Acta Polytechnica doi:10.14311/AP.2016.56.0245 Acta Polytechnica 56(3):245–253, 2016 © Czech Technical University in Prague, 2016 available online at http://ojs.cvut.cz/ojs/index.php/ap TWO-DIMENSIONAL HYBRIDS WITH MIXED BOUNDARY VALUE PROBLEMS Marzena Szajewska, Agnieszka Tereszkiewicz∗ Institute of Mathematics, University of Bialystok, 1M Ciolkowskiego, PL-15-245 Bialystok, Poland ∗ corresponding author: a.tereszkiewicz@uwb.edu.pl Abstract. Boundary value problems are considered on a simplex F in the real Euclidean space R2. The recent discovery of new families of special functions, orthogonal on F , makes it possible to consider not only the Dirichlet or Neumann boundary value problems on F , but also the mixed boundary value problem which is a mixture of Dirichlet and Neumann type, ie. on some parts of the boundary of F a Dirichlet condition is fulfilled and on the other Neumann’s works. Keywords: hybrid functions, Dirichlet boundary value problem, Neumann boundary value problem, mixed boundary value problem. 1. Introduction The boundary value problems, considered in this pa- per, occurring in a real Euclidean space R2 on finite region F ⊂ R2 that is half of a square or half of an equilateral triangle. The main idea of this paper is to study the solutions of Helmholtz equation with the mixed boundary value problems. A surprising variety of recently emerged suitable new families of special functions makes that the realization of this idea is relatively simple and straightforward in any dimension. In addition to the classical boundary value problems of Dirichlet and Neumann type, the new functions, called ‘hybrids’ [6, 10], display properties at the boundary of F, on some parts of the boundary being Dirichlet’s, while on the remaining ones Neumann’s. The boundary value conditions play an important role in describing the physical phenomena. They are used, inter alia, in the theory of elasticity, electrostat- ics and fluid mechanics [2, 4, 16]. In Section 2 we introduce some facts about Weyl groups C2 and G2. In Section 3 we show the exact formulas for four families of special functions for each of the group C2 and G2. The branching rules used to separate variables in Section 4 are described in details for example in the following papers [9, 11, 14]. In Section 5 three types of boundary value problems are considered for four families of special functions described in Section 3. Although for the case A1 ×A1, there is no hybrid functions, the mixed boundary value problem occurs. We present this case in details in Appendix. 2. Weyl group C2 and G2 In this section we recall certain facts about Weyl groups C2 and G2 [1, 3, 5]. We use four bases in R2, namely e-, α-, α̌- and ω- basis. The first one, e-basis, is a natural basis for an Euclidean space. The simple root basis, α-basis, exists for every finite group Figure 1. Shaded triangles represent the fundamental regions F for C2 and G2 group. generated by reflections. The co-root basis α̌ is defined by the formula: α̌i = 2αi 〈αi|αi〉 . The ω-basis is dual to simple root basis. The relation- ship between considered bases is standard for group theory and is expressed by: 〈α̌i|ωj〉 = δij. Below we present the α-basis vectors in Cartesian coordinates for each of considered groups: C2 : α1 := 1√2 (1,−1)e, α2 := 2√ 2 (0, 1)e, G2 : α1 := ( √ 2, 0)e, α2 := ( − 1√ 2 , 1√ 6 ) e . The following notation for coordinates is used: R2 3 λ = (a,b)ω = aω1 + bω2. R2 3 x = (x1,x2)α̌ = (y1,y2)e, where indexes ω, e, and α̌ denote ω-, natural-, and α̌-basis, respectively. 245 http://dx.doi.org/10.14311/AP.2016.56.0245 http://ojs.cvut.cz/ojs/index.php/ap Marzena Szajewska, Agnieszka Tereszkiewicz Acta Polytechnica The fundamental regions F for C2 and G2 group, written in ω-basis, have the vertices FC2 = {0,ω1,ω2}, FG2 = {0, ω1 2 ,ω2} and are shown in Figure 1. The groups C2 and G2 can be reduced to a subgroup A1 ×A1 using a branching rule method described in [11, 14]. For C2 case it is done by the projection matrix PC2 = ( 1 1 0 1 ) (1) acting on the whole orbit of a group. The branching rule is the following: O(a,b) PC2−→ O(a + b)O(b) ∪O(b)O(a + b). (2) The reduction from G2 to A1 × A1 is given by the matrix PG2 = ( 1 1 3 1 ) . (3) The branching rule, in this case, has a form: O(a,b) PG2−−−→ O(a + b)O(3a + b) ∪O(2a + b)O(b) ∪O(a)O(3a + 2b). (4) For group A1 ×A1 we use the following notation for coordinates R2 3 x = (x,y)e ∈ A1 ×A1. 3. C-, S-, Ss-, and Sl-functions of G = C2 or G2 The general formula for special functions correspond- ing to the Weyl group [5] is given by∑ w∈G σ(w)e2πi〈wλ|x〉, where the coordinates x = (x1,x2)α̌ ∈ R2 and weight λ = (a,b)ω are given in α̌- and ω-basis, respectively. The homomorphism σ : G → {±1} (by G we de- note the group C2 or G2) determine the four families of special functions [9], that are of interest to us. The map σ(w) is a product of σ(rl),σ(rs) ∈ {±1}, where rl,rs denote long and short reflections in w, respectively. Consequently, there are four types of homomorphisms σ: C : σ(rl) = σ(rs) = 1, S : σ(rl) = σ(rs) = −1, Sl : σ(rl) = −1, σ(rs) = 1, Ss : σ(rl) = 1, σ(rs) = −1. 3.1. Explicit forms of C- and S-functions In this subsection we provide an exact formulas for the two types of special functions, namely, C- and S-functions for C2 and G2 group. The upper signs in the formulas correspond to C(a,b)(x) functions and the lower ones match up to S(a,b)(x) functions [9, 13]: C2 : ±2 [ cos(2π((a + 2b)x1 + (−a− b)x2)) ± cos(2π((a + 2b)x1 − bx2)) + cos(2π(−ax1 + (a + b)x2)) + cos(2π(ax1 + bx2) ] , G2 : 2 [ cos(2π((2a + b)x1 − (3a + 2b)x2) + cos(2π(ax1 + bx2) ± cos(2π(a + b)x1 − bx2) ± cos(2π(ax1 − (3a + b)x2) ± cos(2π((2a + b)x1 − (3a + b)x2) + cos(2π(a + b)x1 − (3a + 2b)x2) ] . 3.2. Explicit form of Ss and Sl-functions Similarly, as in the previous subsection, we present exact formulas for Sl- and Ss- functions. Again, the upper signs correspond to Ss(a,b)(x) function and the lower ones belong to Sl(a,b)(x) function [9, 13]: C2 : 2 [ ∓cos(2π((a + 2b)x1 − (a + b)x2)) ± cos(2π((a + 2b)x1 − bx2)) − cos(2π(−ax1 + (a + b)x2)) + cos(2π(ax1 + bx2) ] , G2 : 2i [ sin(2π((a + b)x1 − (3a + 2b)x2)) + sin(2π(ax1 + bx2)) ± sin(2π((2a + b)x1 − (3a + 2b)x2)) ∓ sin(2π(ax1 − (3a + b)x2)) − sin(2π((2a + b)x1 − (3a + b)x2)) ∓ sin(2π((a + b)x1 − bx2)) ] . Remark 1. The weight coordinates (a,b)ω for the four families of special functions are different, namely C(a,b)(x) : a,b ∈ Z≥0, S(a,b)(x) : a,b ∈ Z>0, Ss(a,b)(x) : { a ∈ Z>0, b ∈ Z≥0 for C2, a ∈ Z≥0, b ∈ Z>0 for G2, Sl(a,b)(x) : { a ∈ Z≥0, b ∈ Z>0 for C2, a ∈ Z>0, b ∈ Z≥0 for G2, The next remark is a consequence of explicit forms of functions written in Subsections 3.1, 3.2. Remark 2. Four families of special functions are real in case of C2 group. The functions C-, S- are real, and Sl-, Ss- are pure imaginary in case of G2 group. 246 vol. 56 no. 3/2016 Two-Dimensional Hybrids with Mixed Boundary Value Problems 4. Helmholtz differential equation In this section we consider the well-known partial differential equation ∆Ψ(x) = −w2Ψ(x), w −positive real constant called homogeneous Helmholtz equation (see for ex- ample [7, 8, 15] and references therein), where x = (y1,y2)e and ∆ = ∂2 ∂y21 + ∂2 ∂y22 . Remark 3 [5]. The special functions described in the previous section are eigenfunctions of the Laplace operator. The explicit form of the Laplace operator in co- ordinates relative to the ω-basis and α̌-basis is the following C2 : ∆ω = 2∂21 − 2∂1∂2 + ∂ 2 2, ∆α̌ = 12∂ 2 1 + ∂1∂2 + ∂ 2 2, G2 : ∆ω = ∂21 − 3∂1∂2 + 3∂ 2 2, ∆α̌ = 2∂21 + 2∂1∂2 + 2 3∂ 2 2. Since ∆e2πi〈λ|x〉 = −4π2〈λ|λ〉e2πi〈λ|x〉 then we have ∆Ψλ(x) = −4π2〈λ|λ〉Ψλ(x), where Ψλ(x) is one of the functions C,S,Ss or Sl. The inner product of λs is equal C2 : 〈λ|λ〉 = 12a 2 + ab + b2, G2 : 〈λ|λ〉 = 2a2 + 2ab + 23b 2. 4.1. Separation of variables for the Helmholtz equation Using a standard method of separation of variables for the Helmholtz equation [7] ∆Ψ(x) = −w2Ψ(x), x = (y1,y2)e, and searching for the solutions in the form Ψ(x) = X(y1)Y (y2), we have the following differential equa- tion X′′Y + XY ′′ + w2XY = 0. Introducing −k2-separation constant, we get a pair of the ordinary differential equations easy to solve: X′′ + k2X = 0, Y ′′ + (w2 −k2)Y = 0. (5) A basic solution of (5) we can write in the form X1(y1) = cos ky1, Y1(y2) = cos √ w2 −k2y2, X2(y1) = sin ky1, Y2(y2) = sin √ w2 −k2y2, where k 6= 0 and w2 −k2 6= 0. According to the assumptions that k 6= 0 and w2 6= k2 we consider C-, S-, Ss-, and Sl-functions only with positive weights. 4.2. C2 case From the projection matrix PC2 (1) and the branching rule (2) we find two separation constants −k21 and −k22, which have a form −k21 = −2(a + b) 2π2, w2 −k21 = 2b 2π2, −k22 = −2b 2π2, w2 −k22 = 2(a + b) 2π2. Noting that k21 = w2 −k22, as a separation constant we take −k2 = −2(a + b)2π2, w2 −k2 = 2b2π2. Using the branching rule (2) from Section 2 for special functions C,S,Ss,Sl we can rewrite those functions in the form: Ca,b(x) = 4 [ cos(ky1) cos( √ w2 −k2y2) + cos( √ w2 −k2y1) cos(ky2) ] , Sa,b(x) = 4 [ sin( √ w2 −k2y1) sin(ky2) − sin(ky1) sin( √ w2 −k2y2) ] , Ssa,b(x) = 4 [ cos(ky1) cos( √ w2 −k2y2) − cos( √ w2 −k2y1) cos(ky2) ] , Sla,b(x) = −4 [ sin( √ w2 −k2y1) sin(ky2) + sin(ky1) sin( √ w2 −k2y2) ] . By changing the variables by y1 = √ 2x,y2 = √ 2 we get the reduction to A1 ×A1 subgroup Ca,b(x) = Ca+b(x)Cb(y) + Cb(x)Ca+b(y), Sa,b(x) = Sa+b(x)Sb(y) −Sb(x)Sa+b(y), Ssa,b(x) = Sa+b(x)Sb(y) + Sb(x)Sa+b(y), Sla,b(x) = Ca+b(x)Cb(y) −Cb(x)Ca+b(y), (6) The functions Cµ(x), and Sµ(x), on the right side of (6), are defined in Appendix. The coordinates (x,y) ∈ A1 ×A1 are written in α-basis. 4.3. G2 case From the projection matrix PG2 (3) and the branching rule (4) we find three separation constants −k21, −k22, and −k23, which have a form −k21 = −2(2a + b) 2π2, w2 −k21 = 2 3b 2π2, (7) −k22 = −2(a + b) 2π2, w2 −k22 = 2 3 (3a + b) 2π2, −k23 = −2a 2π2, w2 −k23 = 2 3 (3a + 2b) 2π2. Using the branching rule (4) from Section 2 for special functions C, S, Ss, Sl we can rewrite those functions 247 Marzena Szajewska, Agnieszka Tereszkiewicz Acta Polytechnica Figure 2. Fundamental regions of C2 and G2 groups. Sides are marked by s and l symbols which correspond to the reflection orthogonal to the short and long root, respectively. Figure 3. The normal vectors and boundaries are indicated for the Weyl group C2. in the form: Ca,b(x) = 4 [ cos(k1y1) cos( √ w2 −k21y2) + cos(k2y1) cos( √ w2 −k22y2) + cos(k3y1) cos( √ w2 −k23y2) ] , Sa,b(x) = 4 [ −sin(k1y1) sin( √ w2 −k21y2) + sin(k2y1) sin( √ w2 −k22y2) − sin(k3y2) sin( √ w2 −k23y2) ] , Ssa,b(x) = 4i [ −cos(k1y1) sin( √ w2 −k21y2) − cos(k2y1) sin( √ w2 −k22y2) + cos(k3y1) sin( √ w2 −k23y2) ] , Sla,b(x) = 4i [ −sin(k1y1) cos( √ w2 −k21y2) + sin(k2y1) cos( √ w2 −k22y2) + sin(k3y1) cos( √ w2 −k23y2) ] . By changing the variables by y1 = √ 2x,y2 = √ 6 we get the reduction to A1 ×A1 subgroup Ca,b(x) = Ca(x)C3a+2b(y) + Ca+b(x)C3a+b(y) + C2a+b(x)Cb(y), Sa,b(x) = Sa(x)S3a+2b(y) −Sa+b(x)S3a+b(y) + S2a+b(x)Sb(y), Ssa,b(x) = Ca(x)S3a+2b(y) −Ca+b(x)S3a+b(y) −C2a+b(x)Sb(y), Sla,b(x) = Sa(x)C3a+2b(y) −Sa+b(x)C3a+b(y) + S2a+b(x)Cb(y). (8) (D) (N) C2,G2 s l s l Ca,b(x) ∗ ∗ 0 0 Sa,b(x) 0 0 ∗ ∗ (M) (D) (N) C2,G2 s l s l Ssa,b(x) 0 ∗ ∗ 0 Sla,b(x) ∗ 0 0 ∗ Table 1. Behaviour of the functions C, S, Ss and Sl on the boundary ∂F for C2 and G2 group where ∗ denotes any function non-equivalent to 0. The functions Cµ(x), and Sµ(x), on the right side of (8), are defined in Appendix. The coordinates (x,y) ∈ A1 ×A1 are written in α-basis. Proposition 1. Cµ(x), and Sµ(x) functions pre- sented in (8) fulfill the following relationships −k3S3a+2b(x)Ca(x) + k2S3a+b(x)Ca+b(x) −k1Sb(x)C2a+b(x) = √ 3 (√ w2 −k23C3a+2b(x)Sa(x) − √ w2 −k22C3a+b(x)Sa+b(x) − √ w2 −k21Cb(x)S2a+b(x) ) , −k3S3a+2b(x)Sa(x) + k2S3a+b(x)Sa+b(x) + k1Sb(x)S2a+b(x) = √ 3 (√ w2 −k23C3a+2b(x)Ca(x) − √ w2 −k22C3a+b(x)Ca+b(x) − √ w2 −k21Cb(x)C2a+b(x) ) , where ki, √ w2 −ki, i = 1, 2, 3 are defined by (7). 5. Types of boundary conditions In this paper we consider three types of boundary conditions. (1.) The first type, called a Dirichlet boundary condi- tion, defines the value of the function itself: Ψ(x) = f(x), for x ∈ ∂F, (D) where f(x) is a given function defined on the bound- ary. (2.) The second type, called a Neumann boundary condition, defines the value of the normal derivative of the function: ∂Ψ ∂n (x) = f(x), for x ∈ ∂F, (N) where n denotes normal to the boundary ∂F. 248 vol. 56 no. 3/2016 Two-Dimensional Hybrids with Mixed Boundary Value Problems C1,3(x) S1,3(x) Figure 4. The contour plot of C1,3(x), S1,3(x). The triangle denotes the fundamental domain F of the affine Weyl group C2. Sl1,3(x) Ss1,3(x) Figure 5. The contour plot of Sl1,3(x), Ss1,3(x). The triangle denotes the fundamental domain F of the affine Weyl group C2. Figure 6. The normal vectors and boundaries are indicated for the Weyl group G2 (3.) The third type, called a mixed boundary condi- tion, defines the value of the function itself on one part of the boundary and the value of the normal derivative of the function on the other part of the boundary:{ Ψ(x) = f0(x) for x ∈ ∂F0, ∂Ψ ∂n (x) = f1(x) for x ∈ ∂F1, (M) where ∂F = ∂F0 ∪∂F1 and f0(x),f1(x) are given functions, defined on the appropriate boundary. Remark 4 [12]. For the Dirichlet boundary conditions all eigenvalues are positive. For the Neumann boundary condition all eigenval- ues are non-negative. In Table 1 we present how the four types of func- tions, defined in Section 3, behave on the boundary ∂F of the fundamental region F . The fundamental re- gion F for C2 and G2 groups is presented in Figure 2. Symbol s corresponds to the reflection orthogonal to the short root and l corresponds to the reflection orthogonal to the long root. 5.1. C2 case The normal vectors to the fundamental region F of the Weyl group C2 are the following: n1 = (0,−1)e, n2 = ( − 1√ 2 , 1√ 2 ) e , n3 = (1, 0)e. In Figure 3 we present the fundamental region F with indicated boundaries and corresponding normal vectors. The values of the four families of special functions C,S,Ss and Sl satisfying the Dirichlet boundary con- dition (D) on the boundary ∂F of the fundamental Figure 7. A shaded square represents the fundamen- tal region F of A1 × A1. Figure 8. The Boundaries of the fundamental region F of A1 × A1 are indicated. region F are presented in Tables 2, 4, and 5. Tables 3– 5 present the values of the functions satisfying the Neumann boundary condition (N). The examples of functions and their behaviours on the boundary ∂F is presented in Figures 4 and 5. 5.2. G2 case The normal vectors to the fundamental region F of the Weyl group G2 are the following: n1 = (−1, 0)e, n2 = (√3 2 ,− 1 2 ) e , n3 = (1 2, √ 3 2 ) e In Figure 6 we present the fundamental region F with indicated boundaries and corresponding normal vectors. The values of the functions for group G2 fulfilling the Dirichlet boundary condition (D) on the bound- ary of the fundamental region F are presented in Tables 6, 8, and 9. The values of the functions satisfy- ing the Neumann boundary condition (N) are given in Tables 7–9. The examples of functions and their be- haviours on the boundary ∂F is presented in Figures 9 and 10. 249 Marzena Szajewska, Agnieszka Tereszkiewicz Acta Polytechnica Ca,b(x) Dirichlet condition F1 2Ca+b(x) + 2Cb(x) F2 2Ca+b(x)Cb(x) F3 Ca+b(1)Cb(y) + Cb(1)Ca+b(y) Table 2. Values of Ca,b(x) function at the boundary of the fundamental region F in C2 case. Sa,b(x) Neumann condition F1 2i ( kSb(x) − √ w2 −k2Sa+b(x) ) F2 2i (√ w2 −k2Sa+b(x)Cb(x) −kSb(x)Ca+b(x) ) F3 i (√ w2 −k2Ca+b(1)Sb(y) −kCb(1)Sa+b(y) ) Table 3. Values of Sa,b(x) function at the boundary of the fundamental region F in C2 case. Mixed condition Ssa,b(x) Dirichlet condition Neumann condition F1 2(Ca+b(x) −Cb(x)) 0 F2 0 2i ( − √ w2 −k2Sa+b(x)Cb(x) + kSb(x)Ca+b(x) ) F3 C(a+b)(1)Cb(y) −Cb(1)Ca+b(y) 0 Table 4. Values of Ssa,b(x) function at the boundary of the fundamental region F in C2 case. Mixed condition Sla,b(x) Dirichlet condition Neumann condition F1 0 2i ( − √ w2 −k2Sb(x) −kSa+b(x) ) F2 2Sa+b(x)Sb(x) 0 F3 0 i (√ w2 −k2Ca+b(1)Sb(y) + kCb(1)Sa+b(y) ) Table 5. Values of Sla,b(x) function at the boundary of the fundamental region F in C2 case. Ca,b(x) Dirichlet condition F1 2(Cb(y) + C3a+b(y) + C3a+2b(y)) F2 C2a+b(x)Cb(x) + Ca+b(x)C3a+b(x) + Ca(x)C3a+2b(x) F3 C2a+b(x)Cb(x− 1) + Ca+b(x)C3a+b(x− 1) + Ca(x)C3a+2b(x− 1) Table 6. Values of Ca,b(x) function at the boundary of the fundamental region F in G2 case. Sa,b(x) Neumann condition F1 2i(−k3S3a+2b(y) + k2S3a+b(y) −k1Sb(y)) F2 −2i (√ w2 −k23 C3a+2b(x)Sa(x) − √ w2 −k22 C3a+b(x)Sa+b(x) + √ w2 −k21 Cb(x)S2a+b(x) ) F3 2i(k3S3a+2b(x− 1)Ca(x) −k2S3a+b(x− 1)Ca+b(x) + k1Sb(x− 1)C2a+b(x)) Table 7. Values of Sa,b(x) function at the boundary of the fundamental region F in G2 case. 250 vol. 56 no. 3/2016 Two-Dimensional Hybrids with Mixed Boundary Value Problems Mixed condition Ssa,b(x) Dirichlet condition Neumann condition F1 2(−Sb(x) −S3a+b(x) + S3a+2b(x)) 0 F2 0 2i (√ w2 −k23C3a+2b(x)Ca(x) + √ w2 −k22C3a+b(x)Ca+b(x) + √ w2 −k21Cb(x)C2a+b(x) ) F3 −C2a+b(x)Sb(x− 1) −Ca+b(x)S3a+b(x− 1) + Ca(x)S3a+2b(x− 1) Table 8. Values of Ssa,b(x) function at the boundary of the fundamental region F in G2 case. Mixed condition Sla,b(x) Dirichlet condition Neumann condition F1 0 2i(−k3C3a+2b(y) −k2C3a+b(y) + k1Cb(y)) F2 −S2a+b(x)Cb(x) + Sa+b(x)C3a+b(x) + Sa(x)C3a+2b(x) 0 F3 0 2i(k3C3a+2b(x− 1)Ca(x) + k2C3a+b(x− 1)Ca+b(x) −k1Cb(x− 1)C2a+b(x)) Table 9. Values of Sla,b(x) function at the boundary of the fundamental region F in G2 case.