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In some papers G. Adomian has presented a decomposition technique in order to solve different non‐linear equations. The solution is found as an infinite series quickly converging to accurate solutions. The method is well‐suited for physical problems and it avoids linearization, perturbation and other restrictions, methods and assumptions which may change the problem being solved – sometimes seriously – unnecessarily. Proofs of convergence are given by Cherruault and co‐authors. Many numerical studies for physical phenomena, such as Fisher’s equation, Lorentz’s equation and Edem’s equation are given and solved. In this work, the general equation given by ∂ p \over ∂ t = (∇ ⋅(q(x)⋅ ∇p)) + f(x, t) is solved by using decomposition methods, and is compared to other techniques. This equation can be used to describe the motion of a fluid flow in the so‐called reservoir region, where p(x, t) represents the pressure distribution, f(x, t) describes the withdrawal or injection of the fluid, and q(x) is the transmissibility in the reservoir region.

This paper describes a non‐linear reaction‐diffusion equation, which models how a substance spreads in the surface of the cortex so as to avoid a massive destruction of neurones when cerebral tissue is not oxygenated correctly. For the explicit finite differences method, the necessary stability condition is provided by a reaction‐diffusion equation with non‐linearity given by a decreasing function. The solution to the non‐linear reaction‐diffusion equation of the model can be obtained via one of the two methods: the finite differences (explicit schema) method and the 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.

This study aims to analyze and assess 21 years of media coverage (2000–2020) of Flix, a small industrial village located in an rural area on north-eastern Spain, which has endured in these years a severe environmental and industrial crisis, with a strong potential for stigmatization of the place.

The research is conceptualized under the Social Amplification of Risk Framework, a theoretical/conceptual approach aimed at accounting for the huge gaps that often arise between public perception of technological or environmental risks of some technologies, products and places and the expert estimations of these risks. The authors studied the coverage on Flix by a local, a regional and a national newspaper through a content analysis where the corpus (1,524 news pieces) was coded for several variables, including tone, genre and thematic area.

The studied coverage was in general overwhelmingly negative and strongly focused on “bad news” relating to pollution and deindustrialization, although this was much less the case in the local newspaper than in the regional and, in particular, the national newspaper. Thus, a territorially escalated pattern clearly emerges from our research concerning the stigmatization potential of news media coverage for the specific case under scrutiny.

To the authors’ knowledge, this is the first time such a longitudinal study of media coverage and its potential for place stigmatization is performed with this specific territorial perspective.

Faltung equations (closed cycle type) have a wide range of biological applications, nonetheless, they are poorly studied. We use a Volterra‐Kostitzin model (which is a Faltung equation) to study the dynamics of a certain species, where the integral term represents a residual action. The complexity of resolution of this non‐linear equation using classical numeric methods is here solved with the Adomian decomposition method. Our method provides the same graphic solution as others do, such as the numeric method Miladie. However, the decomposition method of Adomian has the advantage that neither time nor space are considered discontinuous and that it gives an analytical solution with a reliable approximation.

In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that appears in the study of the thermal behaviour of a spherical cloud and of the gravitational potential of a polytropic fluid at hydrostatic equilibrium.

In their work, the authors first review Emden's equation and its possible solutions using the Frobenius and power series methods; then, Adomian polynomials are introduced. Afterwards, Emden's equation is solved using Adomian's decomposition method and, finally, they conclude with a comparison of the solution given by Adomian's method with the solution obtained by the other methods, for certain cases where the exact solution is known.

Solving Emden's equation for n in the interval [0, 5] is very interesting for several scientific applications, such as astronomy. However, the exact solution is known only for n=0, n=1 and n=5. The experiments show that Adomian's method achieves an approximate solution which overlaps with the exact solution when n=0, and that coincides with the Taylor expansion of the exact solutions for n=1 and n=5. As a result, the authors obtained quite satisfactory results from their proposal.

The main classical methods for obtaining approximate solutions of Emden's equation have serious computational drawbacks. The authors make a new, efficient numerical implementation for solving this equation, constructing iteratively the Adomian polynomials, which leads to a solution of Emden's equation that extends the range of variation of parameter n compared to the solutions given by both the Frobenius and the power series methods.

In this study, the aim is to solve a class of boundary systems, using the Adomian decomposition method. This method gave very good results for solving algebraic, differential integral and partial differential equations.

To compute an optimal control of non‐linear reaction diffusion equations that are modelling inhibitor problems in the brain.

A new numerical approach that combines a spectral method in time and the Adomian decomposition method in space. The coupling of these two methods is used to solve an optimal control problem in cancer research.

The main conclusion is that the numerical approach we have developed leads to a new way for solving such problems.

Focused research on computing control optimal in non‐linear diffusion reaction equations. The main idea that is developed lies in the approximation of the control space in view of the spectral expansion in the Legendre basis.

Through this work we are convinced that one way to derive efficient numerical optimal control is to associate the Legendre expansion in time and Runge Kutta approximation. We expect to obtain general results from optimal control associated with non‐linear parabolic problem in higher dimension.

Coupling of methods provides a numerical solution of an optical control problem in Cancer research.