Journal of Computational Physics, ISSN 0021-9991, 01/2015, Volume 281, pp. 876 - 895

In this paper, we propose and analyze an efficient operational formulation of spectral tau method for multi-term time–space fractional differential equation...

Operational matrix | Spectral method | Power law wave equation | Advection–diffusion equation | Multi-term time fractional wave–diffusion equations | Telegraph equation | Advection-diffusion equation | Multi-term time fractional wave-diffusion equations | CALCULUS | BOUNDARY-VALUE-PROBLEMS | DIFFUSION EQUATION | PHYSICS, MATHEMATICAL | LOBATTO COLLOCATION METHOD | POLYNOMIALS | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | VOLTERRA INTEGRODIFFERENTIAL EQUATIONS | ERROR ESTIMATION | COEFFICIENTS | Algorithms | Analysis | Methods | Differential equations | Approximation | Discretization | Mathematical analysis | Dirichlet problem | Spectra | Temporal logic

Operational matrix | Spectral method | Power law wave equation | Advection–diffusion equation | Multi-term time fractional wave–diffusion equations | Telegraph equation | Advection-diffusion equation | Multi-term time fractional wave-diffusion equations | CALCULUS | BOUNDARY-VALUE-PROBLEMS | DIFFUSION EQUATION | PHYSICS, MATHEMATICAL | LOBATTO COLLOCATION METHOD | POLYNOMIALS | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | VOLTERRA INTEGRODIFFERENTIAL EQUATIONS | ERROR ESTIMATION | COEFFICIENTS | Algorithms | Analysis | Methods | Differential equations | Approximation | Discretization | Mathematical analysis | Dirichlet problem | Spectra | Temporal logic

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 07/2017, Volume 340, pp. 655 - 669

Several physical phenomena such as transformation of pollutants, energy, particles and many others can be described by the well-known convection–diffusion...

Variable-order time fractional advection–diffusion equation (V-OTFA–DE) | Finite difference scheme | Caputo's variable-order fractional derivative | Positive scheme | Moving least squares (MLS) method | POINT INTERPOLATION METHOD | ALGORITHM | SIMULATION | PHYSICS, MATHEMATICAL | CONSTANT-ORDER | ANOMALOUS DIFFUSION | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Variable-order time fractional advection-diffusion equation (V-OTFA-DE) | MOVING LEAST-SQUARES | CONVERGENCE | FINITE-DIFFERENCE | Mechanical engineering | Methods | Air pollution | Analysis

Variable-order time fractional advection–diffusion equation (V-OTFA–DE) | Finite difference scheme | Caputo's variable-order fractional derivative | Positive scheme | Moving least squares (MLS) method | POINT INTERPOLATION METHOD | ALGORITHM | SIMULATION | PHYSICS, MATHEMATICAL | CONSTANT-ORDER | ANOMALOUS DIFFUSION | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Variable-order time fractional advection-diffusion equation (V-OTFA-DE) | MOVING LEAST-SQUARES | CONVERGENCE | FINITE-DIFFERENCE | Mechanical engineering | Methods | Air pollution | Analysis

Journal Article

Journal of Hydrology, ISSN 0022-1694, 09/2019, p. 124140

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 1/2009, Volume 47, Issue 3, pp. 1760 - 1781

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit...

Mathematical extrapolation | Error rates | Approximation | Partial differential equations | Numerical methods | Random walk | Differential equations | Boundary conditions | Perceptron convergence procedure | Method of lines | Extrapolation method | Fractional derivative of variable order | Nonlinear fractional advection-diffusion equation | Stability and convergence | Finite difference methods | stability and convergence | MATHEMATICS, APPLIED | APPROXIMATION | FELLER SEMIGROUPS | fractional derivative of variable order | nonlinear fractional advection-diffusion equation | finite difference methods | DISPERSION EQUATIONS | DIFFERENTIATION | OPERATORS | method of lines | extrapolation method | Advection-diffusion equation | Extrapolation | Numerical analysis | Stability | Mathematical analysis | Nonlinearity | Convergence

Mathematical extrapolation | Error rates | Approximation | Partial differential equations | Numerical methods | Random walk | Differential equations | Boundary conditions | Perceptron convergence procedure | Method of lines | Extrapolation method | Fractional derivative of variable order | Nonlinear fractional advection-diffusion equation | Stability and convergence | Finite difference methods | stability and convergence | MATHEMATICS, APPLIED | APPROXIMATION | FELLER SEMIGROUPS | fractional derivative of variable order | nonlinear fractional advection-diffusion equation | finite difference methods | DISPERSION EQUATIONS | DIFFERENTIATION | OPERATORS | method of lines | extrapolation method | Advection-diffusion equation | Extrapolation | Numerical analysis | Stability | Mathematical analysis | Nonlinearity | Convergence

Journal Article

Applied Mathematics Letters, ISSN 0893-9659, 03/2020, Volume 101, p. 106074

The fractional advection–diffusion problem coupled with incompressible Navier–Stokes equations is important in science and engineering. In this paper, a fresh...

Fractional advection–diffusion equation | Caputo derivative | Lattice Boltzmann method

Fractional advection–diffusion equation | Caputo derivative | Lattice Boltzmann method

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 06/2017, Volume 46, pp. 536 - 553

In this paper, we investigate the finite volume method (FVM) for a distributed-order space-fractional advection–diffusion (AD) equation. The mid-point...

Finite volume method | Riesz fractional derivatives | Distributed-order equation | Fractional advection–diffusion equation | Stability and convergence | ULTRASLOW DIFFUSION | DIFFERENTIAL-EQUATIONS | DISPERSION EQUATION | NUMERICAL APPROXIMATION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | WAVE EQUATION | ENGINEERING, MULTIDISCIPLINARY | Fractional advection-diffusion equation | BOUNDED DOMAINS | SCHEMES

Finite volume method | Riesz fractional derivatives | Distributed-order equation | Fractional advection–diffusion equation | Stability and convergence | ULTRASLOW DIFFUSION | DIFFERENTIAL-EQUATIONS | DISPERSION EQUATION | NUMERICAL APPROXIMATION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | WAVE EQUATION | ENGINEERING, MULTIDISCIPLINARY | Fractional advection-diffusion equation | BOUNDED DOMAINS | SCHEMES

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 06/2017, Volume 339, pp. 247 - 284

A new positivity preserving variational (PPV) procedure is proposed to solve the convection–diffusion–reaction (CDR) equation. Through the generalization of...

Multi-dimension | Galerkin/Least-Squares | Subgrid Scale | Discrete Upwind | Positivity | ADVECTION-DIFFUSION | LEAST-SQUARES METHOD | GENERALIZED FOURIER ANALYSES | PHYSICS, MATHEMATICAL | TIME STABILIZED METHODS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | NAVIER-STOKES EQUATIONS | INCOMPRESSIBLE FLOWS | COMPUTATIONAL FLUID-DYNAMICS | FINITE-ELEMENT FORMULATION | DOMINATED FLOWS | PETROV-GALERKIN METHOD | Mechanical engineering | Methods

Multi-dimension | Galerkin/Least-Squares | Subgrid Scale | Discrete Upwind | Positivity | ADVECTION-DIFFUSION | LEAST-SQUARES METHOD | GENERALIZED FOURIER ANALYSES | PHYSICS, MATHEMATICAL | TIME STABILIZED METHODS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | NAVIER-STOKES EQUATIONS | INCOMPRESSIBLE FLOWS | COMPUTATIONAL FLUID-DYNAMICS | FINITE-ELEMENT FORMULATION | DOMINATED FLOWS | PETROV-GALERKIN METHOD | Mechanical engineering | Methods

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 02/2016, Volume 274, pp. 208 - 219

Differential quadrature methods based on B-spline functions of degree four and five have been introduced to solve advection–diffusion equation numerically. Two...

B-spline | Advection–diffusion equation | Differential quadrature method | Advection-diffusion equation | MATHEMATICS, APPLIED | POLLUTANTS | DISPERSION | CONVECTION | ALGORITHM | SIMULATION | NUMERICAL-SOLUTION | TRANSPORT | DISTRIBUTED SYSTEM EQUATIONS | Advectiondiffusion equation | INSIGHTS

B-spline | Advection–diffusion equation | Differential quadrature method | Advection-diffusion equation | MATHEMATICS, APPLIED | POLLUTANTS | DISPERSION | CONVECTION | ALGORITHM | SIMULATION | NUMERICAL-SOLUTION | TRANSPORT | DISTRIBUTED SYSTEM EQUATIONS | Advectiondiffusion equation | INSIGHTS

Journal Article

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A Petrov–Galerkin finite element method for the fractional advection–diffusion equation

Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, 09/2016, Volume 309, pp. 388 - 410

This paper presents an in-depth numerical analysis of spatial fractional advection–diffusion equations (FADE) utilizing the finite element method (FEM). A...

Petrov–Galerkin formulation | Non-local diffusion | Fractional advection–diffusion equation | Fractional calculus | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | Fractional advection-diffusion equation | DISPERSION | Petrov-Galerkin formulation | COMPUTATIONAL FLUID-DYNAMICS | FORMULATION | SUPG | Finite element method | Analysis | Methods | Mechanical engineering

Petrov–Galerkin formulation | Non-local diffusion | Fractional advection–diffusion equation | Fractional calculus | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | Fractional advection-diffusion equation | DISPERSION | Petrov-Galerkin formulation | COMPUTATIONAL FLUID-DYNAMICS | FORMULATION | SUPG | Finite element method | Analysis | Methods | Mechanical engineering

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 06/2016, Volume 299, pp. 159 - 175

In this paper, a series of new high-order numerical approximations to th order Caputo derivative is constructed by using th degree interpolation approximation...

Difference scheme | Advection–diffusion equation | Caputo derivative | Convergence | Advection-diffusion equation | MATHEMATICS, APPLIED | Approximation | Mathematical analysis | Dirichlet problem | Truncation errors | Mathematical models | Derivatives

Difference scheme | Advection–diffusion equation | Caputo derivative | Convergence | Advection-diffusion equation | MATHEMATICS, APPLIED | Approximation | Mathematical analysis | Dirichlet problem | Truncation errors | Mathematical models | Derivatives

Journal Article

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Fourier‐based numerical approximation of the Weertman equation for moving dislocations

International Journal for Numerical Methods in Engineering, ISSN 0029-5981, 03/2018, Volume 113, Issue 12, pp. 1827 - 1850

Summary This work addresses the numerical approximation of solutions to a dimensionless form of the Weertman equation, which models a steadily moving...

Peierls‐Nabarro equation | Cauchy‐type nonlinear integro‐differential equation | reaction‐advection‐diffusion equation | preconditioned scheme | fractional Laplacian | discrete Fourier transform | Peierls-Nabarro equation | Cauchy-type nonlinear integro-differential equation | reaction-advection-diffusion equation | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PEIERLS-NABARRO MODEL | ENGINEERING, MULTIDISCIPLINARY | INTEGRAL-EQUATIONS | Nonlinear equations | Fourier transforms | Decay rate | Approximation | Dimensionless numbers | Dislocations | Advection | Numerical analysis | Robustness (mathematics) | Scaling laws | Differential equations | Mathematical models | Time integration | Numerical Analysis | Analysis of PDEs | Mathematics

Peierls‐Nabarro equation | Cauchy‐type nonlinear integro‐differential equation | reaction‐advection‐diffusion equation | preconditioned scheme | fractional Laplacian | discrete Fourier transform | Peierls-Nabarro equation | Cauchy-type nonlinear integro-differential equation | reaction-advection-diffusion equation | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PEIERLS-NABARRO MODEL | ENGINEERING, MULTIDISCIPLINARY | INTEGRAL-EQUATIONS | Nonlinear equations | Fourier transforms | Decay rate | Approximation | Dimensionless numbers | Dislocations | Advection | Numerical analysis | Robustness (mathematics) | Scaling laws | Differential equations | Mathematical models | Time integration | Numerical Analysis | Analysis of PDEs | Mathematics

Journal Article

Computer Physics Communications, ISSN 0010-4655, 01/2019, Volume 234, pp. 40 - 54

Partial differential equations (p.d.e) equipped with spatial derivatives of fractional order capture anomalous transport behaviors observed in diverse fields...

Multiple-Relaxation-Time | Lattice Boltzmann method | Stable process | Fractional advection–diffusion equation | Random walk | CONVECTION | STABILITY | CALCULUS | PHYSICS, MATHEMATICAL | TRANSPORT | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | DISPERSION-EQUATION | Fractional advection-diffusion equation | RANDOM-WALK | LEVY MOTION | PHASE-FIELD MODEL | Anisotropy | Differential equations | Computational Physics | Mathematics | Analysis of PDEs | Physics

Multiple-Relaxation-Time | Lattice Boltzmann method | Stable process | Fractional advection–diffusion equation | Random walk | CONVECTION | STABILITY | CALCULUS | PHYSICS, MATHEMATICAL | TRANSPORT | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | DISPERSION-EQUATION | Fractional advection-diffusion equation | RANDOM-WALK | LEVY MOTION | PHASE-FIELD MODEL | Anisotropy | Differential equations | Computational Physics | Mathematics | Analysis of PDEs | Physics

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 2007, Volume 191, Issue 1, pp. 12 - 20

In this paper, we consider a space–time fractional advection dispersion equation (STFADE) on a finite domain. The STFADE is obtained from the standard...

Implicit difference method | Stability | Fractional advection–diffusion equation | Space–time fractional derivatives | Explicit difference method | Convergence | Space-time fractional derivatives | Fractional advection-diffusion equation | fractional advection-diffusion equation | explicit difference method | MATHEMATICS, APPLIED | implicit difference method | convergence | space-time fractional derivatives | stability | Hydrologic cycle | Analysis | Methods

Implicit difference method | Stability | Fractional advection–diffusion equation | Space–time fractional derivatives | Explicit difference method | Convergence | Space-time fractional derivatives | Fractional advection-diffusion equation | fractional advection-diffusion equation | explicit difference method | MATHEMATICS, APPLIED | implicit difference method | convergence | space-time fractional derivatives | stability | Hydrologic cycle | Analysis | Methods

Journal Article

Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, 12/2017, Volume 327, pp. 352 - 368

In this paper we present an accurate stabilized FIC-FEM formulation for the multidimensional steady-state advection–diffusion–absorption equation. The...

Finite element method | Finite increment calculus | Advection–diffusion–absorption | MAXIMUM PRINCIPLE | LEAST-SQUARES METHOD | FINITE-ELEMENT METHODS | CONVECTION-DOMINATED FLOWS | HIGH REYNOLDS-NUMBERS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | Advection-diffusion-absorption | ENGINEERING, MULTIDISCIPLINARY | INCOMPRESSIBLE FLOWS | POINT METHOD | COMPUTATIONAL FLUID-DYNAMICS | PETROV-GALERKIN METHODS | CALCULUS FORMULATION | Differential equations | Models matemàtics | Mètodes en elements finits | Fluid dynamics | Dinàmica de fluids | Física | Anàlisi numèrica | Matemàtiques i estadística | Mathematical models | Àrees temàtiques de la UPC | Física de fluids

Finite element method | Finite increment calculus | Advection–diffusion–absorption | MAXIMUM PRINCIPLE | LEAST-SQUARES METHOD | FINITE-ELEMENT METHODS | CONVECTION-DOMINATED FLOWS | HIGH REYNOLDS-NUMBERS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | Advection-diffusion-absorption | ENGINEERING, MULTIDISCIPLINARY | INCOMPRESSIBLE FLOWS | POINT METHOD | COMPUTATIONAL FLUID-DYNAMICS | PETROV-GALERKIN METHODS | CALCULUS FORMULATION | Differential equations | Models matemàtics | Mètodes en elements finits | Fluid dynamics | Dinàmica de fluids | Física | Anàlisi numèrica | Matemàtiques i estadística | Mathematical models | Àrees temàtiques de la UPC | Física de fluids

Journal Article

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A new Crank–Nicolson finite element method for the time-fractional subdiffusion equation

Applied Numerical Mathematics, ISSN 0168-9274, 11/2017, Volume 121, pp. 82 - 95

In this paper, a new Crank–Nicolson finite element method for the time-fractional subdiffusion equation is developed, in which a novel time discretization...

Finite element method | Unconditional stability | The time-fractional subdiffusion equation | Subdiffusion | Convergence | FOKKER-PLANCK EQUATION | NONLINEAR SOURCE-TERM | MATHEMATICS, APPLIED | BOUNDARY-CONDITIONS | EXPONENTIAL INTEGRATORS | SUB-DIFFUSION | NUMERICAL ALGORITHM | PARTIAL-DIFFERENTIAL-EQUATIONS | NONUNIFORM TIMESTEPS | RANDOM-WALKS | ADVECTION-DIFFUSION EQUATION | Analysis | Methods

Finite element method | Unconditional stability | The time-fractional subdiffusion equation | Subdiffusion | Convergence | FOKKER-PLANCK EQUATION | NONLINEAR SOURCE-TERM | MATHEMATICS, APPLIED | BOUNDARY-CONDITIONS | EXPONENTIAL INTEGRATORS | SUB-DIFFUSION | NUMERICAL ALGORITHM | PARTIAL-DIFFERENTIAL-EQUATIONS | NONUNIFORM TIMESTEPS | RANDOM-WALKS | ADVECTION-DIFFUSION EQUATION | Analysis | Methods

Journal Article

Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, 09/2018, Volume 506, pp. 135 - 155

In this study vertical distribution of sediment particles in steady uniform turbulent open channel flow over erodible bed is investigated using fractional...

Suspension concentration distribution | Turbulent flow | Rouse equation | Open channel flow | Fractional advection–diffusion equation | VELOCITY | SUSPENDED SEDIMENT | PARTICLES | PHYSICS, MULTIDISCIPLINARY | Fractional advection-diffusion equation | Specific gravity | Turbulence | Analysis

Suspension concentration distribution | Turbulent flow | Rouse equation | Open channel flow | Fractional advection–diffusion equation | VELOCITY | SUSPENDED SEDIMENT | PARTICLES | PHYSICS, MULTIDISCIPLINARY | Fractional advection-diffusion equation | Specific gravity | Turbulence | Analysis

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 11/2018, Volume 76, Issue 10, pp. 2460 - 2476

In the paper, a Crank–Nicolson alternating direction implicit (ADI) Galerkin–Legendre spectral scheme is presented for the two-dimensional Riesz space...

ADI Galerkin–Legendre spectral method | Gauss quadrature | Two-dimensional Riesz space distributed-order advection–diffusion equation | Stability and convergence analysis | FOKKER-PLANCK EQUATION | MATHEMATICS, APPLIED | VARIABLE-ORDER | ADI Galerkin-Legendre spectral method | APPROXIMATION | DIFFERENTIAL-EQUATIONS | TERM | MODEL | WAVE EQUATION | BOUNDED DOMAINS | Two-dimensional Riesz space distributed-order advection-diffusion equation

ADI Galerkin–Legendre spectral method | Gauss quadrature | Two-dimensional Riesz space distributed-order advection–diffusion equation | Stability and convergence analysis | FOKKER-PLANCK EQUATION | MATHEMATICS, APPLIED | VARIABLE-ORDER | ADI Galerkin-Legendre spectral method | APPROXIMATION | DIFFERENTIAL-EQUATIONS | TERM | MODEL | WAVE EQUATION | BOUNDED DOMAINS | Two-dimensional Riesz space distributed-order advection-diffusion equation