Matematik Bölümü Yayın Koleksiyonu

Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/413

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Subject: search.filters.subject.Collocation Method

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  • 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 - Scopus: 17
    A Jacobi Collocation Method for Troesch's Problem in Plasma Physics
    (Editura Academiei Romane, 2014) Doha, E.H.; Baleanu, Dumitru; Baleanu, D.; Bhrawi, A.H.; Hafez, R.M.; Matematik
    In this paper, we propose a numerical approach for solving Troesch's problem which arises in the confinement of a plasma column by radiation pressure. It is also an inherently unstable two-point boundary value problem. The spatial approximation is based on shifted Jacobi-Gauss collocation method in which the shifted Jacobi-Gauss points are used as collocation nodes. The results presented here demonstrate reliability and efficiency of the 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: 68
    Citation - Scopus: 81
    Efficient Sustainable Algorithm for Numerical Solutions of Systems of Fractional Order Differential Equations by Haar Wavelet Collocation Method
    (Elsevier, 2020) Shah, Kamal; Al-Mdallal, Qasem; Jarad, Fahd; Abdeljawad, Thabet; Amin, Rohul
    This manuscript deals a numerical technique based on Haar wavelet collocation which is developed for the approximate solution of some systems of linear and nonlinear fractional order differential equations (FODEs). Based on these techniques, we find the numerical solution to var-ious systems of FODEs. We compare the obtain solution with the exact solution of the considered problems at integer orders. Also, we compute the maximum absolute error to demonstrate the effi-ciency and accuracy of the proposed method. For the illustration of our results we provide four test examples. The experimental rates of convergence for different number of collocation point is calculated which is approximately equal to 2. Fractional derivative is defined in the Caputo sense. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/ 4.0/).
  • Article
    Citation - WoS: 6
    Citation - Scopus: 6
    An Iterative Algorithm for Robust Simulation of the Sylvester Matrix Differential Equations
    (Springer, 2020) Beik, Samaneh Panjeh Ali; Torkzadeh, Leila; Baleanu, Dumitru; Nouri, Kazem
    This paper proposes a new effective pseudo-spectral approximation to solve the Sylvester and Lyapunov matrix differential equations. The properties of the Chebyshev basis operational matrix of derivative are applied to convert the main equation to the matrix equations. Afterwards, an iterative algorithm is examined for solving the obtained equations. Also, the error analysis of the propounded method is presented, which reveals the spectral rate of convergence. To illustrate the effectiveness of the proposed framework, several numerical examples are given.
  • Article
    Citation - WoS: 36
    Citation - Scopus: 37
    Novel Numerical Approach Based on Modified Extended Cubic B-Spline Functions for Solving Non-Linear Time-Fractional Telegraph Equation
    (Mdpi, 2020) Abbas, Muhammad; Iqbal, Azhar; Baleanu, Dumitru; Asad, Jihad H.; Akram, Tayyaba
    The telegraph model describes that the current and voltage waves can be reflected on a wire, that symmetrical wave patterns can form along a line. A numerical study of these voltage and current waves on a transferral line has been proposed via a modified extended cubic B-spline (MECBS) method. The B-spline functions have the flexibility and high order accuracy to approximate the solutions. These functions also preserve the symmetrical property. The MECBS and Crank Nicolson technique are employed to find out the solution of the non-linear time fractional telegraph equation. The time direction is discretized in the Caputo sense while the space dimension is discretized by the modified extended cubic B-spline. The non-linearity in the equation is linearized by Taylor's series. The proposed algorithm is unconditionally stable and convergent. The numerical examples are displayed to verify the authenticity and implementation of the method.
  • Article
    New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method
    (Editura ACAD Romane, 2015) Bhrawy, Ali H.; Zaky, Mahmoud A.; Baleanu, Dumitru
    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. © 2015, Editura Academiei Romane. All rights reserved.