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Subject: search.filters.subject.Shifted Jacobi Polynomials

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  • Article
    Citation - WoS: 29
    Citation - Scopus: 31
    A Novel Spectral Approximation for the Two-Dimensional Fractional Sub-Diffusion Problems
    (Editura Acad Romane, 2015) Bhrawy, A. H.; Baleanu, Dumitru; Zaky, M. A.; Baleanu, D.; Abdelkawy, M. A.; Matematik
    This paper reports a new numerical method that enables easy and convenient discretization of a two-dimensional sub-diffusion equation with fractional derivatives of any order. The suggested method is based on Jacobi tau spectral procedure together with the Jacobi operational matrix for fractional derivatives, described in the Caputo sense. Such approach has the advantage of reducing the problem to the solution of a system of algebraic equations, which may then be solved by any standard numerical technique. The validity and effectiveness of the method are demonstrated by solving two numerical examples, which are presented in the form of tables and graphs to make more easier comparisons with the exact solutions and the results obtained by other methods.
  • 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: 3
    Citation - Scopus: 3
    Numerical Solution of Space-Time Variable Fractional Order Advection-Dispersion Equation Using Jacobi Spectral Collocation Method
    (Univ Putra Malaysia Press, 2020) Moghadam, Soltanpour A.; Baleanu, Dumitru; Arabameri, M.; Barfeie, M.; Baleanu, D.; Soltanpour Moghadam, A.; Matematik
    This article is aimed at studying computational solution of variable order fractional advection-dispersion equation for one-dimensional and two-dimensional spaces utilizing spectral collocation method. In the considered model, the time derivative is Coimbra fractional derivative and space derivative is a Riemann-Liouville derivative. Jacobi polynomials are applied as basic functions in approximation of the solution. The presented approach is an application of the shifted Jacobi-Gauss collocation (SJ-G-C) and the shifted Jacobi-Gauss-Radau collocation (SJ-GR-C) methods using for discretizing along space and time, respectively. Using the related collocation points, the problem would be changed to an algebraic equation system, which can be tackled applying a computational technique. At the end, several examples in one and two dimensional cases have been solved by introduced approach, it would be shown that the proposed numerical algorithm has considerably higher accuracy in contrast to the existing computational schemes including finite difference approach.
  • Article
    Citation - WoS: 13
    Citation - Scopus: 16
    A Robust Scheme for Caputo Variable-Order Time-Fractional Diffusion-Type Equations
    (Springer, 2023) Hosseini, Kamyar; Baleanu, Dumitru; Salahshour, Soheil; Hincal, Evren; Sadri, Khadijeh
    The focus of this work is to construct a pseudo-operational Jacobi collocation scheme for numerically solving the Caputo variable-order time-fractional diffusion-type equations with applications in applied sciences. Modeling scientific phenomena in the context of fluid flow problems, curing reactions of thermosetting systems, solid oxide fuel cells, and solvent diffusion into heavy oils led to the appearance of these equations. For this reason, the numerical solution of these equations has attracted a lot of attention. More precisely, using pseudo-operational matrices and appropriate approximations based on bivariate Jacobi polynomials, the approximate solutions of the variable-order time-fractional diffusion-type equations in the Caputo sense with high accuracy are formally retrieved. Based on orthogonal bivariate Jacobi polynomials and their operational matrices, a sparse algebraic system is generated which makes implementing the proposed approach easy. An error bound is computed for the residual function by proving some theorems. To illustrate the accuracy and efficiency of the scheme, several illustrative examples are considered. The results demonstrate the efficiency of the present method compared to those achieved by the Legendre and Lucas multi-wavelet methods and the Crank-Nicolson compact method.
  • 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: 76
    Citation - Scopus: 82
    On Shifted Jacobi Spectral Approximations for Solving Fractional Differential Equations
    (Elsevier Science inc, 2013) Bhrawy, A. H.; Baleanu, D.; Ezz-Eldien, S. S.; Doha, E. H.
    In this paper, a new formula of Caputo fractional-order derivatives of shifted Jacobi polynomials of any degree in terms of shifted Jacobi polynomials themselves is proved. We discuss a direct solution technique for linear multi-order fractional differential equations (FDEs) subject to nonhomogeneous initial conditions using a shifted Jacobi tau approximation. A quadrature shifted Jacobi tau (Q-SJT) approximation is introduced for the solution of linear multi-order FDEs with variable coefficients. We also propose a shifted Jacobi collocation technique for solving nonlinear multi-order fractional initial value. problems. The advantages of using the proposed techniques are discussed and we compare them with other existing methods. We investigate some illustrative examples of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques. (C) 2013 Elsevier Inc. All rights reserved.
  • Article
    Citation - WoS: 37
    Citation - Scopus: 45
    The Operational Matrix Formulation of the Jacobi Tau Approximation for Space Fractional Diffusion Equation
    (Springer, 2014) Bhrawy, Ali H.; Baleanu, Dumitru; Ezz-Eldien, Samer S.; Doha, Eid H.
    In this article, an accurate and efficient numerical method is presented for solving the space-fractional order diffusion equation (SFDE). Jacobi polynomials are used to approximate the solution of the equation as a base of the tau spectral method which is based on the Jacobi operational matrices of fractional derivative and integration. The main advantage of this method is based upon reducing the nonlinear partial differential equation into a system of algebraic equations in the expansion coefficient of the solution. In order to test the accuracy and efficiency of our method, the solutions of the examples presented are introduced in the form of tables to make a comparison with those obtained by other methods and with the exact solutions easy.
  • Article
    Citation - WoS: 24
    Citation - Scopus: 22
    Shifted Jacobi Spectral Collocation Method With Convergence Analysis for Solving Integro-Differential Equations and System of Integro-Differential Equations
    (inst Mathematics & informatics, 2019) Abdelkawy, Mohamed A.; Amin, Ahmed Z. M.; Baleanu, Dumitru; Doha, Eid H.
    This article addresses the solution of multi-dimensional integro-differential equations (IDEs) by means of the spectral collocation method and taking the advantage of the properties of shifted Jacobi polynomials. The applicability and accuracy of the present technique have been examined by the given numerical examples in this paper. By means of these numerical examples, we ensure that the present technique is simple and very accurate. Furthermore, an error analysis is performed to verify the correctness and feasibility of the proposed method when solving IDE.
  • Article
    Citation - WoS: 203
    Citation - Scopus: 217
    A Spectral Tau Algorithm Based on Jacobi Operational Matrix for Numerical Solution of Time Fractional Diffusion-Wave Equations
    (Academic Press inc Elsevier Science, 2015) Doha, E. H.; Baleanu, D.; Ezz-Eldien, S. S.; Bhrawy, A. H.
    In this paper, an efficient and accurate spectral numerical method is presented for solving second-, fourth-order fractional diffusion-wave equations and fractional wave equations with damping. The proposed method is based on Jacobi tau spectral procedure together with the Jacobi operational matrix for fractional integrals, described in the Riemann-Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The validity and effectiveness of the method are demonstrated by solving five 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. (C) 2014 Elsevier Inc. All rights reserved.