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Let me introduce some scientists in alphabetic order who contributed to and greatly improved the Adomian Decomposition Method to make it available to the mathematical and engineering community.

George Adomian Dr. George Adomian was an American mathematician, theoretical physicist, and electrical engineer of Armenian descent. He received his Ph. He first proposed and considerably developed the Adomian Decomposition Method ADM for solving nonlinear differential equations, both ordinary, and partial, deterministic and stochastic, also integral equations, algebraic and transcendental functional , and matrix equations.

He was a Distinguished Professor academic rank , the David C. He is the author of eight books and over three hundred journal papers. He also attended the secret radar school at MIT that trained the first radar officers for the U.

His obituary was prepared by Randolph Rach and published in R. Rach, Dr. George Adomian — distinguished scientist and mathematician, Kybernetes, Vol. Yves Cherruault Dr. Yves Cherruault was a French mathematician. Professor Cherruault is one of the founders of the field of Biomathematics and an author of seven books and over two hundred journal papers.

Cherruault developed some of the convergence theorems of the ADM. Jun-Sheng Duan Dr. He has made extensive contributions to solutions of differential equations in mathematical physics and engineering using ADM and the Modified Decomposition Method MDM in collaboration with Drs. Rach and Wazwaz. He is the author of more than eighty journal papers.

He enjoys photography and traveling. Randolph Rach Dr. His experience includes research and development in microwave electronics and traveling-wave tube technology, and his research interests span nonlinear system analysis, nonlinear ordinary and partial differential equations, nonlinear integral equations and nonlinear boundary value problems. He published more than one hundred and thirty papers in applied mathematics, and he is an early contributor to the ADM.

He was also the first to propose the modified decomposition method MDM for short. He prepared the comprehensive bibliography on ADM: R. Rach, A bibliography of the theory and applications of the Adomian decomposition method, , Kybernetes, Volume 41, Nos. Sergio E. Serrano Dr. Serrano was born in Santander Colombia in He is a professor of environmental engineering, hydrologic science, applied mathematics, and philosophy at Temple University in Philadelphia.

He has more than one hundred research publications in international science, engineering, and mathematical journals. He is also the author of eight books in environmental engineering, statistics, philosophy, and psychology. Serrano has been awarded four times with nationally-competitive research grants by the National Science Foundation, Washington, DC. Using adaptations and modifications of the ADM, he has developed hundreds of practical engineering models of flood wave propagation, contaminant transport, and groundwater flow in irregular geometries.

HydroScience Inc. Serrano has a passion for hiking. He plays the recorder Renaissance flute. He enjoys alchemy, archaeology, home wine making, herbal medicine, and cooking. He believes that the joy of meaningful living and meaningful relationships can be found in the simplicity of everyday life.

He lives in Philadelphia with his wife of thirty years and his daughter. Abdul-Majid Wazwaz Dr. He was the author and co-author of more than four hundred and fifty papers in applied mathematics and mathematical physics.

He is the author of five books on the subjects of discrete mathematics, integral equations, and partial differential equations. He has contributed extensively to theoretical advances in solitary waves theory, the ADM, and other computational methods. Obviously, this tutorial cannot cover and explain all available improvements for the method.

The basic spirit of the decomposition method consists of three steps. Finally, summation of first finite number of components leads to the approximation to the true solution with any desired level of high precision. If the given differential equation is homogeneous without driving terms , then the ADM finite sum always provides the truncated series version to the true solution. However, for some special classes of inhomogeneous differential equations, the ADM finite sum approximation may include some noise terms that eventually are canceled out with next iterations subject that all iterations are complete.

However, unwanted terms in the ADM applications were first discovered by G. Adomian and R. Rach in The noisy convergence phenomena in decomposition method solutions, Journal of Computational and Applied Mathematics, Volume 15, Issue 3, July , pages Although it is possible to reduce these unwanted noise terms, as shown by Abdul-Majid Wazwaz and other researchers, the issue remains.

However, these noise terms usually do not effect the approximations because they are rapidly damped out numerically. The first proof of convergence of the ADM was given by Cherruault in , who used fixed point theorems for abstract functional equations. Since conditions of these theorems are too restrictive for most physical and engineering applications to be verified in practice, many other articles on convergence of the ADM were published.

In spite of the variety of publications on convergence, computational complexity, improvements, and applications of the ADM, no precise criterion of convergence was formulated in the literature, at least in the context of initial-value problems for ordinary and partial differential equations. When slope function is a holomorphic function , then the formalism of the Cauchy--Kowalevskaya theorem guarantees that solutions of initial-value problems for systems of ordinary differential equations exist and are analytic for small time intervals.

The latter can be successfully applied only for differential equations with polynomial slope functions. At each iterative step, the Adomian decomposition method actually requires solving the same very simple initial value problem with homogeneous initial conditions. The labor involved in such evaluation grows exponentially in general. This drawback becomes negligible in an explicit one step numerical implementation based on the ADM when only a few first terms are taken into account.

It should be noted that the amount of work for the ADM preprocessing is larger than, say, the well known Runge--Kutta or cubic spline algorithms where a user enters only the slope functions and the initial conditions. Usually, the labor required by ADM preprocessing is compensated by a larger step size in the computations than in standard one step algorithms.

Rach in their paper Inversion of nonlinear stochastic operators, Journal of Mathematical Analysis and Applications, Vol.

Previous terms are eliminated by differentiation. Fortunately, Mathematica has two dedicated commands for extracting coefficients from a polynomial: Coefficient that gives a particular coefficient and CoefficientList that provides all coefficients of the given polynomial but not an arbitrary function.

Next to get A1, consider each term where the subscripts of the ui add up 1. In a similar fashion, the general Adomian polynomial An includes any term where the sum of the subscripts of the ui add up to n. It is worth noting that the sum of the subscripts of the term uij is ij and not i.

Once it is achieved, all other terms in the infinite series are solutions of the homogeneous initial value problems. It should be noted that when Adomian method is applied to other problems not necessarily initial value problems for first order differential equations as we consider in this section one can choose another approximation to the nonhomgeneous differential equation: it is important to exclude, depending on x, input term from the given slope function.

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## Mathematics Genealogy Project

About the festival Sasquatch! So Cocolonia hits the streets to join a dance crew who respond to every challenge with a dance. George Adomian March 21, — June 17, was an American mathematician of Armenian descent who developed the Adomian decomposition method ADM for solving nonlinear differential equations, both ordinary and partial. Deaths of notable animals that is, those with their own Wikipedia articles are also reported here.

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## James Adomian

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