COEFFICIENTS OF SINGULARITIES OF THE BIHARMONIC PROBLEM OF NEUMANN TYPE: CASE OF THE CRACK

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1 IJMMS 23:5, PII. S X Hindawi Publishing Corp. COEFFICIENTS OF SINGULARITIES OF THE BIHARMONIC PROBLEM OF NEUMANN TYPE: CASE OF THE CRACK WIDED CHIKOUCHE and AISSA AIBÈCHE Received 13 April 21 This paper concerns the biharmonic problem of Neumann type in a sector V. We give a representation of the solution u of the problem in a form of a series u = α E c α r α φ α, and the functions φ α are solutions of an auxiliary problem obtained by the separation of variables. 2 Mathematics Subject Classification: 35B65, 35C1, 35J Introduction. Let V be a sector of angle ω 2π defined by V = { r cosθ,r sinθ R 2 ;<r <ρ, <θ<ω } 1.1 and the circular boundary part defined by = { ρ cosθ,ρsinθ R 2 ;<θ<ω }. 1.2 We are interested in the study of functions u, belonging to the Sobolev spaces H 2 V, solutions of 2 u = in V, Mu = Tu= for θ =,ω, 1.3 where Mu = ν u+1 ν 1 2 un un 1 n un2, Tu= u n +1 ν d 2 ds 1 un 1 n 2 12 u n n2 2 2 un 1 n 2, ν is a real number called Poisson coefficient <ν<1/2. We show that these functions u are written under the form ur,θ = α E c α r α φ α θ, 1.5

2 36 W. CHIKOUCHE AND A. AIBÈCHE E is the set of solutions of the equation in α sin 2 α 1ω = 1 ν 2 α 1 2 sin 2 ω, Reα> ν For the study of the solutions of 1.6, see, for example, Blum and Rannacher 1] and Grisvard 2]. We are going to calculate the coefficients c α of development 1.5. These calculations have already been done by Tcha-Kondor 3] for the Dirichlet s boundary conditions. He has established, thanks to the Green s formula, a relation of biorthogonality between the functions φ α and φ allowing him to calculate the coefficients c. We follow the same approach. This needs the writing in the domain V of an appropriate Green formula. Using this formula, we establish a relation of biorthogonality between the functions φ α and φ, which is reduced under some conditions to the simple relation obtained by Tcha-Kondor, which enables us to calculate the coefficients c in the particular case of the crack ω = 2π. 2. Separation of variables. Replacing u by r α φ α θ in problem 1.3 leads us to the boundary value problem φ 4 α θ+ α 2 +α 2 2] φ αθ+α 2 α 2 2 φ α θ =, 2.1 να 2 +1 να ] φ α +φ α =, θ =, θ= ω, να 2 31 να+21 ν ] φ α +φ 3 α =, θ =, θ= ω. 2.3 The relation similar to orthogonality for the biharmonic operator is given by the following theorem. Theorem 2.1. Let φ α and φ be solutions of 2.1 withα and solutions of 1.6. So, for α, we have the following relation: ] ω { α φα,φ = 2 2α φ α ν α+ ] +3 ν 2α φ α φ α φ + ν α+ } +3 ν 2 φ ]φ α dθ α 2.4 =. Proof. We use the following Green formula: V v 2 u u 2 v { dx = ut v + u Γ n Mv vtu+ v } n Mu dσ, 2.5

3 COEFFICIENTS OF SINGULARITIES OF THE BIHARMONIC PROBLEM Γ is the boundary of V. For two functions u,v solutions of 1.3, we get the Green s formula in the following form: { ut v + u n Mv vtu+ v } n Mu dσ =. 2.6 On, we have, for the function u α = r α φ α, u α n = u α r = αr α 1 φ α, Mu α = r α 2{ α 2 1 να ] φ α +νφ α}, 2.7 Tu α = r α 3{ α 2 α 2φ α + ν 2α+3 ν ] φ } α. The theorem results from the application of formula 2.6 to the biharmonic functions u α = r α φ α and u = r φ, and by using relations 2.7. Remark 2.2. This relation between the functions φ α and φ is similar to the relation of biorthogonality obtained when the functions φ α and φ fulfill 2.1 with the Dirichlet s boundary conditions φ α = φ α = φ = φ = forθ = and θ = ω. In this case, the relation is simplified because we have ω φ α φ dθ = ω φ αφ dθ. 2.8 Remark 2.3. By a double integration by parts, we get ω φ α φ dθ = ω φ α φ dθ + φ α,φ ] ω φ α,φ ] ω. 2.9 Corollary 2.4. of 1.6; in addition, Let φ α and φ be solutions of 2.1 withα and solutions φα,φ ] ω φ ] ω α,φ =. 2.1 So, for α, we have the following relation: φα,φ ] = ω { α 2 2α φ α +φ 2 α] φ + 2 φ +φ } ]φ α dθ = Remark 2.5. For u α = r α φ α, we have u α 2 r u α r = r α 2 α 2 2α φ α +φ α]. 2.12

4 38 W. CHIKOUCHE AND A. AIBÈCHE LetP be the operator P= 2/r / r. From Corollary 2.4 and Remark 2.5, we deduce the following corollary. Corollary 2.6. Set u α = r α φ α θ and u = r φ, where φ α and φ are solutions of 2.1 withα and solutions of 1.6; in addition, φα,φ ] ω φ ] ω α,φ = If α, we have the following relation: Puα u +u α Pu dσ = Now, using Corollary 2.6, we calculate the coefficients c α of the development of the solution u of 1.3. The calculations will be done in the case of the crack ω = 2π, which is a very important case of singularity of domains. The explicit knowledge of the roots manifestly simplifies the calculations. 3. Case of a crack. The crack corresponds to ω = 2π; if we replace this value in 1.6, we find that solutions α of 1.6 are the real values k/2. In this case, all the roots are of multiplicity 2. We are going to represent u as follows: u = c α u α + { } k d α v α, E = 2,k>2, α E α E u α = r α ϕ α, v α = r α ψ α, 3.1 ϕ α are the even solutions in θ ϕ α θ = r α ] cosα 2θ να 1 να cosαθ, 3.2 and ψ α the odd solutions in θ ψ α θ = r α sinα 2θ 4+1 να 2 ] sinαθ να In this case ω = 2π, we have α = k/2, then ϕ αω = ϕ α =, ψ α ω = ψ α = ; 3.4

5 COEFFICIENTS OF SINGULARITIES OF THE BIHARMONIC PROBLEM hence, ϕ α,ϕ ] ω = ϕ α,ϕ ]ω = and ψ α,ψ ] ω = ψ α,ψ ]ω =. But ϕ α,ψ ] ω = and ϕ α,ψ ]ω. From here comes the idea of decomposing the solution u of 1.3 to its even and odd parts with respect to k u = u 1 +u 2, u i = cα u α +d α v α, i= 1,2, α E i { } 2n+1 E 1 ={n, n > 1}, E 2 =, 2n> Calculation of c and d. We consider the integrals Pui u +u i Pu dσ, Pui v +u i Pv dσ, 3.6 if α E 1, then ϕ α ω = ϕ α, ψ αω = ψ α, if α E 2, then ϕ α ω = ϕ α, ψ αω = ψ α. 3.7 Equations 3.4 and 3.7 allow us to apply Corollary 2.6 to functions u α and u resp., u α, v and v α, v ; then, we obtain Pui u +u i Pu dσ = 2c u Pu dσ +d Pv u +v Pu dσ, Pui v +u i Pv dσ = c Pu v +u Pv dσ +2d v Pv dσ. 3.8 Direct calculation gives us Pu v +u Pv dσ =, u Pu dσ = 2ρ2 1 ω ] 1 ν3+ν 8, 1 ν 2 v Pv dσ = 2ρ2 1 ω ] 1 ν3+ν ν So, we have just established the following proposition. Proposition 3.1. Let u be the solution of 1.3 written in the form u = u 1 +u 2, 3.1 where u i = α E i cα u α +d α v α. 3.11

6 31 W. CHIKOUCHE AND A. AIBÈCHE Suppose that the series that gives u i is uniformly convergent; so, if E i, i = 1,2, then 1 ν 2 ρ 1 2 c = 4ω 1 ν3+ν 8 ] Pui u +u i Pu dσ, 1 ν 2 ρ d = 4ω 1 ν3+ν 2+8 ] Pui v +u i Pv dσ. Remark 3.2. We have ζ H 3/2, thetraceofu on and χ H 1/2, the trace of Pu on. Ifζ belongs to the space H 4 ],2π and χ to H 2 ],2π, then we have a uniform convergence of the series in V ρ for all ρ ρ, 3] Independence of the coefficients. We are going to prove that the coefficients c resp., d of the development of the solution u of 1.3 are independent from ρ. We have the following result. Theorem 3.3. The coefficients c and d are independent from ρ. Proof. In order to prove that c is independent from ρ, wearegoingto show that its derivative with respect to ρ is null, and by observing the expression of c Proposition 3.1, we just have to prove that γ = ρ 1 2 Pui u +u i Pu dσ 3.13 has the null derivative with respect to ρ. By derivation in regard to r, we have γ = ω { ui r r 2 ϕ + 2 u i 2 2 u i r r ] u i r 1 ϕ r + u i r r ϕ u ir ϕ +ϕ ] } dθ On, we have u i r 1 2 u i = Tu i +1 ν r 3 θ 1 3 u i, 2 r 2 r θ 2 2 u i 2 2 u i r = Mu 2 i ν ] 1 u i r r ] u i. r 2 θ

7 COEFFICIENTS OF SINGULARITIES OF THE BIHARMONIC PROBLEM Reinjecting these formulas in the expression of γ,weobtain γ = ω Mui r 1 ϕ +Tu i r 2 ϕ dθ + + ω ω { 2 1 ν ] } ϕ +ϕ u i r 1 ν 3 u i r θ ϕ 2 r dθ { 2 1 ν 1 ] 2 u i θ ϕ 2 u i 2 2 ϕ +ϕ ] } r 1 dθ By a double integration by parts, we verify that ω ω 2 u ω i θ ϕ 2 dθ = u i ϕ dθ, u i r θ 2 ϕ dθ = ω u i r ϕ dθ Reinjecting in the expression of γ and putting ρ1 2 ρ in factor, we obtain γ = ρ1 2 ω Mui r 1 ϕ +Tui r ϕ ] ρdθ +ρ 1 2 ω +ρ 1 2 ω 2 1 ν ] ϕ +νϕ r 2 ] u i r ρdθ { 2 2ϕ + 2 ν+3 ν ] ϕ } r 3 u i ρdθ By taking account of 2.7, whose expressions appear explicitly in γ,weobtain } γ { = u ρ1 2 Mu i n +Tu u i iu dσ + Mu n +Tu u i dσ = 3.2 since we come back to the Green s formula 2.6 applied to u i and u. We follow the same analysis to prove the independence of d with respect to ρ Convergence of the series. We write c α and d α in the form c α = I i ρ α, d α = j i ρ a, 3.21

8 312 W. CHIKOUCHE AND A. AIBÈCHE where 1 ν 2 α I i = ρ 4ω 1 ν3+να 8 ] { Pui ϕ α +u i α 2 2α ϕ α +ϕ α] ρ 2 } dσ, 1 ν 2 α J i = ρ 4ω 1 ν3+να 2+8 ] { Pui ψ α +u i α 2 2α ψ α +ψ α] ρ 2 } dσ The solution u of 1.3 is then written as follows: u = u 1 +u 2, 3.23 u i = r α r α ] I i ϕ α + J i ψ α. ρ ρ 3.24 α E i We have the following result. Theorem 3.4. The series 3.24 converges as soon as r<ρ. Proof. Set N i,α = = ω ω { Pui ϕ α +u i α 2 2α ϕ α +ϕ α] ρ 2 } dθ Pu i ϕ α dθ + α 2 2α ρ 2 ω u i ϕ α dθ +ρ 2 ω u i ϕ α dθ We show that N i,α is a product of 1/α by limited term for α large. According to 3.17, we have ω ω u i ϕ α dθ = u i ϕ α dθ Replacing ϕ α by its expression and integrating by parts, we get ω u i ϕ α dθ = 1 α { α ω α 2 u i sinα 2θ dθ 4 1 να 1 να ω } u i sinαθ dθ On the other hand, by a triple integration by parts, we have α 2 2α ω u i ϕ α dθ = 1 { α 2 α α α 2 α ω u i sinα 2θ dθ 4 1 να 1 να ω } u i sinαθ dθ. 3.28

9 COEFFICIENTS OF SINGULARITIES OF THE BIHARMONIC PROBLEM Also, by an integration by parts, we get u i 2 u i ϕ α dθ = 1 ω ui r r α 2 θ 2 2 u i r r θ να ω ui α 1 να θ 2 r ω Then, we deduce the existence of a constant C so as sinα 2θ dθ 2 u i sinαθ dθ. r θ 3.29 Ni,α C α. 3.3 Using this last inequality and remarking that ϕ α is limited, as well as the term 1 ν 2 α 4ω 1 νν +3α 8 ] 3.31 for large α, we deduce the existence of a constant C so as c α r α ϕ α C r α, 3.32 α ρ α E i α E i which converges as soon as r<ρ. In the same way, we prove the convergence of the series α E i d α r α ψ α. References 1] H. Blum and R. Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners, Math. Methods Appl. Sci , no. 4, ] P. Grisvard, Singularities in Boundary Value Problems, Recherches en Mathématiques Appliquées, vol. 22, Masson, Paris, 1992 French. 3] O. Tcha-Kondor, Nouvelles séries trigonométriques adaptées à l étude de problèmes aux limites pour l équation biharmonique. Étude du cas de la fissure New trigonometric series adapted to the study of boundary value problems for the biharmonic equation. The case of the crack], C. R. Acad. Sci. Paris Sér. I Math , no. 5, French. Wided Chikouche: Département de Mathématiques, Centre Universitaire de Jijel, Ouled Aissa, BP 98, 18 Jijel, Algeria address: chikouche@mail.univ-jijel.dz Aissa Aibèche: Département de Mathématiques, Centre Universitaire de Jijel, Ouled Aissa, BP 98, 18 Jijel, Algeria address: aibeche@mail.univ-jijel.dz

10 Mathematical Problems in Engineering Special Issue on Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios Call for Papers Thinking about nonlinearity in engineering areas, up to the 7s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points. Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from Qualitative Theory of Differential Equations, allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers. This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment. Authors should follow the Mathematical Problems in Engineering manuscript format described at Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at mts.hindawi.com/ according to the following timetable: Guest Editors José Roberto Castilho Piqueira, Telecommunication and Control Engineering Department, Polytechnic School, The University of São Paulo, São Paulo, Brazil; piqueira@lac.usp.br Elbert E. Neher Macau, Laboratório Associado de Matemática Aplicada e Computação LAC, Instituto Nacional de Pesquisas Espaciais INPE, São Josè dos Campos, São Paulo, Brazil ; elbert@lac.inpe.br Celso Grebogi, Center for Applied Dynamics Research, King s College, University of Aberdeen, Aberdeen AB24 3UE, UK; grebogi@abdn.ac.uk Manuscript Due December 1, 28 First Round of Reviews March 1, 29 Publication Date June 1, 29 Hindawi Publishing Corporation