Y. Cherruault

New numerical study of Adomian method applied to a diffusion model

We prove in this paper the convergence of Adomian method applied to linear or non‐linear diffusion equations. The results show that the convergence of this method is not influenced by the choice of the linear inversible operator L in the equation to be solved. Furthermore we give some particular examples about a new canonical form where the initial value u0 of Adomian series is chosen in some special form which verifies the initial and boundary conditions. Then Adomian series converges to exact solution or all approximated (truncated series) solutions verify these conditions.

Further remarks on convergence of decomposition method

The decomposition method solves a wide class of nonlinear functional equations. This method uses a series solution with rapid convergence. This paper is intended as a useful review and clarification of related issues.

New results of convergence of Adomian’s method

New results about convergence of Adomian’s method are presented. This method was developed by G. Adomian for solving non‐linear functional equations of different kinds. New conditions for obtaining convergence of the decomposition series are given. In a similar way, the convergence of a regularisation method which can, for example, be applied to Fredholm integral equations of the first kind, is proved.

A new formulation of Adomian method

A new approach of the decomposition method (Adomian) in which the Adomian scheme is obtained in a more natural way than in the classical presentation, is given. A new condition for obtaining convergence of the decomposition series is also included.

The decomposition method applied to the Cauchy problem

In this paper, we use the decomposition method for solving the Cauchy problem without using the canonical form of Adomian. We also give proof of convergence by using a new formulation of the Adomain polynomials and we compare our technique with the Picard method.

Analytical approximate solution of a SIR epidemic model with constant vaccination strategy by homotopy perturbation method

Purpose – The purpose of this paper is to introduce an efficient method for solving susceptible‐infected‐removed (SIR) epidemic model. A SIR model that monitors the temporal dynamics of a childhood disease in the presence of preventive vaccine. The qualitative analysis reveals the vaccination reproductive number for disease control and eradication. It introduces homotopy perturbation method (HPM) to overcome these problems.Design/methodology/approach – The paper considers HPM to solve differential system which describes SIR epidemic model. The essential idea of this method is to introduce a homotopy parameter, say p, which takes values from 0 to 1. When p=0, the system of equations usually reduces to a sufficiently simplified form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation. One of the most remarkable features of the HPM is that usu...

Improvement of convergence of Adomian’s method using Padé approximants

We show that Pade approximants considerably improve convergence of Adomians decomposition. The power of the method proposed is demonstrated through two illustrative examples from the field of nonlinear optics.

Functional equations generating space-densifying curves

In this paper, infinite-dimensional vector spaces of α-dense curves are generated by means of the functional equations f(x) + f(2x) = 0 and f(x) + f(2x) + f(3x) = 0. For both equations, a basis of continuous solutions has been constructed, obtaining in this way the main achievement which is the fact of that these bases may α-densify a large class of compact sets of the plane for arbitrary small α. Finally, a constructive method to find the noncontinuous solutions of the second functional equation is also settled.

The theoretic calculation time associated to α‐dense curves

The theoretic calculation time associated to every α‐dense curve into a fixed H of Rn is inversely proportional to the discretization step depending on the length of the curve and, more directly, of the derivatives of its coordinate functions. For a given degree of density α, it is interesting to seek curves into H which may minimize the theoretic calculation time and then to solve the practical problem of computing approximations for global optimization of a given continuous function defined in H, by means of its restriction over a family of curves with the same degree of density into the cube H.

The decomposition method applied to differential systems

The decomposition method is used for solving differential systems in biology and medicine. A comparison is given between the Runge‐Kutta method and the decomposition technique. New relationships for calculating Adomian’s polynomials are used for solving the differential systems governing the competition between species and based on the Lotka‐Volterra model.