A mixed boundary value problem for the stationary heat transfer partial differential equation with variable coefficient is reduced to some systems of direct segregated parametrix-based Boundary-Domain Integral Equations (BDIEs). We use a parametrix different from the one employed by Mikhailov (2002) and Chkadua, Mikhailov, Natroshvili (2009). Mapping properties of the potential type integral operators appearing in these equations are presented in appropriate Sobolev spaces. We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed

Mikhailov, SE

Portillo, CF

English

A mixed boundary value problem for the stationary heat transfer partial di erential equation with variable coe cient is reduced to some systems of direct segregated parametrix-based Boundary-Domain Integral
Equations (BDIEs). We use a parametrix di erent from the one employed by Mikhailov (2002) and Chkadua, Mikhailov, Natroshvili (2009). Mapping properties of the potential type integral operators appearing in these equations are presented in appropriate Sobolev spaces. We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain
integral operators are also analysed

Mikhailov, S

Brunel University Research Archive

Proceedings of the 10th UK Conference on Boundary Integral Methods, University of Brighton,UK, 13-14 July 2015, (Edited by P. Harris), ISBN 978-1-910172-05-6, 2015, 76-849 A New Family of Boundary-Domain Integral Equationsfor a Mixed Elliptic BVP with Variable CoefficientS.E. Mikhailov and C.F. PortilloBrunel University LondonSergey.Mikhailov@brunel.ac.uk, Carlos.Fresneda-Portillo@brunel.ac.ukAbstractA mixed boundary value problem for the stationary heat transfer par-tial differential equation with variable coefficient is reduced to some sys-tems of direct segregated parametrix-based Boundary-Domain IntegralEquations (BDIEs). We use a parametrix different from the one em-ployed by Mikhailov (2002) and Chkadua, Mikhailov, Natroshvili (2009).Mapping properties of the potential type integral operators appearing inthese equations are presented in appropriate Sobolev spaces. We prove theequivalence between the original BVP and the corresponding BDIE sys-tem. The invertibility and Fredholm properties of the boundary-domainintegral operators are also analysed.9.1 Preliminaries and the BVPThe domains. Let Ω = Ω+ be a bounded simply connected domain, Ω− :=R3 rΩ+ the complementary (unbounded) subset of Ω. The boundary S := ∂Ωis simply connected, closed and infinitely differentiable, S ∈ C∞. Furthermore,S := SN ∪ SD where both SN and SD are non-empty, connected disjoint mani-folds of S. The border of these two submanifolds is also infinitely differentiable,∂SN = ∂SD ∈ C∞.PDE. We consider the following partial differential equation:Au(x) := A(x)[u(x)] :=3∑i=1∂∂xi(a(x)∂u(x)∂xi)= f(x), x ∈ Ω, (9.1)where u(x) is an unknown function, a(x) ∈ C∞ is a variable coefficient and fis a given function on Ω. It is easy to see that if a ≡ 1 then, the operator Abecomes the Laplace operator ∆.Function spaces. We will use the following function spaces in this paper (seee.g. [5, 3, 4] for more details). Let D′(Ω) be the Schwartz distribution space;W sp (Ω) and Ws(S), s ≥ 0, the Sobolev-Slobodetski spaces; Hs(Ω) and Hs(S)with s ∈ R, the Bessel potential spaces; the space HsK(R3) consisting of allthe distributions of Hs(R3) whose support is inside of a compact set K ⊂ R3;Hsloc(Ω−) the spaces consisting of distributions in Hs(K) for every compactK ⊂ Ω−, s ∈ R; H˜s(S1) = {g ∈ Hs(S) : supp(g) ∈ S1}; Hs(S1) = {g|S1 : g ∈1Hs(S)}, where the notation g|S1 = rS1g is used to indicate the restriction of thefunction g from S to S1. Note that Hsloc(Ω−) = W s2,loc(Ω−) and Hr(S) = W r2 (S)for r ≥ 0). We will also make use of the space, see e.g. [2, 1],H1,0(Ω;A) := {u ∈ H1(Ω) : Au ∈ L2(Ω)}which is a Hilbert space with the norm defined by‖ u ‖2H1,0(Ω;A):=‖ u ‖2H1(Ω) + ‖ Au ‖2L2(Ω).Traces and conormal derivatives. For a scalar function w ∈ Hs(Ω±), s >1/2, the trace operator γ± = γ±S , acting on w is well defined and γ±w ∈Hs−12 (S) (see, e.g., [5, 7]). For u ∈ Hs(Ω), s > 3/2, we can define on S theconormal derivative operator, T±, in the trace sense:T±[u(x)] :=3∑i=1a(x)ni(x)(∂u∂xi)±= a(x)(∂u(x)∂n(x))±,where n(x) is the exterior unit normal vector to the domain Ω at point x ∈ S.Moreover, for any function u ∈ H1,0(Ω;A), we can extend the definitionto the canonical conormal derivative T±u ∈ H− 12 (Ω), associated with the firstGreen identity, cf. [2, 5, 7],〈T±u,w〉S := ±∫Ω±[(γ−1ω)Au+ E(u, γ−1w)]dx, for all w ∈ H 12 (S), (9.2)where γ−1 : H12 (S)→ H1(R3) is a continuous right inverse to the trace operatorwhereas the function E is defined asE(u, v)(x) :=n∑i=1a(x)∂u(x)∂xi∂v(x)∂xi,and 〈, 〉S represents the L2−based dual form on S.Boundary value problem We aim to derive boundary-domain integral equa-tion systems for the following mixed boundary value problem. Given f ∈ L2(Ω),φ0 ∈ H 12 (SD) and ψ0 ∈ H− 12 (SN ), we seek a function u ∈ H1(Ω) such thatAu = f in Ω, (9.3a)rSDγ+u = φ0 on SD, (9.3b)rSNT+u = ψ0 on SN , (9.3c)where equation (9.3a) is understood in the weak sense, the Dirichlet condition(9.3b) is understood in the trace sense and the Neumann condition (9.3c) isunderstood in the functional sense (9.2).2By Lemma 3.4 of [2] (cf. also Theorem 3.9 in [7]), the first Green identityholds for any u ∈ H1,0(Ω;A) and v ∈ H1(Ω),〈T±u, γ+v〉S := ±∫Ω[vAu+ E(u, v)]dx. (9.4)The following assertion is well know and can be proved, e.g., using the Lax-Milgram lemma.Theorem 9.1.1. The boundary value problem (9.3) has one and only one so-lution.9.2 Parametrices and remaindersDefinition 9.2.1. Let P (x, y) ∈ D′(Ω) be a distribution of two variables. Wesay that P (x, y) is a parametrix (or Levi function) for the operator A(x) ifA(x)[P (x, y)] = δ(x− y) +R(x, y), (9.5)where δ(·) is the Dirac delta distribution and remainder R(x, y) possesses a weak(integrable) singularity at x = y, i.e., R(x, y) = O(|x− y|−k) with k < 3.The notion of parametrix is well known, see e.g. [6] and references therein.For a given operator A, the parametrix is not unique. For example, theparametrixP y(x, y) =1a(y)P∆(x− y), x, y ∈ R3,was employed in [6, 1], for the operator A defined in (9.1), whereP∆(x− y) = −14pi|x− y|is the fundamental solution of the Laplace operator. The remainder correspond-ing to the parametrix Py isRy(x, y) =3∑i=11a(y)∂a(x)∂xi∂∂xiP∆(x− y) , x, y ∈ R3. (9.6)In this paper, for the same operator A defined in (9.1), we will use anotherparametrix,P (x, y) = P x(x, y) =1a(x)P∆(x− y), x, y ∈ R3, (9.7)while the corresponding remainder isR(x, y) = Rx(x, y) = −3∑i=1∂∂xi(1a(x)∂a(x)∂xiP∆(x, y))= −3∑i=1∂∂xi(∂ ln a(x)∂xiP∆(x, y)), x, y ∈ R3.3Note that the both remainders Rx and Ry are weakly singular:Rx(x, y), Ry(x, y) = O(|x− y|−2).This is due to the smoothness of the variable coefficient a.9.3 Volume and surface potentialsThe volume parametrix-based Newton-type potential and the remainder poten-tial are respectively defined, for y ∈ R3, asPρ(y) :=∫ΩP (x, y)ρ(x) dxRρ(y) :=∫ΩR(x, y)ρ(x) dx.The parametrix-based single layer and double layer surface potentials aredefined for y ∈ R3 : y /∈ S, asV ρ(y) := −∫SP (x, y)ρ(x) dS(x),Wρ(y) := −∫ST+x P (x, y)ρ(x) dS(x).We also define the following pseudo-differential operators associated withdirect values of the single and double layer potentials and with their conormalderivatives, for y ∈ S,Vρ(y) := −∫SP (x, y)ρ(x) dS(x),Wρ(y) := −∫STxP (x, y)ρ(x) dS(x)W ′ρ(y) := −∫STyP (x, y)ρ(x) dS(x),L±ρ(y) := T±y Wρ(y).The operators P,R, V,W,V,W,W ′ and L can be expressed in terms the vol-ume and surface potentials and operators associated with the Laplace operator,4as followsPρ = P∆(ρa), (9.8)Rρ = −∇ · [P∆(ρ)∇ ln a] , (9.9)V ρ = V∆(ρa), (9.10)Vρ = V∆(ρa), (9.11)Wρ = W∆ρ− V∆(ρ∂ ln a∂n), (9.12)Wρ =W∆ρ− V∆(ρ∂ ln a∂n), (9.13)W ′ρ = aW ′∆(ρa), (9.14)L±ρ = Lˆρ− aT±∆V∆(ρ∂ ln a∂n), (9.15)Lˆρ := aL∆ρ. (9.16)The symbols with the subscript ∆ denote the analogous surface potentialsfor the constant coefficient case, a ≡ 1. Furthermore, by the Liapunov-Taubertheorem, L+∆ρ = L−∆ρ = L∆ρ.Using relations (9.10)-(9.16) it is now rather simple to obtain, similar to[1], the mapping properties, jump relations and invertibility results for theparametrix-based surface and volume potentials, provided in Theorems 9.3.1-9.3.8, from the well-known properties of their constant-coefficient counterparts(associated with the Laplace equation).Theorem 9.3.1. Let s ∈ R. Then, the following operators are continuous,P : H˜s(Ω) −→ Hs+2(Ω), s ∈ R, (9.17)P : Hs(Ω) −→ Hs+2(Ω), s > −12, (9.18)R : H˜s(Ω) −→ Hs+1(Ω), s ∈ R, (9.19)R : Hs(Ω) −→ Hs+1(Ω), s > −12. (9.20)Corollary 9.3.2. Let s > 12 , let S1 a non-empty submanifold of S with smoothboundary. Then, the following operators are compact:R : Hs(Ω) −→ Hs(Ω),rS1R : Hs(Ω) −→ Hs−12 (S1),rS1T+R : Hs(S1) −→ Hs− 32 (S1).5Theorem 9.3.3. Let s ∈ R. Then the following operators are continuous:V : Hs(S) −→ Hs+ 32 (Ω),W : Hs(S) −→ Hs+ 12 (Ω).Theorem 9.3.4. Let s ∈ R, the following operators are continuous:V : Hs(S) −→ Hs+1(S),W : Hs(S) −→ Hs+1(S),W ′ : Hs(S) −→ Hs+1(S),L± : Hs(S) −→ Hs−1(S).Theorem 9.3.5. Let ρ ∈ H− 12 (S), τ ∈ H 12 (S). Then the following operatorsjump relations hold:γ±V ρ = Vρ,γ±Wτ = ∓12τ +Wτ,T±V ρ = ±12ρ+W ′ρ.Theorem 9.3.6. Let s ∈ R, let S1 and S2 two non-empty manifolds withsmooth boundaries, ∂S1 and ∂S2, respectively. Then, the following operatorsare compact:rS2V : H˜s(S1) −→ Hs(S2),rS2W : H˜s(S1) −→ Hs(S2),rS2W ′ : H˜s(S1) −→ Hs(S2).Theorem 9.3.7. Let S1 be a non-empty simply connected submanifold of Swith infinitely smooth boundary curve, and 0 < s < 1. Then, the operatorsrS1V : H˜s−1(S1) −→ Hs(S1),V : H˜s−1(S) −→ Hs(S)are invertible.Theorem 9.3.8. Let S1 be a non-empty simply connected submanifold of Swith infinitely smooth boundary curve, and 0 < s < 1. Then, the operatorrS1Lˆ : H˜s(S1) −→ Hs−1(S1)is invertible and the operatorrS1(L± − Lˆ) : H˜s(S1) −→ Hs−1(S1)is compact.69.4 Third Green identities and integral relationsIn this section we provide the results similar to the ones in [1] but for our,different, parametrix (9.7).Let u, v ∈ H1,0(Ω;A). Subtracting from the first Green identity (9.4) itscounterpart with the swapped u and v, we arrive at the second Green identity,see e.g. [5], ∫Ω(A(v)u−A(u)v) dx =∫S[T+(v)u− T+(u)v] dS(x). (9.21)Taking now v(x) := P (x, y), we obtain from (9.21) and (9.5) by the standardlimiting procedures (cf. [8]) the third Green identity for any function u ∈H1,0(Ω;A):u+Ru− V T+u+Wγ+u = PAu in Ω. (9.22)If u ∈ H1,0(Ω;A) is a solution of the partial differential equation (9.3a), thenfrom (9.22) we obtain:u+Ru− V T+(u) +Wγ+u = Pf inΩ, (9.23)12γ+u+ γ+Ru− VT+(u) +Wγ+u = γ+Pf on S, (9.24)12T+(u) + T+Ru−W ′T+(u) + L+γ+u = T+Pf on S. (9.25)For some distributions f , Ψ and Ψ, we consider a more general, indirectintegral relation associated with the third Green identity (9.23):u+Ru− VΨ +WΦ = Pf in Ω. (9.26)Lemma 9.4.1. Let u ∈ H1(Ω), f ∈ L2(Ω), Ψ ∈ H− 12 (S) and Φ ∈ H 12 (S)satisfying the relation (9.26). Then u belongs to H1,0(Ω,A) and solves theequation Au = f in Ω, and also the following identity is satisfied,V (Ψ− T+u)−W (Φ− γ+v) = 0 in Ω. (9.27)Proof. To prove that u ∈ H1,0(Ω;A) we take into account that by hypothesisu ∈ H1(Ω), and so there is only left to prove that Au ∈ L2(Ω).Due to (9.8), (9.10) and (9.12), equation (9.26) impliesu =Pf −Ru+ VΨ−WΦ=P∆(fa)−Ru+ V∆(Ψa)−W∆Φ + V∆(∂ ln a∂nΦ). (9.28)We note that Ru ∈ H2(Ω) due to the mapping properties (9.20). Moreover, V∆and W∆ in (9.28) are harmonic potentials, while P∆ is the Newtonian potentialfor the Laplacian, i.e. ∆P∆(fa)=fa. Consequently, ∆u =fa−∆Ru ∈ L2(Ω).Hence, Au ∈ L2(Ω) and u ∈ H1,0(Ω;A).7Since u ∈ H1,0(Ω;A), the third Green identity (9.23) is valid for the functionu, and we proceed subtracting (9.23) from (9.26) to obtainW (γ+u− Φ)− V (T+u−Ψ) = P(Au− f). (9.29)Recalling again the relations (9.8), (9.10) and (9.12), and applying the Laplaceoperator to both sides of (9.29) we obtainAu− f = 0, (9.30)i.e., u solves (9.3a). Finally, substituting (9.30) into (9.29), we prove (9.27).Lemma 9.4.2. Let Ψ∗ ∈ H− 12 (S). IfVΨ∗(y) = 0, y ∈ Ω (9.31)then Ψ∗(y) = 0.Proof. Taking the trace of (9.31)gives:VΨ∗(y) = V4(Ψ∗a)(y) = 0, y ∈ Ω,from where the result follows due to the invertibility of the operator V4 (cf.Theorem 9.3.7).9.5 BDIE system for the mixed problemWe aim to obtain a segregated boundary-domain integral equation system formixed BVP (9.3). To this end, let the functions Φ0 ∈ H 12 (S) and Ψ0 ∈ H− 12 (S)be respective continuations of the boundary functions φ0 ∈ H 12 (SD) and ψ0 ∈H−12 (SN ) to the whole S. Let us now representγ+v = Φ0 + φ, T+(v) = Ψ0 + ψ on S, (9.32)where φ ∈ H˜ 12 (SN ) and ψ ∈ H˜− 12 (SD) are unknown boundary functions.To obtain one of the possible boundary-domain integral equation systems weemploy (9.23) in the domain Ω and (9.24) on S, substituting there γ+u = Φ0 +φand T+u = Ψ0 + ψ and further considering the unknown functions φ and ψas formally independent (segregated) of u in Ω. Consequently, we obtain thefollowing system M12 of two equations for three unknown functions,u+Ru− V ψ +Wφ = F0 in Ω, (9.33a)12φ+ γ+Ru− Vψ +Wφ = γ+F0 − Φ0 on S, (9.33b)whereF0 = Pf + VΨ0 −WΦ0. (9.34)8We remark that F0 belongs to the space H1(Ω) in virtue of the mappingproperties of the surface and volume potentials, see Theorems 9.3.1 and 9.3.3.The system M12, given by (9.33a)-(9.33b) can be written in matrix notationasM12X = F12, (9.35)where X represents the vector containing the unknowns of the system,X = (u, ψ, φ)> ∈ H1(Ω)× H˜− 12 (SD)× H˜ 12 (SN ), (9.36)the right hand side vector isF12 := [F0, γ+F0 −Ψ0]> ∈ H1(Ω)×H 12 (S),and matrix operator M12 is defined by:M12 =[I +R −V Wγ+R −V 12I +W]. (9.37)We note that the mapping properties of the operators involved in the matriximply the continuity of the operatorM12 : H1(Ω)× H˜− 12 (SD)× H˜ 12 (SN ) −→ H1(Ω)×H 12 (S).Theorem 9.5.1 (BDIE-BVP Equivalence). Let f ∈ L2(Ω). Let Φ0 ∈ H 12 (S)and Ψ0 ∈ H− 12 (S) be some fixed extensions of φ0 ∈ H 12 (SD) and ψ0 ∈ H− 12 (SN )respectively.1. If some u ∈ H1(Ω) solves the BVP (9.3), then the triple (u, ψ, φ)> ∈H1(Ω)× H˜− 12 (SD)× H˜ 12 (SN ) whereφ = γ+u− Φ0, ψ = T+u−Ψ0, on S, (9.38)solves the BDIE system M12.2. If a triple (u, ψ, φ)> ∈ H1(Ω) × H˜− 12 (SD) × H˜ 12 (SN ) solves the BDIEsystem then u solves the BVP and the functions ψ, φ satisfy (9.38).3. The system M12 is uniquely solvable.Proof. First, let us prove item 1. Let u ∈ H1(Ω) be a solution of the boundaryvalue problem (9.3) and let φ, ψ be defined by (9.38). Then, due to (9.3b) and(9.3c), we have(ψ, φ) ∈ H˜− 12 (SD)× H˜ 12 (SN ).Then, it immediately follows from the third Green identities (9.23) and (9.24)that the triple (u, φ, ψ) solves BDIE system M12.9Item 2. Let the triple (u, ψ, φ)> ∈ H1(Ω) × H˜− 12 (SD) × H˜ 12 (SN ) solve theBDIE system. Taking the trace of the equation (9.33a) and substract it fromthe equation (9.33b), we obtainφ = γ+u− Φ0 on S. (9.39)This means that the first condition in (9.38) is satisfied. Now, restrictingequation (9.39) to SD, we observe that φ vanishes as supp(φ) ⊂ SN . Hence,φ0 = Φ0 = γ+u on SD and consequently, the Dirichlet condition of the BVP(9.3b) is satisfied.We proceed using the Lemma 9.4.1 in the first equation of the systemM12,(9.33a), with Ψ = ψ + Ψ0 and Φ = φ+ Φ0 which implies that u is a solution ofthe equation (9.3a) and also the following equality:V (Ψ0 + ψ − T+u) +W (Φ0 + φ− γ+u) = 0 in Ω. (9.40)In virtue of (9.39), the second term of the previous equation vanishes. Hence,V (Ψ0 + ψ − T+u) = 0 in Ω. (9.41)Now, in virtue of Lemma 9.4.2 we obtainΨ0 + ψ − T+u = 0 on S. (9.42)Since ψ vanishes on SN , we have Ψ0 = ψ0, and equation (9.42) implies that usatisfies the Neumann condition (9.3c).Since every solution of the BVP is a solution of the BDIEs M12, then theBDIEs has at least one solution. However, every solution of the homogeneousBDIEs can, by item 2, be related with solution of the homogeneous BVP, whichcan be only the trivial solution. This implies that the homogeneous BDIEsolution can be only trivial, which completes the proof of item 3.Lemma 9.5.1. (F0, γ+F0 − Φ0) = 0 if and only if (f,Φ0,Ψ0) = 0Proof. It is trivial that if (f,Φ0,Ψ0) = 0 then (F0, γ+F0−Φ0) = 0. Conversely,supposing that (F0, γ+F0 − Φ0) = 0, then taking into account equation (9.34)and applying Lemma 9.4.1 with F0 = 0 as u, we deduce that f = 0 and VΨ0 −WΦ0 = 0 in Ω. Now, the second equality, γ+F0 − Φ0 = 0, implies that Φ0 = 0on S and applying Lemma 9.4.2 gives Ψ0 = 0 on S.Theorem 9.5.2. The operatorM12 : H1(Ω)× H˜− 12 (SD)× H˜ 12 (SN ) −→ H1(Ω)×H 12 (S),is invertible.Proof. Let M120 be the matrix operator defined byM120 :=[I −V W0 −V 12I]. (9.43)The operator M120 is also bounded due to the mapping properties of the oper-ators involved. Furthermore, the operator10M12 −M120 : H1(Ω)× H˜−12 (SD)× H˜ 12 (SN ) −→ H1(Ω)×H 12 (S)is compact due to the compact mapping properties of the operators R and W,(cf. Theorem 9.3.2 and Theorem 9.3.6).Let us prove that the operatorM120 is invertible.For this purpose, we considerthe following system with arbitrary right hand side F˜ = [F˜1, F˜2]> ∈ H1(Ω) ×H12 (S) and let X = (u, ψ, φ)> ∈ H1(Ω)× H˜− 12 (SD)× H˜ 12 (SN ) be the vector ofunknownsM120 X = F˜ . (9.44)Writing (9.44) component-wise,u− V ψ +Wφ = F˜1, in Ω, (9.45a)12φ− Vψ = F˜2, on S. (9.45b)Equation (9.45b) restricted to SD gives:−rSDVψ = rSD F˜2. (9.46)Due to the invertibility of the operator V (cf. Lemma 9.3.7), equation (9.46)is uniquely solvable on SD. Equation (9.46) means that (Vψ + F˜2) ∈ H˜ 12 (SN ).Thus, the unique solvability of (9.46) implies that φ is also uniquely determinedby the equationφ = (2Vψ + 2F˜2) ∈ H˜ 12 (SN ). (9.47)Consequently, u also is uniquely determined by the first equation (9.45a) of thesystem.u = V ψ −Wφ+ F˜1.Furthermore, since V ψ, Wφ ∈ H1(Ω), we have u ∈ H1(Ω).Thus, the operator M120 is invertible and the operator M12 is a zero indexFredholm operator due to the compactness of the operator M12 −M120 . Hencethe Fredholm property and the injectivity ofM12 due item 3 of to Lemma 9.5.1imply the invertibility of operator M12.References[1] O. Chkadua and S.E. Mikhailov and D. Natroshvili, Analysis of directboundary-domain integral equations for variable-coefficient for a mixedBVP with variable coefficient, I: Equivalence and invertibility J. IntegralEquations and Appl., 21(2009) 499-543.[2] M. Costabel, Boundary integral operators on Lipschitz domains. Elemen-tary results, SIAM J. of Mathematical Analysis, 19 (1988) 613-626.[3] R.Dautray and J. Lions, Mathematical Analysis and Numerical Methods forScience and Technology, Vol. 4 Integral Equations and Numerical Methods,Springer, 1990.11[4] G.C. Hsiao and W.L. Wendland, Boundary Integral Equations, Springer,2008.[5] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,Cambridge University Press, 2000.[6] S.E. Mikhailov, Localized boundary-domain integral formulations for prob-lems with variable coefficients, Engineering Analysis with Boundary Ele-ments, 26 (2002) 681–690.[7] S.E. Mikhailov, Traces, extensions and co-normal derivatives for ellipticsystems on Lipschitz domains, J. of Mathematical Analysis and Appl., 378(2012) 324-342.[8] C. Miranda, Partial Differential Equations of Elliptic Type, 2nd edition.Springer, 1970.12

A new family of boundary-domain Integral equations for a mixed elliptic BVP with variable coefficient

A New Family of Boundary-Domain Integral Equations for a Mixed Elliptic BVP with Variable Coefficient

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A New Family of Boundary-Domain Integral Equations for a Mixed Elliptic BVP with Variable Coefficient

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