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Register a new userA Brief Introduction to the Adomian Decomposition Method, with Applications in Astronomy and Astrophysics
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The Adomian Decomposition Method (ADM) is a very effective approach for solving broad classes of nonlinear partial and ordinary differential equations, with important applications in different fields of applied mathematics, engineering, physics and biology. It is the goal of the present paper to provide a clear and pedagogical introduction to the Adomian Decomposition Method and to some of its applications. In particular, we focus our attention to a number of standard first-order ordinary differential equations (the linear, Bernoulli, Riccati, and Abel) with arbitrary coefficients, and present in detail the Adomian method for obtaining their solutions. In each case we compare the Adomian solution with the exact solution of some particular differential equations, and we show their complete equivalence. The second order and the fifth order ordinary differential equations are also considered. An important extension of the standard ADM, the Laplace-Adomian Decomposition Method is also introduced through the investigation of the solutions of a specific second order nonlinear differential equation. We also present the applications of the method to the Fisher-Kolmogorov second order partial nonlinear differential equation, which plays an important role in the description of many physical processes, as well as three important applications in astronomy and astrophysics, related to the determination of the solutions of the Kepler equation, of the Lane-Emden equation, and of the general relativistic equation describing the motion of massive particles in the spherically symmetric and static Schwarzschild geometry.
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The biharmonic equation, as well as its nonlinear and inhomogeneous generalizations, plays an important role in engineering and physics. In particular the focusing biharmonic nonlinear Schr{o}dinger equation, and its standing wave solutions, have been intensively investigated. In the present paper we consider the applications of the Laplace-Adomian and Adomian Decomposition Methods for obtaining semi-analytical solutions of the generalized biharmonic equations of the type $Delta ^{2}y+alpha Delta y+omega y+b^{2}+gleft( yright) =f$, where $alpha $, $omega $ and $b$ are constants, and $g$ and $f$ are arbitrary functions of $y$ and the independent variable, respectively. After introducing the general algorithm for the solution of the biharmonic equation, as an application we consider the solutions of the one-dimensional and radially symmetric biharmonic standing wave equation $Delta ^{2}R+R-R^{2sigma +1}=0$, with $sigma = {rm constant}$. The one-dimensional case is analyzed by using both the Laplace-Adomian and the Adomian Decomposition Methods, respectively, and the truncated series solutions are compared with the exact numerical solution. The power series solution of the radial biharmonic standing wave equation is also obtained, and compared with the numerical solution.
A fractional Adomian decomposition method for fractional nonlinear differential equations is proposed. The iteration procedure is based on Jumaries fractional derivative. An example is given to elucidate the solution procedure, and the results are compared with the exact solution, revealing high accuracy and efficiency.
Laboratory astrophysics and complementary theoretical calculations are the foundations of astronomy and astrophysics and will remain so into the foreseeable future. The impact of laboratory astrophysics ranges from the scientific conception stage for ground-based, airborne, and space-based observatories, all the way through to the scientific return of these projects and missions. It is our understanding of the under-lying physical processes and the measurements of critical physical parameters that allows us to address fundamental questions in astronomy and astrophysics. In this regard, laboratory astrophysics is much like detector and instrument development at NASA, NSF, and DOE. These efforts are necessary for the success of astronomical research being funded by the agencies. Without concomitant efforts in all three directions (observational facilities, detector/instrument development, and laboratory astrophysics) the future progress of astronomy and astrophysics is imperiled. In addition, new developments in experimental technologies have allowed laboratory studies to take on a new role as some questions which previously could only be studied theoretically can now be addressed directly in the lab. With this in mind we, the members of the AAS Working Group on Laboratory Astrophysics, have prepared this State of the Profession Position Paper on the laboratory astrophysics infrastructure needed to ensure the advancement of astronomy and astrophysics in the next decade.
A concise introduction to quantum entanglement in multipartite systems is presented. We review entanglement of pure quantum states of three--partite systems analyzing the classes of GHZ and W states and discussing the monogamy relations. Special attention is paid to equivalence with respect to local unitaries and stochastic local operations, invariants along these orbits, momentum map and spectra of partial traces. We discuss absolutely maximally entangled states and their relation to quantum error correction codes. An important case of a large number of parties is also analysed and entanglement in spin systems is briefly reviewed.
We introduce and motivate generative modeling as a central task for machine learning and provide a critical view of the algorithms which have been proposed for solving this task. We overview how generative modeling can be defined mathematically as trying to make an estimating distribution the same as an unknown ground truth distribution. This can then be quantified in terms of the value of a statistical divergence between the two distributions. We outline the maximum likelihood approach and how it can be interpreted as minimizing KL-divergence. We explore a number of approaches in the maximum likelihood family, while discussing their limitations. Finally, we explore the alternative adversarial approach which involves studying the differences between an estimating distribution and a real data distribution. We discuss how this approach can give rise to new divergences and methods that are necessary to make adversarial learning successful. We also discuss new evaluation metrics which are required by the adversarial approach.
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