© Hindawi Publishing Corp. COEFFICIENTS OF SINGULARITIES OF THE BIHARMONIC PROBLEM OF NEUMANN TYPE: CASE OF THE CRACK

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.

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 β allowing him to calculate the coefficients c β .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 β , which is reduced under some conditions to the simple relation obtained by Tcha-Kondor, which enables us to calculate the coefficients c β in the particular case of the crack (ω = 2π).

Separation of variables.
Replacing u by r α φ α (θ) in problem (1.3) leads us to the boundary value problem ) 3) The relation similar to orthogonality for the biharmonic operator is given by the following theorem.
Corollary 2.6.Set u α = r α φ α (θ) and u β = r β φ β , where φ α and φ β are solutions of (2.1) with α and β solutions of (1.6); in addition, (2.13) If α ≠ β, we have the following relation: 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.

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: and ψ α the odd solutions in θ In this case (ω = 2π), we have α = k/2, then From here comes the idea of decomposing the solution u of (1.3) to its even and odd parts with respect to k (3.5)

Calculation of c β and d β .
We consider the integrals Equations (3.4) and (3.7) allow us to apply Corollary 2.6 to functions u α and u β (resp., u α , v β and v α , v β ); then, we obtain (3.8) Direct calculation gives us (3.9) So, we have just established the following proposition.

Independence of the coefficients.
We are going to prove that the coefficients c β (resp., d β ) 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 β and d β are independent from ρ.
Proof.In order to prove that c β is independent from ρ, we are going to show that its derivative with respect to ρ is null, and by observing the expression of c β (Proposition 3.1), we just have to prove that has the null derivative with respect to ρ.
By derivation in regard to r , we have On Σ, we have (3.15) Reinjecting these formulas in the expression of γ β , we obtain By a double integration by parts, we verify that Reinjecting in the expression of γ β and putting ρ 1−2β • ρ in factor, we obtain By taking account of (2.7), whose expressions appear explicitly in γ β , we obtain since we come back to the Green's formula (2.6) applied to u i and u β .We follow the same analysis to prove the independence of d β with respect to ρ.

Convergence of the series. We write c α and d α in the form
where The solution u of (1.3) is then written as follows: We have the following result.

Proof.
Set (3.25) We show that N i,α is a product of 1/α by limited term for α large.
According to (3.17), we have Replacing ϕ α by its expression and integrating by parts, we get On the other hand, by a triple integration by parts, we have (3.28) Also, by an integration by parts, we get sin αθ dθ. (3.29) Then, we deduce the existence of a constant C 0 so as Using this last inequality and remarking that ϕ α is limited, as well as the term for large α, we deduce the existence of a constant C so as 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 .

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, 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 http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: