Article

On the solution of boundary value problem. The decomposition method

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

Abstract

The method developed by Adomian, the decomposition method, supplies approximate solution to the original nonlinear and/or stochastic ordinary, partial, or integral differential equations. The approach may be used to find analytical approximations of the solution of the original equations. Since no linearization or perturbation is used and also since the solution can be approximated as close a needed, this approximate analytical solution forms a very useful tool for analyzing boundary value problems encountered in practice. It should be pointed out that these boundary value problems are usually unstable and very difficult to solve numerically. Furthermore, even for nonlinear problems without the consideration of the boundary value difficulties, linearization or perturbation approaches frequently alter the characteristics of the original problem completely. In this paper, we wish to show the difficulties in handling nonlinear boundary value problems and how the decomposition method can be used to analyze these type of problems.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

... This statement seems to have been known for 1 We denote by π all the projections of the fiber bundles E, DE, SE and S(R ⊕ E) to B. a long time. Nevertheless, the reader will find here two proofs, one that uses the Hodge-Morrey-Friedrichs decomposition as in [15] and one which is more elementary. In Section 6 of [1], Bott and Chern give a slight extension of the result from [6] to the case when the rank of the bundle equals the dimension of the manifold (see Proposition 6.3 from [1]). ...
... The subscript D is meant to suggest Dirichlet boundary conditions as they are usually called ( [15]). It is easy to infer from the short exact sequence of chain complexes ...
... is non-degenerate. At this point, the Hodge-Morrey-Friederichs decomposition theorem on a manifold with boundary as developed in [15] proves useful. In particular, Theorem 2.6.1 in [15] saya that every absolute deRham cohomology class in M can be represented by a harmonic field with Neuman boundary conditions, i.e. a form ω which satisfies: ...
Article
We use the mapping cone for the relative deRham cohomology of a manifold with boundary in order to show that the Chern-Gauss-Bonnet Theorem for oriented Riemannian vector bundles over such manifolds is a manifestation of Lefschetz Duality in any of the two embodiments of the latter. We explain how Thom isomorphism fits into this picture, complementing thus the classical results about Thom forms with compact support. When the rank is odd, we construct, by using secondary transgression forms introduced here, a new closed pair of forms on the disk bundle associated to a vector bundle, pair which is Lefschetz dual to the zero section.
... . Remarquons que par homogénéité de J, on peut supposer que pour tout i, M |Φ i | 2 dv(g) = 1. Or puisque l'opérateur de Dirac D sur H − définit un opérateur de Fredohlm, il existe une constante réelle C 0 > 0 telle que pour tout ψ ∈ C − \ {0}, on ait (voir [Sch95] par exemple) : ...
... ψ ∈ C − . Le champ de spineurs Φ vérifie au sens des distributions l'équation D 2 Φ = αΦà l'intérieur de M. En utilisant le théorème d'injection de Sobolev et les estimations elliptiques classiques (voir[Sch95] par exemple), on vérifie facilement que Φ est lisseà l'intérieur de M. Les théorèmes de trace permettent de conclure que Φ est lisse jusqu'au bord. Une intégration par partie permet donc d'obtenir :M D 2 Φ, ψ dv(g) + ∂M γ(ν)DΦ, ψ ds(g) = α M Φ, ψ dv(g),pour tout ψ ∈ C − . ...
Thesis
Full-text available
La principale motivation des travaux de cette thèse est d'étudier l'aspect conforme du spectre de l'opérateur de Dirac sur une variété à bord. Dans un premier temps, on donnera des estimations de la première valeur propre de l'opérateur de Dirac fondamental de la variété M sous deux conditions à bord locales prenant en compte leurs propriétés conformes. Une étude détaillée de ces conditions à bord permet alors de clore cette première partie par une estimation classique du spectre de l'opérateur de Dirac, raffinant un résultat antèrieur de O. Hijazi, S. Montiel et A. Roldan. Dans un second temps, on construit un invariant spinoriel conforme à partir de la première valeur propre de l'opérateur de Dirac sous une des conditions à bord étudiée dans le premier chapitre. Cet invariant peut être vu comme l'analogue de l'invariant de Yamabe dans le cadre spinoriel. Une étude approfondie de cet invariant conduit de manière naturelle à la construction de la fonction de Green de l'opérateur de Dirac.
... . Remarquons que par homogénéité de J, on peut supposer que pour tout i, M |Φ i | 2 dv(g) = 1. Or puisque l'opérateur de Dirac D sur H − définit un opérateur de Fredohlm, il existe une constante réelle C 0 > 0 telle que pour tout ψ ∈ C − \ {0}, on ait (voir [Sch95] par exemple) : ...
... ψ ∈ C − . Le champ de spineurs Φ vérifie au sens des distributions l'équation D 2 Φ = αΦà l'intérieur de M. En utilisant le théorème d'injection de Sobolev et les estimations elliptiques classiques (voir[Sch95] par exemple), on vérifie facilement que Φ est lisseà l'intérieur de M. Les théorèmes de trace permettent de conclure que Φ est lisse jusqu'au bord. Une intégration par partie permet donc d'obtenir :M D 2 Φ, ψ dv(g) + ∂M γ(ν)DΦ, ψ ds(g) = α M Φ, ψ dv(g),pour tout ψ ∈ C − . ...
Article
Full-text available
In this thesis, we study the conformal aspect of the spectrum of the Dirac operator on manifolds with boundary. First, we prove some lower bounds for the first eigenvalue of the Dirac operator under two local boundary conditions using the conformal covariance of these operators. A carefully treatment of these boundary conditions leads to a classical estimation of the eigenvalues of the Dirac operator under one of the preceding boundary conditions which improves a previous result of O. Hijazi, S. Montiel and A. Roldán. In a second time, we construct a spinorial conformal invariant defined from the first eigenvalue of the Dirac operator under the generalized chiral bag boundary condition. This invariant can be seen as an analogous of the Yamabe invariant in the setting of spin geometry. A detailed study of this invariant leads to the construction of the Green function for the Dirac operator.
... For the theory of Sobolev spaces on Riemannian manifolds the reader is referred to [11] and [24]. Using the above characterization of A 0 we will show in Section 4 that for thin domains close to spheres, a spectral gap condition is satisfied, which can be used to prove existence of inertial manifolds. ...
Article
Full-text available
In this paper we survey some recent results on parabolic equa- tions on curved squeezed domains. More specifically, consider the family of semilinear Neumann boundary value problems
... )ψ 0 − G − q ( . )ψ 0 is in the kernel of the Dirac operator (using the classical regularity theorems, see [Sch95] for example). Since the Dirac operator (under the chiral bag boundary condition) is supposed to be invertible, this spinor vanishes identically and so unicity follows directly. ...
Article
Full-text available
In this paper, we define the Green function for the Dirac operator under two local boundary conditions: the condition associated with a chirality operator (also called the chiral bag boundary condition) and the \MIT bag boundary condition. Then we give some applications of these constructions for each Green function. From the existence of the chiral Green function, we derive an inequality on a spin conformal invariant which, in particular, solve the Yamabe problem on manifolds with boundary in some cases. Finally, using the \MIT Green function, we give a simple proof of a positive mass theorem previously proved by Escobar.
... We will now recall a few classical definitions and results about Sobolev spaces on Riemannian manifolds. For more details on this subject, the reader is referred to [3], [7] and [17]. ...
Article
Full-text available
In this paper we study a family of semilinear reaction-diffusion equations on thin spatial domains, lying close to a lower dimensional submanifold M. As the thickness tends to zero, the domains collapse onto (a subset of) M. As it was proved in a previous paper (M. Prizzi, M. Rinaldi and K. P. Rybakowski, Curved thin domains and parabolic equations, Stud. Math. 151), the above family has a limit equation, which is an abstract semilinear parabolic equation defined on a certain abstract limit phase space. One of the objectives of this paper is to give more manageable characterizations of the limit phase space. Under additional hypotheses, we also give a simple description of the limit equation. If, in addition, M is a sphere and the nonlinearity of the above equations is dissipative, we prove that, if the thickness is small enough, the corresponding equation possesses an inertial manifold, i.e. an invariant manifold containing the attractor of the equation. We thus obtain the existence of inertial manifolds for reaction-diffusion equations on certain classes of thin domains of genuinely high dimension.
Article
Full-text available
We classify the structures and transition/evolutions of Taylor vortices with perturbations in one of the following categories: a) Hamiltonian vector fields, b) divergence-free vector fields, and c) solutions of the Navier-Stokes equations on two-dimensional torus. This is a part of a project oriented toward to developing a geometric theory of incompressible fluid flows in the physical space.
Article
Full-text available
We give a div-curl type lemma for the wedge product of closed differential forms on R^n when they have coefficients respectively in a Hardy space and L^infinity or BMO. In this last case, the wedge product belongs to an appropriate Hardy-Orlicz space.
ResearchGate has not been able to resolve any references for this publication.