2018, 1, ISBN 9781315167183, Volume 1, xxxvii, 273 pages

The main focus of the book is to implement wavelet based transform methods for solving problems of fractional order partial differential equations arising in...

Wavelets (Mathematics) | Fractional differential equations | Differential equations, Partial | Numerical solutions | Differential equations | Differential Equations | Applied Mathematics | Mathematical Physics | Computational Numerical Analysis

Wavelets (Mathematics) | Fractional differential equations | Differential equations, Partial | Numerical solutions | Differential equations | Differential Equations | Applied Mathematics | Mathematical Physics | Computational Numerical Analysis

Book

Optical and Quantum Electronics, ISSN 0306-8919, 3/2018, Volume 50, Issue 3, pp. 1 - 20

In this study, some new traveling wave solutions for fractional partial differential equations (PDEs) have been developed. The time-fractional Burgers...

Burgers equations | Fractional modified Reimann-Liouville derivative | ( $$\frac{G^{'}}{G^{2}}$$ G ′ G 2 )-Expansion method | Whitham Broer Kuap equations | Optics, Lasers, Photonics, Optical Devices | Characterization and Evaluation of Materials | Computer Communication Networks | Physics | Traveling wave solution | Electrical Engineering | Biological population model | (G′G2)-Expansion method | QUANTUM SCIENCE & TECHNOLOGY | 1ST INTEGRAL METHOD | EXP-FUNCTION METHOD | FUNCTIONAL VARIABLE METHOD | OPTICS | (G '/G)-Expansion method | ENGINEERING, ELECTRICAL & ELECTRONIC | Water waves | Mortality | Methods | Differential equations

Burgers equations | Fractional modified Reimann-Liouville derivative | ( $$\frac{G^{'}}{G^{2}}$$ G ′ G 2 )-Expansion method | Whitham Broer Kuap equations | Optics, Lasers, Photonics, Optical Devices | Characterization and Evaluation of Materials | Computer Communication Networks | Physics | Traveling wave solution | Electrical Engineering | Biological population model | (G′G2)-Expansion method | QUANTUM SCIENCE & TECHNOLOGY | 1ST INTEGRAL METHOD | EXP-FUNCTION METHOD | FUNCTIONAL VARIABLE METHOD | OPTICS | (G '/G)-Expansion method | ENGINEERING, ELECTRICAL & ELECTRONIC | Water waves | Mortality | Methods | Differential equations

Journal Article

Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 07/2017, Volume 48, pp. 509 - 519

In this paper, we pursue the general form of the fractional reduced differential transform method (DTM) to (N+1)-dimensional case, so that fractional order...

Fractional couple Burgers equation | (N+1)-dimensional fractional reduced DTM | Fractional Zakharov–Kuznetsov equation | Fractional wave like problem | Fractional calculus | APPROXIMATE SOLUTIONS | MATHEMATICS, APPLIED | PHYSICS, FLUIDS & PLASMAS | HOMOTOPY PERTURBATION METHOD | PHYSICS, MATHEMATICAL | HEAT-LIKE | VARIATIONAL ITERATION METHOD | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | WAVE-LIKE EQUATIONS | Fractional Zakharov-Kuznetsov equation | SYSTEMS | Methods | Differential equations

Fractional couple Burgers equation | (N+1)-dimensional fractional reduced DTM | Fractional Zakharov–Kuznetsov equation | Fractional wave like problem | Fractional calculus | APPROXIMATE SOLUTIONS | MATHEMATICS, APPLIED | PHYSICS, FLUIDS & PLASMAS | HOMOTOPY PERTURBATION METHOD | PHYSICS, MATHEMATICAL | HEAT-LIKE | VARIATIONAL ITERATION METHOD | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | WAVE-LIKE EQUATIONS | Fractional Zakharov-Kuznetsov equation | SYSTEMS | Methods | Differential equations

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 02/2016, Volume 307, pp. 243 - 261

Efficient spectral–Galerkin algorithms are developed to solve multi-dimensional fractional elliptic equations with variable coefficients in conserved form as...

Fractional PDE | Error estimates | Spectral–Galerkin method | Preconditioned iterative method | Spectral-Galerkin method | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | 2ND-ORDER | APPROXIMATION | DIFFUSION | TIME | PHYSICS, MATHEMATICAL | FINITE-ELEMENT-METHOD | Algorithms | Analysis | Methods | Differential equations | Errors | Partial differential equations | Mathematical analysis | Mathematical models | Spectra | Boundaries | Estimates

Fractional PDE | Error estimates | Spectral–Galerkin method | Preconditioned iterative method | Spectral-Galerkin method | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | 2ND-ORDER | APPROXIMATION | DIFFUSION | TIME | PHYSICS, MATHEMATICAL | FINITE-ELEMENT-METHOD | Algorithms | Analysis | Methods | Differential equations | Errors | Partial differential equations | Mathematical analysis | Mathematical models | Spectra | Boundaries | Estimates

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 03/2018, Volume 357, pp. 125 - 141

While there is currently a lot of enthusiasm about “big data”, useful data is usually “small” and expensive to acquire. In this paper, we present a new...

Small data | Uncertainty quantification | Fractional equations | System identification | Bayesian modeling | Probabilistic machine learning | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | SYSTEMS | SPECTRAL PROPERTIES | IDENTIFICATION | PHYSICS, MATHEMATICAL | IMMERSED BOUNDARY METHOD | Magneto-electric machines | Analysis | Differential equations | Machine learning | Big data | Information management | Machinery

Small data | Uncertainty quantification | Fractional equations | System identification | Bayesian modeling | Probabilistic machine learning | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | SYSTEMS | SPECTRAL PROPERTIES | IDENTIFICATION | PHYSICS, MATHEMATICAL | IMMERSED BOUNDARY METHOD | Magneto-electric machines | Analysis | Differential equations | Machine learning | Big data | Information management | Machinery

Journal Article

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 | SYSTEM | CALCULUS | PHYSICS, MATHEMATICAL | LOBATTO COLLOCATION METHOD | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | VOLTERRA INTEGRODIFFERENTIAL EQUATIONS | DIFFUSION | 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 | SYSTEM | CALCULUS | PHYSICS, MATHEMATICAL | LOBATTO COLLOCATION METHOD | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | VOLTERRA INTEGRODIFFERENTIAL EQUATIONS | DIFFUSION | Algorithms | Analysis | Methods | Differential equations | Approximation | Discretization | Mathematical analysis | Dirichlet problem | Spectra | Temporal logic

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 11/2018, Volume 336, pp. 433 - 453

In this paper, we consider a new fractional function based on Legendre and Laguerre polynomials for solving a class of linear and nonlinear time-space...

Pseudo-operational matrix of integration | Operational matrix of integration | Fractional partial differential equation | Fractional-order Legendre–Laguerre functions | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | COLLOCATION METHOD | INTEGRATION | CALCULUS | Fractional-order Legendre-Laguerre functions | WAVELET OPERATIONAL MATRIX | Differential equations

Pseudo-operational matrix of integration | Operational matrix of integration | Fractional partial differential equation | Fractional-order Legendre–Laguerre functions | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | COLLOCATION METHOD | INTEGRATION | CALCULUS | Fractional-order Legendre-Laguerre functions | WAVELET OPERATIONAL MATRIX | Differential equations

Journal Article

Nonlinear Dynamics, ISSN 0924-090X, 07/2017, Volume 89, Issue 1, pp. 321 - 331

The symmetry method is developed to study space-time fractional nonlinear partial differential equations. Also, the Noether operators are extended for...

Conservation laws | Erdélyi–Kober operators | Nonlinear self-adjointness | Symmetry analysis | Space–time fractional partial differential equations | INVARIANCE | CALCULUS | Erdelyi-Kober operators | SELF-ADJOINTNESS | ENGINEERING, MECHANICAL | INTEGRALS | ORDER | MECHANICS | Space-time fractional partial differential equations | MODELS | COMPACTONS | DYNAMICS | Laws, regulations and rules | Environmental law | Analysis | Differential equations | Nonlinear equations | Partial differential equations | Mathematical analysis | Nonlinear differential equations | Symmetry

Conservation laws | Erdélyi–Kober operators | Nonlinear self-adjointness | Symmetry analysis | Space–time fractional partial differential equations | INVARIANCE | CALCULUS | Erdelyi-Kober operators | SELF-ADJOINTNESS | ENGINEERING, MECHANICAL | INTEGRALS | ORDER | MECHANICS | Space-time fractional partial differential equations | MODELS | COMPACTONS | DYNAMICS | Laws, regulations and rules | Environmental law | Analysis | Differential equations | Nonlinear equations | Partial differential equations | Mathematical analysis | Nonlinear differential equations | Symmetry

Journal Article

Physics Letters A, ISSN 0375-9601, 06/2012, Volume 376, Issue 28-29, pp. 2045 - 2048

In this Letter, the fractional derivatives in the sense of modified Riemann–Liouville derivative and the Bäcklund transformation of fractional Riccati equation...

Fractional differential equation | Bäcklund transformation | Fractional Riccati equation | Exact solution | Backlund transformation | PHYSICS, MULTIDISCIPLINARY | Differential equations | Partial differential equations | Mathematical analysis | Solid state physics | Nonlinearity | Transformations | Derivatives | Riccati equation

Fractional differential equation | Bäcklund transformation | Fractional Riccati equation | Exact solution | Backlund transformation | PHYSICS, MULTIDISCIPLINARY | Differential equations | Partial differential equations | Mathematical analysis | Solid state physics | Nonlinearity | Transformations | Derivatives | Riccati equation

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 07/2016, Volume 39, Issue 10, pp. 2461 - 2476

In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear...

time‐fractional partial differential equations (TFPDEs) | time‐fractional sine‐Gordon equations | boundary elements method (BEM) | time‐fractional telegraph equation | radial basis functions (RBFs) | boundary integral equation method | time‐fractional Fokker–Planck equations | time‐fractional Klein–Gordon equations | dual reciprocity boundary elements method (DRBEM) | time-fractional telegraph equation | time-fractional partial differential equations (TFPDEs) | time-fractional sine-Gordon equations | time-fractional Fokker–Planck equations | time-fractional Klein–Gordon equations | CONVECTION FLOW | MATHEMATICS, APPLIED | ELEMENT-METHOD | TELEGRAPH EQUATION | time-fractional Fokker-Planck equations | SUB-DIFFUSION EQUATION | HIGH-ORDER | SPACE | SCHEME | NUMERICAL-SOLUTION | DRBEM SOLUTION | time-fractional Klein-Gordon equations | SINE-GORDON | Methods | Differential equations | Partial differential equations | Reciprocity | Integral equations | Mathematical analysis | Nonlinearity | Mathematical models | Derivatives | Boundaries

time‐fractional partial differential equations (TFPDEs) | time‐fractional sine‐Gordon equations | boundary elements method (BEM) | time‐fractional telegraph equation | radial basis functions (RBFs) | boundary integral equation method | time‐fractional Fokker–Planck equations | time‐fractional Klein–Gordon equations | dual reciprocity boundary elements method (DRBEM) | time-fractional telegraph equation | time-fractional partial differential equations (TFPDEs) | time-fractional sine-Gordon equations | time-fractional Fokker–Planck equations | time-fractional Klein–Gordon equations | CONVECTION FLOW | MATHEMATICS, APPLIED | ELEMENT-METHOD | TELEGRAPH EQUATION | time-fractional Fokker-Planck equations | SUB-DIFFUSION EQUATION | HIGH-ORDER | SPACE | SCHEME | NUMERICAL-SOLUTION | DRBEM SOLUTION | time-fractional Klein-Gordon equations | SINE-GORDON | Methods | Differential equations | Partial differential equations | Reciprocity | Integral equations | Mathematical analysis | Nonlinearity | Mathematical models | Derivatives | Boundaries

Journal Article

International Journal of Computer Mathematics, ISSN 0020-7160, 07/2018, Volume 95, Issue 6-7, pp. 1048 - 1099

In this review paper, we are mainly concerned with the finite difference methods, the Galerkin finite element methods, and the spectral methods for fractional...

Galerkin finite element methods | spectral methods | fast algorithms | 65M06 | 26A33 | 35R11 | fractional partial differential equations | 65M60 | Finite difference methods | MATHEMATICS, APPLIED | DIFFUSION-WAVE EQUATION | BOUNDARY-VALUE-PROBLEMS | SPECTRAL COLLOCATION METHOD | HIGH-ORDER APPROXIMATION | DIFFERENCE/SPECTRAL APPROXIMATIONS | DISTRIBUTED-ORDER | COMPACT ADI SCHEME | DISCONTINUOUS GALERKIN METHODS | FINITE-ELEMENT-METHOD | NONLOCAL DIFFUSION | Finite element method | Algorithms | Partial differential equations | Mathematical analysis | Numerical methods | Galerkin method | Spectral methods | Finite difference method

Galerkin finite element methods | spectral methods | fast algorithms | 65M06 | 26A33 | 35R11 | fractional partial differential equations | 65M60 | Finite difference methods | MATHEMATICS, APPLIED | DIFFUSION-WAVE EQUATION | BOUNDARY-VALUE-PROBLEMS | SPECTRAL COLLOCATION METHOD | HIGH-ORDER APPROXIMATION | DIFFERENCE/SPECTRAL APPROXIMATIONS | DISTRIBUTED-ORDER | COMPACT ADI SCHEME | DISCONTINUOUS GALERKIN METHODS | FINITE-ELEMENT-METHOD | NONLOCAL DIFFUSION | Finite element method | Algorithms | Partial differential equations | Mathematical analysis | Numerical methods | Galerkin method | Spectral methods | Finite difference method

Journal Article

Stochastic Processes and their Applications, ISSN 0304-4149, 09/2015, Volume 125, Issue 9, pp. 3301 - 3326

We consider non-linear time-fractional stochastic heat type equation ∂tβut(x)=−ν(−Δ)α/2ut(x)+It1−β[σ(u)W⋅(t,x)] in (d+1) dimensions, where ν>0,β∈(0,1), α∈(0,2]...

Time-fractional stochastic partial differential equations | Fractional Duhamel’s principle | Caputo derivatives | Walsh isometry | Fractional Duhamel's principle | STATISTICS & PROBABILITY | DUHAMELS PRINCIPLE | Analysis | Differential equations

Time-fractional stochastic partial differential equations | Fractional Duhamel’s principle | Caputo derivatives | Walsh isometry | Fractional Duhamel's principle | STATISTICS & PROBABILITY | DUHAMELS PRINCIPLE | Analysis | Differential equations

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 01/2018, Volume 264, Issue 2, pp. 1377 - 1410

We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions....

Reflected stable processes | Stable processes | Fractional differential equations | Nonlocal operators | Feller processes | MATHEMATICS | APPROXIMATION | Markov processes | Algorithms | Analysis | Differential equations | Matematisk analys | Computational Mathematics | Beräkningsmatematik | Sannolikhetsteori och statistik | Mathematical Analysis | Probability Theory and Statistics

Reflected stable processes | Stable processes | Fractional differential equations | Nonlocal operators | Feller processes | MATHEMATICS | APPROXIMATION | Markov processes | Algorithms | Analysis | Differential equations | Matematisk analys | Computational Mathematics | Beräkningsmatematik | Sannolikhetsteori och statistik | Mathematical Analysis | Probability Theory and Statistics

Journal Article

Nonlinear Dynamics, ISSN 0924-090X, 01/2019, Volume 95, Issue 1, pp. 361 - 368

The nonlinear space-time fractional differential equations (FDE) of Burgers' type play an important role for describing many phenomena in applied sciences....

Fractional partial differential equations | Kudryashov method | Burgers | CLASSIFICATION | MECHANICS | ENGINEERING, MECHANICAL | Methods | Differential equations | Ordinary differential equations | Traveling waves | Partial differential equations | Mathematical analysis

Fractional partial differential equations | Kudryashov method | Burgers | CLASSIFICATION | MECHANICS | ENGINEERING, MECHANICAL | Methods | Differential equations | Ordinary differential equations | Traveling waves | Partial differential equations | Mathematical analysis

Journal Article