Journal of Computational and Applied Mathematics, ISSN 0377-0427, 08/2017, Volume 319, pp. 493 - 513

In recent years, numerical methods have been introduced to solve two-dimensional Volterra and Fredholm integral equations. In this study, a numerical scheme is...

Three-dimensional integral equations | Shifted Jacobi polynomials | Operational matrices | Convergence | MATHEMATICS, APPLIED | SPREAD | Mechanical engineering

Three-dimensional integral equations | Shifted Jacobi polynomials | Operational matrices | Convergence | MATHEMATICS, APPLIED | SPREAD | Mechanical engineering

Journal Article

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Numerical solution of fractional delay differential equation by shifted Jacobi polynomials

International Journal of Computer Mathematics, ISSN 0020-7160, 03/2017, Volume 94, Issue 3, pp. 471 - 492

In this paper, the fractional delay differential equation (FDDE) is considered for the purpose to develop an approximate scheme for its numerical solutions....

33C45 | delay differential equations | fractional differential equations | operational matrix | shifted Jacobi polynomials | Newton's iterative method | 34A08 | POPULATION | MATHEMATICS, APPLIED | DYNAMICS | MODEL | IDENTIFICATION | Polynomials | Differential equations | Matrices (mathematics) | Lasers | Mathematical analysis | Mathematical models | Matrix methods | Delay

33C45 | delay differential equations | fractional differential equations | operational matrix | shifted Jacobi polynomials | Newton's iterative method | 34A08 | POPULATION | MATHEMATICS, APPLIED | DYNAMICS | MODEL | IDENTIFICATION | Polynomials | Differential equations | Matrices (mathematics) | Lasers | Mathematical analysis | Mathematical models | Matrix methods | Delay

Journal Article

CMES - Computer Modeling in Engineering and Sciences, ISSN 1526-1492, 2018, Volume 115, Issue 1, pp. 67 - 84

Journal Article

Bulletin of the Malaysian Mathematical Sciences Society, ISSN 0126-6705, 2014, Volume 37, Issue 4, pp. 983 - 996

A new shifted Jacobi operational matrix (SJOM) of fractional integration of any order is introduced and applied together with spectral tau method for solving...

Operational matrix | Multiterm FDEs | Riemann-Liouville derivative | Shifted Jacobi polynomials | Tau method | SYSTEM | multi-term FDEs | SERIES APPROACH | BOUNDARY-VALUE-PROBLEMS | DIFFERENTIAL-EQUATIONS | IDENTIFICATION | shifted Jacobi polynomials | CHEBYSHEV POLYNOMIALS | MATHEMATICS | ORDERS | COLLOCATION

Operational matrix | Multiterm FDEs | Riemann-Liouville derivative | Shifted Jacobi polynomials | Tau method | SYSTEM | multi-term FDEs | SERIES APPROACH | BOUNDARY-VALUE-PROBLEMS | DIFFERENTIAL-EQUATIONS | IDENTIFICATION | shifted Jacobi polynomials | CHEBYSHEV POLYNOMIALS | MATHEMATICS | ORDERS | COLLOCATION

Journal Article

Hacettepe Journal of Mathematics and Statistics, ISSN 1303-5010, 2016, Volume 45, Issue 2, pp. 311 - 335

In this paper, a numerical method is developed for solving linear and nonlinear integro-partial differential equations in terms of the two variables Jacobi...

Operational matrix | Best approximation | Shifted Jacobi polynomials | Collocation method | Integro–partial differential equations | MATHEMATICS | NUMERICAL-SOLUTION | APPROXIMATION | VOLTERRA INTEGRODIFFERENTIAL EQUATIONS | STATISTICS & PROBABILITY | Integro-partial differential equations

Operational matrix | Best approximation | Shifted Jacobi polynomials | Collocation method | Integro–partial differential equations | MATHEMATICS | NUMERICAL-SOLUTION | APPROXIMATION | VOLTERRA INTEGRODIFFERENTIAL EQUATIONS | STATISTICS & PROBABILITY | Integro-partial differential equations

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 07/2015, Volume 293, pp. 142 - 156

In this paper, an efficient and accurate spectral numerical method is presented for solving second-, fourth-order fractional diffusion-wave equations and...

Operational matrix | Shifted Jacobi polynomials | Tau method | Fractional diffusion-wave equations | Caputo derivative | SCHEME | ORDER | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | SUBDIFFUSION | PARTIAL-DIFFERENTIAL-EQUATIONS | HOMOTOPY PERTURBATION METHOD | PHYSICS, MATHEMATICAL | Algorithms | Damping | Approximation | Mathematical analysis | Exact solutions | Tables | Mathematical models | Spectra | Diffusion

Operational matrix | Shifted Jacobi polynomials | Tau method | Fractional diffusion-wave equations | Caputo derivative | SCHEME | ORDER | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | SUBDIFFUSION | PARTIAL-DIFFERENTIAL-EQUATIONS | HOMOTOPY PERTURBATION METHOD | PHYSICS, MATHEMATICAL | Algorithms | Damping | Approximation | Mathematical analysis | Exact solutions | Tables | Mathematical models | Spectra | Diffusion

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 01/2016, Volume 40, Issue 2, pp. 832 - 845

In this study, we propose shifted fractional-order Jacobi orthogonal functions (SFJFs) based on the definition of the classical Jacobi polynomials. We derive a...

Nonlinear fractional differential equations | Tau method | Shifted fractional Jacobi polynomials | Systems of fractional differential equations | Pseudo-spectral methods | Differential equations

Nonlinear fractional differential equations | Tau method | Shifted fractional Jacobi polynomials | Systems of fractional differential equations | Pseudo-spectral methods | Differential equations

Journal Article

Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 2012, Volume 17, Issue 1, pp. 62 - 70

► A new numerical scheme based on general Jacobi–Gauss collocation method is introduced for Lane–Emden type equation. ► The Jacobi–Gauss collocation method has...

Shifted Jacobi polynomials | Jacobi–Gauss quadrature | Collocation method | Second-order initial value problems | Lane–Emden type equation | Lane-Emden type equation | Jacobi-Gauss quadrature | MATHEMATICS, APPLIED | PHYSICS, FLUIDS & PLASMAS | DIFFERENTIAL-EQUATIONS | PHYSICS, MATHEMATICAL | INITIAL-VALUE PROBLEMS | VARIATIONAL ITERATION METHOD | POLYNOMIALS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | SINGULAR IVPS | HOMOTOPY-PERTURBATION METHOD | APPROXIMATE SOLUTION | SPECTRAL-GALERKIN ALGORITHMS | Approximation | Collocation | Infinity | Mathematical analysis | Images | Nonlinearity | Mathematical models | Spectral methods

Shifted Jacobi polynomials | Jacobi–Gauss quadrature | Collocation method | Second-order initial value problems | Lane–Emden type equation | Lane-Emden type equation | Jacobi-Gauss quadrature | MATHEMATICS, APPLIED | PHYSICS, FLUIDS & PLASMAS | DIFFERENTIAL-EQUATIONS | PHYSICS, MATHEMATICAL | INITIAL-VALUE PROBLEMS | VARIATIONAL ITERATION METHOD | POLYNOMIALS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | SINGULAR IVPS | HOMOTOPY-PERTURBATION METHOD | APPROXIMATE SOLUTION | SPECTRAL-GALERKIN ALGORITHMS | Approximation | Collocation | Infinity | Mathematical analysis | Images | Nonlinearity | Mathematical models | Spectral methods

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 01/2018, Volume 317, pp. 49 - 67

Based on Jacobi polynomials, an operational method is proposed to solve the generalized Abel’s integral equations (a class of singular integral equations)....

Abel’s integral equation | Shifted Jacobi polynomials | Error estimation | Collocation method

Abel’s integral equation | Shifted Jacobi polynomials | Error estimation | Collocation method

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 2019, Volume 78, Issue 3, pp. 889 - 904

Herein, we propose a numerical scheme to solve spectrally hyperbolic partial differential equations (HPDEs) using Galerkin method and approximate the solutions...

Hyperbolic partial differential equations | Galerkin method | Shifted Jacobi polynomials | SYSTEM | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | VARIABLE-COEFFICIENTS | COLLOCATION METHOD | TELEGRAPH EQUATION | GAUSS-RADAU SCHEME | Methods | Differential equations | Partial differential equations | Polynomials

Hyperbolic partial differential equations | Galerkin method | Shifted Jacobi polynomials | SYSTEM | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | VARIABLE-COEFFICIENTS | COLLOCATION METHOD | TELEGRAPH EQUATION | GAUSS-RADAU SCHEME | Methods | Differential equations | Partial differential equations | Polynomials

Journal Article

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On shifted Jacobi spectral approximations for solving fractional differential equations

Applied Mathematics and Computation, ISSN 0096-3003, 04/2013, Volume 219, Issue 15, pp. 8042 - 8056

► A new formula of fractional-order derivatives of shifted Jacobi polynomials is proved. ► A Jacobi spectral tau approximation for solving linear FDEs with...

Multi-term fractional differential equations | Shifted Jacobi polynomials | Jacobi–Gauss–Lobatto quadrature | Caputo derivative | Spectral methods | Nonlinear fractional initial value problems | Jacobi-Gauss-Lobatto quadrature | EXISTENCE | MATHEMATICS, APPLIED | TAU METHOD | OPERATIONAL MATRIX | ORDER | NUMERICAL-SOLUTION | COLLOCATION METHOD | COEFFICIENTS | EFFICIENT | Algorithms | Differential equations | Universities and colleges

Multi-term fractional differential equations | Shifted Jacobi polynomials | Jacobi–Gauss–Lobatto quadrature | Caputo derivative | Spectral methods | Nonlinear fractional initial value problems | Jacobi-Gauss-Lobatto quadrature | EXISTENCE | MATHEMATICS, APPLIED | TAU METHOD | OPERATIONAL MATRIX | ORDER | NUMERICAL-SOLUTION | COLLOCATION METHOD | COEFFICIENTS | EFFICIENT | Algorithms | Differential equations | Universities and colleges

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 05/2016, Volume 39, Issue 7, pp. 1765 - 1779

This paper presents a shifted fractional‐order Jacobi orthogonal function (SFJF) based on the definition of the classical Jacobi polynomial. A new fractional...

shifted fractional‐order Jacobi orthogonal function | time‐fractional partial differential equations | Tau method | operational matrix | shifted Jacobi polynomials | Caputo derivative | time-fractional partial differential equations | shifted fractional-order Jacobi orthogonal function | MATHEMATICS, APPLIED | APPROXIMATION | CAUCHY-PROBLEM | SCHEME | DIFFUSION-WAVE | FINITE-DIFFERENCE METHODS | REGULARIZATION METHOD | SPECTRAL ELEMENT METHODS | EQUATION | Algorithms | Approximation | Discretization | Partial differential equations | Integrals | Mathematical analysis | Mathematical models | Polynomials

shifted fractional‐order Jacobi orthogonal function | time‐fractional partial differential equations | Tau method | operational matrix | shifted Jacobi polynomials | Caputo derivative | time-fractional partial differential equations | shifted fractional-order Jacobi orthogonal function | MATHEMATICS, APPLIED | APPROXIMATION | CAUCHY-PROBLEM | SCHEME | DIFFUSION-WAVE | FINITE-DIFFERENCE METHODS | REGULARIZATION METHOD | SPECTRAL ELEMENT METHODS | EQUATION | Algorithms | Approximation | Discretization | Partial differential equations | Integrals | Mathematical analysis | Mathematical models | Polynomials

Journal Article

Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 10/2012, Volume 17, Issue 10, pp. 3802 - 3810

► A shifted Jacobi tau method is introduced for linear high-order multi-point boundary value problems (BVPs). ► Extension of the tau method for multi-point...

Gauss quadrature | Shifted Jacobi polynomials | Tau method | High-order differential equation | Nonlinear boundary value problems | Collocation method | Multi-point boundary value problem | EXISTENCE | INTERVAL | MATHEMATICS, APPLIED | PHYSICS, FLUIDS & PLASMAS | TAU-METHOD | DIFFERENTIAL-EQUATIONS | PHYSICS, MATHEMATICAL | POLYNOMIALS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | COEFFICIENTS | APPROXIMATE SOLUTION | ERROR | FREDHOLM INTEGRODIFFERENTIAL EQUATIONS | GALERKIN ALGORITHMS | Methods | Differential equations | Universities and colleges | Boundary value problems | Approximation | Mathematical analysis | Nonlinearity | Mathematical models | Differentiation | Convergence

Gauss quadrature | Shifted Jacobi polynomials | Tau method | High-order differential equation | Nonlinear boundary value problems | Collocation method | Multi-point boundary value problem | EXISTENCE | INTERVAL | MATHEMATICS, APPLIED | PHYSICS, FLUIDS & PLASMAS | TAU-METHOD | DIFFERENTIAL-EQUATIONS | PHYSICS, MATHEMATICAL | POLYNOMIALS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | COEFFICIENTS | APPROXIMATE SOLUTION | ERROR | FREDHOLM INTEGRODIFFERENTIAL EQUATIONS | GALERKIN ALGORITHMS | Methods | Differential equations | Universities and colleges | Boundary value problems | Approximation | Mathematical analysis | Nonlinearity | Mathematical models | Differentiation | Convergence

Journal Article

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Jacobi Collocation Approximation for Solving Multi-dimensional Volterra Integral Equations

International Journal of Nonlinear Sciences and Numerical Simulation, ISSN 1565-1339, 07/2017, Volume 18, Issue 5, pp. 411 - 425

This paper addresses the solution of one- and two-dimensional Volterra integral equations (VIEs) by means of the spectral collocation method. The novel...

Volterra integral equation | spectral collocation method | system of Volterra integral equation | error analysis | shifted Jacobi polynomials | MATHEMATICS, APPLIED | SERIES | ALGORITHM | INTEGRODIFFERENTIAL EQUATIONS | PHYSICS, MATHEMATICAL | WAVE TYPE EQUATIONS | NUMERICAL-SOLUTION | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | COMPUTATIONAL METHOD | SYSTEMS | 2ND KIND | FREDHOLM | HYBRID | Error analysis | Collocation | Mathematical analysis | Integral equations | Feasibility studies | Polynomials | Error correction | Volterra integral equations

Volterra integral equation | spectral collocation method | system of Volterra integral equation | error analysis | shifted Jacobi polynomials | MATHEMATICS, APPLIED | SERIES | ALGORITHM | INTEGRODIFFERENTIAL EQUATIONS | PHYSICS, MATHEMATICAL | WAVE TYPE EQUATIONS | NUMERICAL-SOLUTION | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | COMPUTATIONAL METHOD | SYSTEMS | 2ND KIND | FREDHOLM | HYBRID | Error analysis | Collocation | Mathematical analysis | Integral equations | Feasibility studies | Polynomials | Error correction | Volterra integral equations

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 2011, Volume 35, Issue 12, pp. 5662 - 5672

In this paper, we state and prove a new formula expressing explicitly the derivatives of shifted Chebyshev polynomials of any degree and for any...

Multi-term fractional differential equations | Gauss quadrature | Nonlinear fractional differential equations | Tau method | Collocation method | Shifted Chebyshev polynomials | NUMERICAL-SOLUTION | IVPS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | HOMOTOPY-PERTURBATION METHOD | GALERKIN ALGORITHMS | JACOBI-POLYNOMIALS | Analysis | Methods | Universities and colleges | Approximation | Mathematical analysis | Differential equations | Chebyshev approximation | Initial value problems | Mathematical models | Derivatives | Spectral methods

Multi-term fractional differential equations | Gauss quadrature | Nonlinear fractional differential equations | Tau method | Collocation method | Shifted Chebyshev polynomials | NUMERICAL-SOLUTION | IVPS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | HOMOTOPY-PERTURBATION METHOD | GALERKIN ALGORITHMS | JACOBI-POLYNOMIALS | Analysis | Methods | Universities and colleges | Approximation | Mathematical analysis | Differential equations | Chebyshev approximation | Initial value problems | Mathematical models | Derivatives | Spectral methods

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 05/2015, Volume 279, pp. 80 - 96

In this paper, a class of linear and nonlinear functional integro-differential equations are considered that can be found in the various fields of sciences...

Jacobi operational matrices | Operational collocation method | Shifted Jacobi polynomials | Error estimation | Functional integro-differential equations | MATHEMATICS, APPLIED | CONVERGENCE ANALYSIS | COLLOCATION METHOD | Differential equations

Jacobi operational matrices | Operational collocation method | Shifted Jacobi polynomials | Error estimation | Functional integro-differential equations | MATHEMATICS, APPLIED | CONVERGENCE ANALYSIS | COLLOCATION METHOD | Differential equations

Journal Article

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Jacobi collocation scheme for variable-order fractional reaction-subdiffusion equation

Computational and Applied Mathematics, ISSN 0101-8205, 9/2018, Volume 37, Issue 4, pp. 5315 - 5333

We developed a numerical scheme to solve the variable-order fractional linear subdiffusion and nonlinear reaction-subdiffusion equations using the shifted...

Variable-order fractional equations | Computational Mathematics and Numerical Analysis | Fractional nonlinear reaction-subdiffusion equation | 33C45 | Fractional subdiffusion equation | Mathematics | Convergence analysis | 34A08 | Shifted Jacobi polynomials | Mathematical Applications in Computer Science | 65M70 | Applications of Mathematics | Mathematical Applications in the Physical Sciences | MATHEMATICS, APPLIED | APPROXIMATION | BOUNDARY-VALUE-PROBLEMS | DIFFERENTIAL-EQUATIONS | DIFFUSION EQUATION | INITIAL-VALUE PROBLEMS | POLYNOMIALS | OPERATIONAL MATRIX | NUMERICAL-SOLUTION | ANOMALOUS SUBDIFFUSION | FINITE-DIFFERENCE

Variable-order fractional equations | Computational Mathematics and Numerical Analysis | Fractional nonlinear reaction-subdiffusion equation | 33C45 | Fractional subdiffusion equation | Mathematics | Convergence analysis | 34A08 | Shifted Jacobi polynomials | Mathematical Applications in Computer Science | 65M70 | Applications of Mathematics | Mathematical Applications in the Physical Sciences | MATHEMATICS, APPLIED | APPROXIMATION | BOUNDARY-VALUE-PROBLEMS | DIFFERENTIAL-EQUATIONS | DIFFUSION EQUATION | INITIAL-VALUE PROBLEMS | POLYNOMIALS | OPERATIONAL MATRIX | NUMERICAL-SOLUTION | ANOMALOUS SUBDIFFUSION | FINITE-DIFFERENCE

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 01/2018, Volume 317, pp. 49 - 67

Based on Jacobi polynomials, an operational method is proposed to solve the generalized Abel's integral equations (a class of singular integral equations)....

Shifted Jacobi polynomials | Abel's integral equation | Error estimation | Collocation method | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | WAVELETS | 1ST KIND | Plasma physics | Atmospheric physics | Analysis | Methods | Algorithms | Mechanical engineering

Shifted Jacobi polynomials | Abel's integral equation | Error estimation | Collocation method | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | WAVELETS | 1ST KIND | Plasma physics | Atmospheric physics | Analysis | Methods | Algorithms | Mechanical engineering

Journal Article

Nonlinear Analysis: Modelling and Control, ISSN 1392-5113, 2019, Volume 24, Issue 3, pp. 332 - 352

This article addresses the solution of multi-dimensional integro-differential equations (IDEs) by means of the spectral collocation method and taking the...

Spectral collocation method | Gauss quadrature | Shifted Jacobi polynomials | Integro-differential equation | Jacobi | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | integro-differential equation | spectral collocation method | DIFFERENTIAL-DIFFERENCE EQUATION | shifted Jacobi polynomials | PETROV-GALERKIN ELEMENTS | Jacobi-Gauss quadrature | Jacobi–Gauss quadrature

Spectral collocation method | Gauss quadrature | Shifted Jacobi polynomials | Integro-differential equation | Jacobi | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | integro-differential equation | spectral collocation method | DIFFERENTIAL-DIFFERENCE EQUATION | shifted Jacobi polynomials | PETROV-GALERKIN ELEMENTS | Jacobi-Gauss quadrature | Jacobi–Gauss quadrature

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 09/2015, Volume 38, Issue 14, pp. 3022 - 3032

In this paper, a shifted Jacobi–Gauss collocation spectral algorithm is developed for solving numerically systems of high‐order linear retarded and advanced...

collocation method | system of differential‐difference equations | Jacobi–Gauss quadrature | shifted Jacobi polynomials | system of differential-difference equations | Jacobi-Gauss quadrature | POLYNOMIALS | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | VARIABLE-COEFFICIENTS | BOUNDARY-VALUE-PROBLEMS | DELAY EQUATION | MULTI-PANTOGRAPH EQUATION | APPROXIMATE SOLUTION | Analysis | Algorithms | Approximation | Initial conditions | Collocation | Mathematical analysis | Mathematical models | Spectra | Convergence

collocation method | system of differential‐difference equations | Jacobi–Gauss quadrature | shifted Jacobi polynomials | system of differential-difference equations | Jacobi-Gauss quadrature | POLYNOMIALS | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | VARIABLE-COEFFICIENTS | BOUNDARY-VALUE-PROBLEMS | DELAY EQUATION | MULTI-PANTOGRAPH EQUATION | APPROXIMATE SOLUTION | Analysis | Algorithms | Approximation | Initial conditions | Collocation | Mathematical analysis | Mathematical models | Spectra | Convergence

Journal Article

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