a 2D unbounded domain

On the existence and uniqueness of solution of boundary‐domain integral equations for the Dirichlet problem for the nonhomogeneous heat transfer equation defined on a 2D unbounded domain

Boundary-domain integral equations for the diffusion equation in inhomogeneous media based on a new family of parametrices

A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains

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