arXiv subjects

Esmail Babolian

Publications and source records attributed to Esmail Babolian.

7 recordsLinked to original sources

arXiv2017

Bernstein dual-Petrov-Galerkin method: application to 2D time fractional diffusion equation

In this paper, we develop a Bernstein dual-Petrov-Galerkin method for the numerical simulation of a two-dimensional fractional diffusion equation. A spectral discretization is applied by introducing suitable combinations of dual Bernstein polynomials as the test functions and the Bernstein polynomials as the trial ones. We derive the exact sparse operational matrix of differentiation for the dual Bernstein basis which provides a matrix based approach for the spatial discretization. It is shown that the method leads to banded linear systems that can be solved efficiently. The stability and convergence of the proposed method is discussed. Finally, some numerical examples are provided to support the theoretical claims and to show the accuracy and efficiency of the method.

math.NA↗

arXiv2017

A Fractional Gauss-Jacobi quadrature rule for approximating fractional integrals and derivatives

We introduce an efficient algorithm for computing fractional integrals and derivatives and apply it for solving problems of the calculus of variations of fractional order. The proposed approximations are particularly useful for solving fractional boundary value problems. As an application, we solve a special class of fractional Euler-Lagrange equations. The method is based on Hale and Townsend algorithm for finding the roots and weights of the fractional Gauss-Jacobi quadrature rule and the predictor-corrector method introduced by Diethelm for solving fractional differential equations. Illustrative examples show that the given method is more accurate than the one introduced in [Comput. Math. Appl. 66 (2013), no. 5, 597--607], which uses the Golub-Welsch algorithm for evaluating fractional directional integrals.

math.NA↗

arXiv2016

Bernstein modal basis: application to the spectral Petrov-Galerkin method for fractional partial differential equations

In the spectral Petrov-Galerkin methods, the trial and test functions are required to satisfy particular boundary conditions. By a suitable linear combination of orthogonal polynomials, a basis, that is called the modal basis, is obtained. In this paper, we extend this idea to the non-orthogonal dual Bernstein polynomials. A compact general formula is derived for the modal basis functions based on dual Bernstein polynomials. Then, we present a Bernstein-spectral Petrov-Galerkin method for a class of time fractional partial differential equations with Caputo derivative. It is shown that the method leads to banded sparse linear systems for problems with constant coefficient. Some numerical examples are provided to show the efficiency and the spectral accuracy of the method.

math.NA↗

arXiv2014

Solving Abel integral equations of first kind via fractional calculus

We give a new method for numerically solving Abel integral equations of first kind. An estimation for the error is obtained. The method is based on approximations of fractional integrals and Caputo derivatives. Using trapezoidal rule and Computer Algebra System Maple, the exact and approximation values of three Abel integral equations are found, illustrating the effectiveness of the proposed approach.

math.CA↗

arXiv2014

A New Fast Numerical Method for One-Dimensional Nonlinear Sine-Gordon Equation Using Multivariate Padé approximation

This paper has two purposes. First we present a new definition of the multivariate Padé approximation, a new fast numerical method. Then numerical solution of the one-dimensional (1D) time-dependent nonlinear Sine-Gordon equation (SGE) is considered by this method. Numerical results are obtained for various cases involving undamped SGE. The results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme. It is shown that the technique is easy to apply for multidimensional problems.

math.NA↗

arXiv2014

$(n D+1)$ Padé approximation

To generalize the concept of Padé approximation for functions to more than one variable, several definitions have been introduced. All definitions have advantages and disadvantages. The advantages of these approaches has been discussed in many articles. One of the main disadvantages of these methods are low convergence rate and the loss of information in computing with low degrees. In this work we present a new definition of the multivariate Padé approximation, treated the more than one variable to the coefficients. Our definition is designed to avoid such disadvantages with retaining the advantages of simple and easily computation. The main result obtained as consequence of this definition is some properties and convergence results of the Montessus De Ballore theorem.

math.NA↗

arXiv2008

A degenerate kernel method for eigenvalue problems of a class of non-compact operators

We consider the eigenvalue problem of certain kind of non-compact linear operators given as the sum of a multiplication and a kernel operator. A degenerate kernel method is used to approximate isolated eigenvalues. It is shown that entries of the corresponding matrix of this method can be evaluated exactly. The convergence of the method is proved; it is proved that the convergence rate is $O(h)$. By some numerical examples, we confirm the results.

math.NA↗