Applied Mathematics Letters, ISSN 0893-9659, 11/2017, Volume 73, pp. 120 - 127

The main aim of the current paper is to develop a high-order numerical scheme to solve the space–time tempered fractional diffusion-wave equation. The...

Unconditional stability | Space–time tempered fractional diffusion-wave equation | Integro-differential equation | Finite difference method | Convergence | SCHEME | MATHEMATICS, APPLIED | APPROXIMATION | Space time tempered fractional | DIFFERENTIAL-EQUATIONS | diffusion-wave equation | Methods | Differential equations

Unconditional stability | Space–time tempered fractional diffusion-wave equation | Integro-differential equation | Finite difference method | Convergence | SCHEME | MATHEMATICS, APPLIED | APPROXIMATION | Space time tempered fractional | DIFFERENTIAL-EQUATIONS | diffusion-wave equation | Methods | Differential equations

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 06/2017, Volume 73, Issue 12, pp. 2561 - 2572

In this paper, a multi-dimensional α-fractional diffusion–wave equation is introduced and the properties of its fundamental solution are studied. This equation...

Anomalous wave propagation | Mellin–Barnes integral | Entropy production rate | Fractional diffusion–wave equation | Phase velocity | Anomalous diffusion | MATHEMATICS, APPLIED | MODELS | Mellin-Barnes integral | Fractional diffusion-wave equation | ENTROPY

Anomalous wave propagation | Mellin–Barnes integral | Entropy production rate | Fractional diffusion–wave equation | Phase velocity | Anomalous diffusion | MATHEMATICS, APPLIED | MODELS | Mellin-Barnes integral | Fractional diffusion-wave equation | ENTROPY

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 11/2014, Volume 277, pp. 1 - 15

In this paper, compact finite difference schemes for the modified anomalous fractional sub-diffusion equation and fractional diffusion-wave equation are...

Weighted and shifted Grünwald difference operator | Compact difference scheme | Fractional diffusion-wave equation | Modified anomalous fractional sub-diffusion equation | Weighted and shifted grünwald difference operator | NUMERICAL-METHODS | NONLINEAR SOURCE-TERM | APPROXIMATIONS | STABILITY | ALGORITHM | HIGH-ORDER | SUBDIFFUSION EQUATION | PHYSICS, MATHEMATICAL | Weighted and shifted Grunwald difference operator | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | HIGH SPATIAL ACCURACY | NEUMANN BOUNDARY-CONDITIONS | Mathematics - Numerical Analysis

Weighted and shifted Grünwald difference operator | Compact difference scheme | Fractional diffusion-wave equation | Modified anomalous fractional sub-diffusion equation | Weighted and shifted grünwald difference operator | NUMERICAL-METHODS | NONLINEAR SOURCE-TERM | APPROXIMATIONS | STABILITY | ALGORITHM | HIGH-ORDER | SUBDIFFUSION EQUATION | PHYSICS, MATHEMATICAL | Weighted and shifted Grunwald difference operator | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | HIGH SPATIAL ACCURACY | NEUMANN BOUNDARY-CONDITIONS | Mathematics - Numerical Analysis

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 06/2018, Volume 41, Issue 9, pp. 3476 - 3494

The main purpose of the current paper is to propose a new numerical scheme based on the spectral element procedure for simulating the neutral delay...

finite difference scheme and energy method | fractional damped diffusion‐wave equation | spectral element method | distributed‐order fractional equation | fractional delay PDE | stability and convergence analysis | distributed-order fractional equation | fractional damped diffusion-wave equation | MATHEMATICS, APPLIED | 2ND-ORDER | STABILITY | TIME-DELAY | POLYNOMIALS | PREDATOR-PREY SYSTEM | SCHEME | NUMERICAL-SOLUTION | PARTIAL-DIFFERENTIAL-EQUATIONS | DYNAMICS | COLLOCATION | Finite element method | Computer simulation | Wave equations | Spectral element method | Spectra | Delay | Convergence

finite difference scheme and energy method | fractional damped diffusion‐wave equation | spectral element method | distributed‐order fractional equation | fractional delay PDE | stability and convergence analysis | distributed-order fractional equation | fractional damped diffusion-wave equation | MATHEMATICS, APPLIED | 2ND-ORDER | STABILITY | TIME-DELAY | POLYNOMIALS | PREDATOR-PREY SYSTEM | SCHEME | NUMERICAL-SOLUTION | PARTIAL-DIFFERENTIAL-EQUATIONS | DYNAMICS | COLLOCATION | Finite element method | Computer simulation | Wave equations | Spectral element method | Spectra | Delay | Convergence

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 09/2018, Volume 339, pp. 179 - 192

This paper is concerned with the fractional evolution equation with a discrete distribution of Caputo time-derivatives such that the largest and the smallest...

Propagation function | Cosine family | Bernstein function | Time-fractional diffusion–wave equation | Solution operator | Mathematics - Analysis of PDEs

Propagation function | Cosine family | Bernstein function | Time-fractional diffusion–wave equation | Solution operator | Mathematics - Analysis of PDEs

Journal Article

Applied Mathematics Letters, ISSN 0893-9659, 06/2017, Volume 68, pp. 87 - 93

This paper focuses on providing the high order algorithms for the space–time tempered fractional diffusion-wave equation. The designed schemes are...

Space–time tempered fractional diffusion-wave equation | Integro-differential equation | Numerical stability and convergence | MATHEMATICS, APPLIED | diffusion-wave equation | Space-time tempered fractional | Analysis | Methods | Algorithms

Space–time tempered fractional diffusion-wave equation | Integro-differential equation | Numerical stability and convergence | MATHEMATICS, APPLIED | diffusion-wave equation | Space-time tempered fractional | Analysis | Methods | Algorithms

Journal Article

Applied Numerical Mathematics, ISSN 0168-9274, 01/2019, Volume 135, pp. 30 - 46

In this paper, we focus our attention on the development of the high-order numerical algorithm for the time–space tempered fractional diffusion-wave equation...

Fractional–compact difference operator | Riesz derivative | Time–space tempered fractional diffusion-wave equation | Tempered fractional integral | Fractional-compact difference operator | SCHEME | MATHEMATICS, APPLIED | FINITE-DIFFERENCE METHOD | Time-space tempered fractional diffusion-wave equation | DERIVATIVES | Algorithms

Fractional–compact difference operator | Riesz derivative | Time–space tempered fractional diffusion-wave equation | Tempered fractional integral | Fractional-compact difference operator | SCHEME | MATHEMATICS, APPLIED | FINITE-DIFFERENCE METHOD | Time-space tempered fractional diffusion-wave equation | DERIVATIVES | Algorithms

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 10/2015, Volume 298, pp. 652 - 660

In this paper, we derive and analyse a compact difference scheme for a distributed-order time-fractional diffusion-wave equation. This equation is approximated...

Compact difference scheme | Stability | Distributed-order fractional derivative | Diffusion-wave equation | Convergence | NUMERICAL-METHODS | VARIABLE-ORDER | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | PHYSICS, MATHEMATICAL

Compact difference scheme | Stability | Distributed-order fractional derivative | Diffusion-wave equation | Convergence | NUMERICAL-METHODS | VARIABLE-ORDER | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | PHYSICS, MATHEMATICAL

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 07/2019, Volume 352, pp. 235 - 248

In this paper, we generalize a one-dimensional fractional diffusion-wave equation to a one-dimensional variable-order space-time fractional nonlinear...

Tau-collocation method | Operational matrix of variable-order fractional derivative (OMV-OFD) | Chebyshev cardinal functions | Variable-order space-time fractional nonlinear diffusion-wave equation (V-OS-TFND-WE) | Variable-order space-time fractional nonlinear diffusion-wave equation (V-OS-TEND-WE) | MATHEMATICS, APPLIED | DIFFERENCE-METHODS | APPROXIMATION | STABILITY | OPTIMIZATION METHOD | SCHEME | NUMERICAL-SOLUTION | LEGENDRE WAVELETS | BOUNDED DOMAINS | CONVERGENCE

Tau-collocation method | Operational matrix of variable-order fractional derivative (OMV-OFD) | Chebyshev cardinal functions | Variable-order space-time fractional nonlinear diffusion-wave equation (V-OS-TFND-WE) | Variable-order space-time fractional nonlinear diffusion-wave equation (V-OS-TEND-WE) | MATHEMATICS, APPLIED | DIFFERENCE-METHODS | APPROXIMATION | STABILITY | OPTIMIZATION METHOD | SCHEME | NUMERICAL-SOLUTION | LEGENDRE WAVELETS | BOUNDED DOMAINS | CONVERGENCE

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 11/2017, Volume 74, Issue 10, pp. 2449 - 2465

The aim of this paper is to develop an efficient numerical treatment for the two-dimensional fractional nonlinear reaction–diffusion-wave equation with the...

Alternating direction implicit method | Stability and convergence | Nonlinear fractional differential equation | Legendre–Galerkin spectral method | FINITE DIFFERENCE/SPECTRAL APPROXIMATIONS | MATHEMATICS, APPLIED | ADVECTION-DISPERSION EQUATION | ELEMENT-METHOD | 2ND-ORDER | SINE-GORDON EQUATION | DIFFERENTIAL-EQUATIONS | SUBDIFFUSION EQUATION | NUMERICAL APPROXIMATION | SPACE | SOURCE-TERM | Legendre-Galerkin spectral method | Analysis | Methods | Differential equations

Alternating direction implicit method | Stability and convergence | Nonlinear fractional differential equation | Legendre–Galerkin spectral method | FINITE DIFFERENCE/SPECTRAL APPROXIMATIONS | MATHEMATICS, APPLIED | ADVECTION-DISPERSION EQUATION | ELEMENT-METHOD | 2ND-ORDER | SINE-GORDON EQUATION | DIFFERENTIAL-EQUATIONS | SUBDIFFUSION EQUATION | NUMERICAL APPROXIMATION | SPACE | SOURCE-TERM | Legendre-Galerkin spectral method | Analysis | Methods | Differential equations

Journal Article

Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, 06/2019, Volume 350, pp. 154 - 168

This paper is concerned with the moving least squares (MLS) meshless approach for the numerical solution of two-dimensional (2D) variable-order time fractional...

Nonlinear diffusion-wave equation | Moving least squares (MLS) | Time fractional | Variable-order | MESHLESS METHOD | 2-DIMENSIONAL LEGENDRE WAVELETS | STABILITY | PLATES | INTERPOLATION | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | MOVING LEAST-SQUARES | PARTIAL-DIFFERENTIAL-EQUATIONS | FINITE-DIFFERENCE | GORDON | Mechanical engineering | Differential equations | Finite element method | Domains | Approximation | Partial differential equations | Mathematical analysis | Meshless methods | Wave equations | Finite difference method

Nonlinear diffusion-wave equation | Moving least squares (MLS) | Time fractional | Variable-order | MESHLESS METHOD | 2-DIMENSIONAL LEGENDRE WAVELETS | STABILITY | PLATES | INTERPOLATION | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | MOVING LEAST-SQUARES | PARTIAL-DIFFERENTIAL-EQUATIONS | FINITE-DIFFERENCE | GORDON | Mechanical engineering | Differential equations | Finite element method | Domains | Approximation | Partial differential equations | Mathematical analysis | Meshless methods | Wave equations | Finite difference method

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

SIAM Journal on Scientific Computing, ISSN 1064-8275, 2016, Volume 38, Issue 1, pp. A146 - A170

We consider initial/boundary value problems for the subdiffusion and diffusion-ave equations involving a Caputo fractional derivative in time. We develop two...

Convolution quadrature | Finite element method | Fractional diffusion | Error estimate | Diffusion wave | diffusion wave | MATHEMATICS, APPLIED | APPROXIMATIONS | STABILITY | GALERKIN METHOD | convolution quadrature | error estimate | fractional diffusion | finite element method | ORDER PARABOLIC EQUATIONS | INTEGRODIFFERENTIAL EQUATION | DIFFERENCE METHOD | EVOLUTION EQUATION | ERROR ANALYSIS | FINITE-ELEMENT-METHOD

Convolution quadrature | Finite element method | Fractional diffusion | Error estimate | Diffusion wave | diffusion wave | MATHEMATICS, APPLIED | APPROXIMATIONS | STABILITY | GALERKIN METHOD | convolution quadrature | error estimate | fractional diffusion | finite element method | ORDER PARABOLIC EQUATIONS | INTEGRODIFFERENTIAL EQUATION | DIFFERENCE METHOD | EVOLUTION EQUATION | ERROR ANALYSIS | FINITE-ELEMENT-METHOD

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 12/2015, Volume 290, pp. 174 - 195

In this paper we apply a high order difference scheme and Galerkin spectral technique for the numerical solution of multi-term time fractional partial...

Convergence and stability | Energy method | Multi-term time fractional diffusion-wave equations | Solvability | High order compact finite difference | Galerkin spectral method | equations | Multi-term time fractional diffusion-wave | MATHEMATICS, APPLIED | APPROXIMATIONS | STABILITY | DISCONTINUOUS GALERKIN METHOD | SPACE | SCHEME | SUBDIFFUSION | FLUID | PARTIAL-DIFFERENTIAL-EQUATIONS | CONVERGENCE | COLLOCATION | Analysis | Algorithms | Numerical analysis | Mathematical analysis | Mathematical models | Derivatives | Estimates | Galerkin methods | Spectral methods | Finite difference method

Convergence and stability | Energy method | Multi-term time fractional diffusion-wave equations | Solvability | High order compact finite difference | Galerkin spectral method | equations | Multi-term time fractional diffusion-wave | MATHEMATICS, APPLIED | APPROXIMATIONS | STABILITY | DISCONTINUOUS GALERKIN METHOD | SPACE | SCHEME | SUBDIFFUSION | FLUID | PARTIAL-DIFFERENTIAL-EQUATIONS | CONVERGENCE | COLLOCATION | Analysis | Algorithms | Numerical analysis | Mathematical analysis | Mathematical models | Derivatives | Estimates | Galerkin methods | Spectral methods | Finite difference method

Journal Article

Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, 01/2019, Volume 52, Issue 1, p. 15201

We study generalized diffusion-wave equation in which the second order time derivative is replaced by an integro-differential operator. It yields time...

Mittag-Leffler function | diffusion-wave equation | anomalous diffusion | PHYSICS, MULTIDISCIPLINARY | PHYSICS, MATHEMATICAL | SINGLE MOLECULES | PRABHAKAR | TRANSPORT | MODELS | RANDOM-WALKS | DYNAMICS | FRACTIONAL DIFFUSION | FUNDAMENTAL SOLUTION | Physics - Statistical Mechanics

Mittag-Leffler function | diffusion-wave equation | anomalous diffusion | PHYSICS, MULTIDISCIPLINARY | PHYSICS, MATHEMATICAL | SINGLE MOLECULES | PRABHAKAR | TRANSPORT | MODELS | RANDOM-WALKS | DYNAMICS | FRACTIONAL DIFFUSION | FUNDAMENTAL SOLUTION | Physics - Statistical Mechanics

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 2011, Volume 61, Issue 8, pp. 2227 - 2231

The homotopy perturbation method is applied to the generalized fourth-order fractional diffusion–wave equations. The problem is formulated in the Caputo sense....

Caputo fractional derivative | He’s homotopy perturbation method | Mittag-Leffler function | Generalized fourth-order fractional diffusion–wave equations | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Generalized fourth-order fractional diffusion-wave equations | He's homotopy perturbation method | HOMOTOPY PERTURBATION METHOD

Caputo fractional derivative | He’s homotopy perturbation method | Mittag-Leffler function | Generalized fourth-order fractional diffusion–wave equations | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Generalized fourth-order fractional diffusion-wave equations | He's homotopy perturbation method | HOMOTOPY PERTURBATION METHOD

Journal Article

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, ISSN 1364-5021, 06/2009, Volume 465, Issue 2106, pp. 1869 - 1891

A single-order time-fractional diffusion-wave equation is generalized by introducing a time distributed-order fractional derivative and forcing term, while a...

Differential operators | Mathematical theorems | Volterra equations | Differential equations | Scalars | Laplace transformation | Laplacians | Banach space | Cauchy problem | Distributed-order fractional derivative | Diffusion-wave equation | Volterra equation | Fractional derivative | MULTIDISCIPLINARY SCIENCES | fractional derivative | distributed-order fractional derivative | diffusion-wave equation | FRACTIONAL DIFFUSION | MULTIDIMENSIONAL SOLUTIONS

Differential operators | Mathematical theorems | Volterra equations | Differential equations | Scalars | Laplace transformation | Laplacians | Banach space | Cauchy problem | Distributed-order fractional derivative | Diffusion-wave equation | Volterra equation | Fractional derivative | MULTIDISCIPLINARY SCIENCES | fractional derivative | distributed-order fractional derivative | diffusion-wave equation | FRACTIONAL DIFFUSION | MULTIDIMENSIONAL SOLUTIONS

Journal Article

Physics Letters A, ISSN 0375-9601, 01/2015, Volume 379, Issue 3, pp. 71 - 76

In this paper, an efficient and accurate computational method based on the Legendre wavelets (LWs) is proposed for solving the time fractional diffusion-wave...

Riemann–Liouville integral | Hat functions (HFs) | Fractional operational matrix (FOM) | Legendre wavelets (LWs) | Caputo derivative | Fractional diffusion-wave equation (FDWE) | Caputo derivative Riemann-Liouville integral | 2-DIMENSIONAL LEGENDRE WAVELETS | PHYSICS, MULTIDISC

Riemann–Liouville integral | Hat functions (HFs) | Fractional operational matrix (FOM) | Legendre wavelets (LWs) | Caputo derivative | Fractional diffusion-wave equation (FDWE) | Caputo derivative Riemann-Liouville integral | 2-DIMENSIONAL LEGENDRE WAVELETS | PHYSICS, MULTIDISC