97 research outputs found
(R1951) Numerical Solution for a Class of Nonlinear Emden-Fowler Equations by Exponential Collocation Method
In this research, exponential approximation is used to solve a class of nonlinear Emden-Fowler equations. This method is based on the matrix forms of exponential functions and their derivatives using collocation points. To demonstrate the usefulness of the method, we apply it to some different problems. The numerical approximate solutions are compared with available (existing) exact (analytical) solutions to show the accuracy of the proposed method. The method has been checked with several examples to show its validity and reliability. The reported examples illustrate that the method is reasonably efficient and accurate
Numerical Simulation
Nowadays mathematical modeling and numerical simulations play an important role in life and natural science. Numerous researchers are working in developing different methods and techniques to help understand the behavior of very complex systems, from the brain activity with real importance in medicine to the turbulent flows with important applications in physics and engineering. This book presents an overview of some models, methods, and numerical computations that are useful for the applied research scientists and mathematicians, fluid tech engineers, and postgraduate students
JDNN: Jacobi Deep Neural Network for Solving Telegraph Equation
In this article, a new deep learning architecture, named JDNN, has been
proposed to approximate a numerical solution to Partial Differential Equations
(PDEs). The JDNN is capable of solving high-dimensional equations. Here, Jacobi
Deep Neural Network (JDNN) has demonstrated various types of telegraph
equations. This model utilizes the orthogonal Jacobi polynomials as the
activation function to increase the accuracy and stability of the method for
solving partial differential equations. The finite difference time
discretization technique is used to overcome the computational complexity of
the given equation. The proposed scheme utilizes a Graphics Processing Unit
(GPU) to accelerate the learning process by taking advantage of the neural
network platforms. Comparing the existing methods, the numerical experiments
show that the proposed approach can efficiently learn the dynamics of the
physical problem
Approximate Universal Relations for Neutron Stars and Quark Stars
Neutron stars and quark stars are ideal laboratories to study fundamental
physics at supra nuclear densities and strong gravitational fields.
Astrophysical observables, however, depend strongly on the star's internal
structure, which is currently unknown due to uncertainties in the equation of
state. Universal relations, however, exist among certain stellar observables
that do not depend sensitively on the star's internal structure. One such set
of relations is between the star's moment of inertia (), its tidal Love
number (Love) and its quadrupole moment (), the so-called I-Love-Q
relations. Similar relations hold among the star's multipole moments, which
resemble the well-known black hole no-hair theorems. Universal relations break
degeneracies among astrophysical observables, leading to a variety of
applications: (i) X-ray measurements of the nuclear matter equation of state,
(ii) gravitational wave measurements of the intrinsic spin of inspiraling
compact objects, and (iii) gravitational and astrophysical tests of General
Relativity that are independent of the equation of state. We here review how
the universal relations come about and all the applications that have been
devised to date.Comment: 89 pages, 38 figures; review article submitted to Physics Report
Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations
In this manuscript, we introduce a spectral technique for approximating the variable-order fractional Riccati equation (VO-FRDEs). Firstly, the solution and its space fractional derivatives is expanded as shifted Chebyshev polynomials series. Then we determine the expansion coefficients by reducing the VO-FRDEs and its conditions to a system of algebraic equations. We show the accuracy and applicability of our numerical approach through four numerical examples.
 
Integrated neuro-evolution-based computing solver for dynamics of nonlinear corneal shape model numerically
[EN]In this study, bio-inspired computational techniques have been exploited to get the numerical solution of a nonlinear two-point boundary value problem arising in the modelling of the corneal shape. The computational process of modelling and optimization makes enormously straightforward to obtain accurate approximate solutions of the corneal shape models through artificial neural networks, pattern search (PS), genetic algorithms (GAs), simulated annealing (SA), active-set technique (AST), interior-point technique, sequential quadratic programming and their hybrid forms based on GA–AST, PS–AST and SA–AST. Numerical results show that the designed solvers provide a reasonable precision and efficiency with minimal computational cost. The efficacy of the proposed computing strategies is also investigated through a descriptive statistical analysis by means of histogram illustrations, probability plots and one-way analysis of variance
A post-Newtonian diagnosis of quasiequilibrium configurations of neutron star-neutron star and neutron star-black hole binaries
We use a post-Newtonian diagnostic tool to examine numerically generated
quasiequilibrium initial data sets for non-spinning double neutron star and
neutron star-black hole binary systems. The PN equations include the effects of
tidal interactions, parametrized by the compactness of the neutron stars and by
suitable values of ``apsidal'' constants, which measure the degree of
distortion of stars subjected to tidal forces. We find that the post-Newtonian
diagnostic agrees well with the double neutron star initial data, typically to
better than half a percent except where tidal distortions are becoming extreme.
We show that the differences could be interpreted as representing small
residual eccentricity in the initial orbits. In comparing the diagnostic with
preliminary numerical data on neutron star-black hole binaries, we find less
agreement.Comment: 17 pages, 6 tables, 8 figure
A Godunov-Type Solver for the Numerical Approximation of Gravitational Flows
International audienceWe present a new numerical method to approximate the solutions of an Euler-Poisson model, which is inherent to astrophysical flows where gravity plays an important role. We propose a discretization of gravity which ensures adequate coupling of the Poisson and Euler equations, paying particular attention to the gravity source term involved in the latter equations. In order to approximate this source term, its discretization is introduced into the approximate Riemann solver used for the Euler equations. A relaxation scheme is involved and its robustness is established. The method has been implemented in the software HERACLES and several numerical experiments involving gravitational flows for astrophysics highlight the scheme
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