Scopus İndeksli Yayınlar Koleksiyonu

Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8651

Browse

Filters

2013 - 20159

Settings

Author: search.filters.author.Bhrawy, A. H.Subject: search.filters.subject.Collocation Method

Search Results

Now showing 1 - 9 of 9
  • Article
    Citation - WoS: 118
    Citation - Scopus: 125
    Numerical Simulation of Time Variable Fractional Order Mobile-Immobile Advection-Dispersion Model
    (Editura Acad Romane, 2015) Abdelkawy, M. A.; Baleanu, Dumitru; Zaky, M. A.; Bhrawy, A. H.; Baleanu, D.; Matematik
    This paper reports a novel numerical technique for solving the time variable fractional order mobile-immobile advection-dispersion (TVFO-MIAD) model with the Coimbra variable time fractional derivative, which is preferable for modeling dynamical systems. The main advantage of the proposed method is that two different collocation schemes are investigated for both temporal and spatial discretizations of the TVFO-MIAD model. The problem with its boundary and initial conditions is then reduced to a system of algebraic equations that is far easier to be solved. Numerical results are consistent with the theoretical analysis and indicate the high accuracy and effectiveness of this algorithm.
  • Article
    Citation - WoS: 13
    Citation - Scopus: 16
    Generalized Laguerre-Gauss Scheme for First Order Hyperbolic Equations on Semi-Infinite Domains
    (Editura Acad Romane, 2015) Bhrawy, A. H.; Baleanu, Dumitru; Hafez, R. M.; Alzahrani, E. O.; Baleanu, D.; Alzahrani, A. A.; Matematik
    In this article, we develop a numerical approximation for first-order hyperbolic equations on semi-infinite domains by using a spectral collocation scheme. First, we propose the generalized Laguerre-Gauss-Radau collocation scheme for both spatial and temporal discretizations. This in turn reduces the problem to the obtaining of a system of algebraic equations. Second, we use a Newton iteration technique to solve it. Finally, the obtained results are compared with the exact solutions, highlighting the performance of the proposed numerical method.
  • Article
    Citation - WoS: 59
    Citation - Scopus: 63
    A New Generalized Laguerre-Gauss Collocation Scheme for Numerical Solution of Generalized Fractional Pantograph Equations
    (Editura Acad Romane, 2014) Bhrawy, A. H.; Baleanu, Dumitru; Al-Zahrani, A. A.; Alhamed, Y. A.; Baleanu, D.; Matematik
    The manuscript is concerned with a generalization of the fractional pantograph equation which contains a linear functional argument. This type of equation has applications in many branches of physics and engineering. A new spectral collocation scheme is investigated to obtain a numerical solution of this equation with variable coefficients on a semi-infinite domain. This method is based upon the generalized Laguerre polynomials and Gauss quadrature integration. This scheme reduces solving the generalized fractional pantograph equation to a system of algebraic equations. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.
  • Article
    Citation - WoS: 27
    Citation - Scopus: 28
    An Efficient Collocation Technique for Solving Generalized Fokker-Planck Type Equations With Variable Coefficients
    (Editura Acad Romane, 2014) Bhrawy, A. H.; Baleanu, Dumitru; Ahmed, Engy A.; Baleanu, D.; Matematik
    This paper proposes an efficient numerical integration process for the generalized Fokker-Planck equation with variable coefficients. For spatial discretization the Jacobi-Gauss-Lobatto collocation (J-GL-C) method is implemented in which the Jacobi-Gauss-Lobatto points are used as collocation nodes for spatial derivatives. This approach has the advantage of obtaining the solution in terms of the Jacobi parameters alpha and beta. Using the above technique, the problem is reduced to the solution of a system of ordinary differential equations in tithe. This system can be also solved by standard numerical techniques. Our results demonstrate that the proposed method is a powerful algorithm for solving nonlinear partial differential equations.
  • Article
    Citation - WoS: 13
    Citation - Scopus: 14
    An Accurate Legendre Collocation Scheme for Coupled Hyperbolic Equations With Variable Coefficients
    (Editura Acad Romane, 2014) Doha, E. H.; Baleanu, Dumitru; Bhrawy, A. H.; Baleanu, D.; Abdelkawy, M. A.; Matematik
    The study of numerical solutions of nonlinear coupled hyperbolic partial differential equations (PDEs) with variable coefficients subject to initial-boundary conditions continues to be a major research area with widespread applications in modern physics and technology. One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (NPDEs) as well as PDEs with variable coefficients. A numerical solution based on a Legendre collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients. This approach, which is based on Legendre polynomials and Gauss-Lobatto quadrature integration, reduces the solving of nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equations that is far easier to solve. The obtained results show that the proposed numerical algorithm is efficient and very accurate.
  • Article
    Citation - WoS: 89
    Citation - Scopus: 124
    New Numerical Approximations for Space-Time Fractional Burgers' Equations Via a Legendre Spectral-Collocation Method
    (Editura Acad Romane, 2015) Bhrawy, A. H.; Zaky, M. A.; Baleanu, D.
    Burgers' equation is a fundamental partial differential equation in fluid mechanics. This paper reports a new space-time spectral algorithm for obtaining an approximate solution for the space-time fractional Burgers' equation (FBE) based on spectral shifted Legendre collocation (SLC) method in combination with the shifted Legendre operational matrix of fractional derivatives. The fractional derivatives are described in the Caputo sense. We propose a spectral shifted Legendre collocation method in both temporal and spatial discretizations for the space-time FBE. The main characteristic behind this approach is that it reduces such problem to that of solving a system of nonlinear algebraic equations that can then be solved using Newton's iterative method. Numerical results with comparisons are given to confirm the reliability of the proposed method for FBE.
  • Article
    Citation - WoS: 20
    Citation - Scopus: 19
    A Highly Accurate Jacobi Collocation Algorithm for Systems of High-Order Linear Differential-Difference Equations With Mixed Initial Conditions
    (Wiley, 2015) Doha, E. H.; Baleanu, D.; Hafez, R. M.; Bhrawy, A. H.
    In this paper, a shifted Jacobi-Gauss collocation spectral algorithm is developed for solving numerically systems of high-order linear retarded and advanced differential-difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi-Gauss interpolation nodes as collocation nodes. The system of differential-difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright (C) 2015 John Wiley & Sons, Ltd.
  • Article
    Citation - WoS: 19
    Citation - Scopus: 18
    Efficient Jacobi-Gauss Collocation Method for Solving Initial Value Problems of Bratu Type
    (Pleiades Publishing inc, 2013) Bhrawy, A. H.; Baleanu, D.; Hafez, R. M.; Doha, E. H.
    In this paper, we propose the shifted Jacobi-Gauss collocation spectral method for solving initial value problems of Bratu type, which is widely applicable in fuel ignition of the combustion theory and heat transfer. The spatial approximation is based on shifted Jacobi polynomials J(n)((alpha, beta))(x) with alpha, beta is an element of (-1, infinity), x is an element of [0, 1] and n the polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes. Illustrative examples have been discussed to demonstrate the validity and applicability of the proposed technique. Comparing the numerical results of the proposed method with some well-known results show that the method is efficient and gives excellent numerical results.
  • Article
    Citation - WoS: 99
    Citation - Scopus: 110
    A New Jacobi Rational-Gauss Collocation Method for Numerical Solution of Generalized Pantograph Equations
    (Elsevier, 2014) Bhrawy, A. H.; Baleanu, D.; Hafez, R. M.; Doha, E. H.
    This paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this article, a new spectral collocation method is applied to solve the generalized pantograph equation with variable coefficients on a semi-infinite domain. This method is based on Jacobi rational functions and Gauss quadrature integration. The Jacobi rational-Gauss method reduces solving the generalized pantograph equation to a system of algebraic equations. Reasonable numerical results are obtained by selecting few Jacobi rational-Gauss collocation points. The proposed Jacobi rational-Gauss method is favorably compared with other methods. Numerical results demonstrate its accuracy, efficiency, and versatility on the half-line. (C) 2013 IMACS. Published by Elsevier B.V. All rights reserved.