119 research outputs found
Spectral methods in general relativistic astrophysics
We present spectral methods developed in our group to solve three-dimensional
partial differential equations. The emphasis is put on equations arising from
astrophysical problems in the framework of general relativity.Comment: 51 pages, elsart (Elsevier Preprint), 19 PostScript figures,
submitted to Journal of Computational & Applied Mathematic
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
A New Unconditionally Stable Method for Telegraph Equation Based on Associated Hermite Orthogonal Functions
The present paper proposes a new unconditionally stable method to solve telegraph equation by using associated Hermite (AH) orthogonal functions. Unlike other numerical approaches, the time variables in the given equation can be handled analytically by AH basis functions. By using the Galerkin’s method, one can eliminate the time variables from calculations, which results in a series of implicit equations. And the coefficients of results for all orders can then be obtained by the expanded equations and the numerical results can be reconstructed during the computing process. The precision and stability of the proposed method are proved by some examples, which show the numerical solution acquired is acceptable when compared with some existing methods
An approach based on the pseudospectral method for fractional telegraph equations
We aim to implement the pseudospectral method on fractional Telegraph equation. To implement this method, Chebyshev cardinal functions (CCFs) are considered bases. Introducing a matrix representation of the Caputo fractional derivative (CFD) via an indirect method and applying it via the pseudospectral method helps to reduce the desired problem to a system of algebraic equations. The proposed method is an effective and accurate numerical method such that its implementation is easy. Some examples are provided to confirm convergence analysis, effectiveness and accuracy
Numerical Solution of Telegraph Equation by Using LT Inversion Technique
Orientadora: Ana Margarida GraelmMonografia (licenciatura) - Universidade Federal do Paraná. Setor de Ciências Biológicas. Curso de Educação Física
High-Order Numerical Solution of Second-Order One-Dimensional Hyperbolic Telegraph Equation Using a Shifted Gegenbauer Pseudospectral Method
We present a high-order shifted Gegenbauer pseudospectral method (SGPM) to
solve numerically the second-order one-dimensional hyperbolic telegraph
equation provided with some initial and Dirichlet boundary conditions. The
framework of the numerical scheme involves the recast of the problem into its
integral formulation followed by its discretization into a system of
well-conditioned linear algebraic equations. The integral operators are
numerically approximated using some novel shifted Gegenbauer operational
matrices of integration. We derive the error formula of the associated
numerical quadratures. We also present a method to optimize the constructed
operational matrix of integration by minimizing the associated quadrature error
in some optimality sense. We study the error bounds and convergence of the
optimal shifted Gegenbauer operational matrix of integration. Moreover, we
construct the relation between the operational matrices of integration of the
shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive
the global collocation matrix of the SGPM, and construct an efficient
computational algorithm for the solution of the collocation equations. We
present a study on the computational cost of the developed computational
algorithm, and a rigorous convergence and error analysis of the introduced
method. Four numerical test examples have been carried out in order to verify
the effectiveness, the accuracy, and the exponential convergence of the method.
The SGPM is a robust technique, which can be extended to solve a wide range of
problems arising in numerous applications.Comment: 36 pages, articl
- …