2015, ISBN 1482254301, xxv, 458

The theory of linear Volterra integro-differential equations has been developing rapidly in the last three decades. This book provides an easy to read concise...

Volterra equations | Integro-differential equations | Mathematics

Volterra equations | Integro-differential equations | Mathematics

Book

Neural Computing and Applications, ISSN 0941-0643, 7/2017, Volume 28, Issue 7, pp. 1591 - 1610

In this article, we propose the reproducing kernel Hilbert space method to obtain the exact and the numerical solutions of fuzzy Fredholm–Volterra...

Data Mining and Knowledge Discovery | Reproducing kernel Hilbert space method | 47B32 | Fuzzy integrodifferential equations | Computational Science and Engineering | Strongly generalized differentiability | Computational Biology/Bioinformatics | Gram–Schmidt process | Computer Science | Image Processing and Computer Vision | 34K28 | Artificial Intelligence (incl. Robotics) | 46S40 | Probability and Statistics in Computer Science | EXISTENCE | SYSTEM | Gram-Schmidt process | INCLUSIONS | BOUNDARY-VALUE-PROBLEMS | DIFFERENTIAL-EQUATIONS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | UNIQUENESS | NUMERICAL-SOLUTION | TURNING-POINT PROBLEMS | GLOBAL-SOLUTIONS | Annealing | Algorithms

Data Mining and Knowledge Discovery | Reproducing kernel Hilbert space method | 47B32 | Fuzzy integrodifferential equations | Computational Science and Engineering | Strongly generalized differentiability | Computational Biology/Bioinformatics | Gram–Schmidt process | Computer Science | Image Processing and Computer Vision | 34K28 | Artificial Intelligence (incl. Robotics) | 46S40 | Probability and Statistics in Computer Science | EXISTENCE | SYSTEM | Gram-Schmidt process | INCLUSIONS | BOUNDARY-VALUE-PROBLEMS | DIFFERENTIAL-EQUATIONS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | UNIQUENESS | NUMERICAL-SOLUTION | TURNING-POINT PROBLEMS | GLOBAL-SOLUTIONS | Annealing | Algorithms

Journal Article

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

1977, Lecture notes in biomathematics, ISBN 0387084495, Volume 20, 196

Book

Publicacions Matemàtiques, ISSN 0214-1493, 1/2014, Volume 58, Issue 1, pp. 133 - 154

Aim of this paper is to show that weak solutions of the following fractional Laplacian equation $\left\{ {_{u = g}^{{{\left( { - \Delta } \right)}^s}u =...

Regularity theory | Weak solutions | Fractional Laplacian | Integrodifferential operators | Viscosity solutions | MATHEMATICS | regularity theory | fractional Laplacian | weak solutions | INTEGRODIFFERENTIAL EQUATIONS | viscosity solutions | OPERATORS | 35R09 | 49N60 | 35D30 | 45K05

Regularity theory | Weak solutions | Fractional Laplacian | Integrodifferential operators | Viscosity solutions | MATHEMATICS | regularity theory | fractional Laplacian | weak solutions | INTEGRODIFFERENTIAL EQUATIONS | viscosity solutions | OPERATORS | 35R09 | 49N60 | 35D30 | 45K05

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 2018, Volume 56, Issue 1, pp. 1 - 23

We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional...

Convolution quadrature | Discrete fractional grönwall inequality | L1 scheme | Error estimate | Nonlinear fractional diffusion equation | DIFFUSION-WAVE EQUATIONS | MATHEMATICS, APPLIED | discrete fractional Gronwall inequality | TIME DISCRETIZATIONS | DIFFERENCE SCHEME | convolution quadrature | nonlinear fractional diffusion equation | error estimate | MAXIMAL REGULARITY | INTEGRODIFFERENTIAL EQUATION | DISCRETE | FRACTIONAL DIFFUSION

Convolution quadrature | Discrete fractional grönwall inequality | L1 scheme | Error estimate | Nonlinear fractional diffusion equation | DIFFUSION-WAVE EQUATIONS | MATHEMATICS, APPLIED | discrete fractional Gronwall inequality | TIME DISCRETIZATIONS | DIFFERENCE SCHEME | convolution quadrature | nonlinear fractional diffusion equation | error estimate | MAXIMAL REGULARITY | INTEGRODIFFERENTIAL EQUATION | DISCRETE | FRACTIONAL DIFFUSION

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 06/2019, Volume 267, Issue 1, pp. 547 - 586

We prove some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator...

Double phase functionals | Quasilinear nonlocal operators | Hölder continuity | Fractional Sobolev spaces | Viscosity solutions | MATHEMATICS | Holder continuity | INTEGRODIFFERENTIAL EQUATIONS | ELLIPTIC-EQUATIONS | MINIMIZERS

Double phase functionals | Quasilinear nonlocal operators | Hölder continuity | Fractional Sobolev spaces | Viscosity solutions | MATHEMATICS | Holder continuity | INTEGRODIFFERENTIAL EQUATIONS | ELLIPTIC-EQUATIONS | MINIMIZERS

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 04/2013, Volume 81, pp. 70 - 86

The fractional stochastic differential equations have wide applications in various fields of science and engineering. This paper addresses the issue of...

Existence result | Fractional stochastic differential equation | Resolvent operators | MATHEMATICS | MATHEMATICS, APPLIED | INTEGRODIFFERENTIAL EQUATIONS | EVOLUTION INCLUSIONS | MILD SOLUTIONS | Differential equations

Existence result | Fractional stochastic differential equation | Resolvent operators | MATHEMATICS | MATHEMATICS, APPLIED | INTEGRODIFFERENTIAL EQUATIONS | EVOLUTION INCLUSIONS | MILD SOLUTIONS | Differential equations

Journal Article

Transactions of the American Mathematical Society, ISSN 0002-9947, 05/2019, Volume 371, Issue 5, pp. 3417 - 3450

Let X=(X_t)_{t \ge 0} be a stochastic process which has a (not necessarily stationary) independent increment on a probability space (\Omega , \mathbb{P}). In...

Diffusion equation for jump process | Pseudo-differential operator | theory | Non-stationary increment | MATHEMATICS | pseudo-differential operator | INTEGRODIFFERENTIAL EQUATIONS | non-stationary increment | OPERATORS | L-p-theory

Diffusion equation for jump process | Pseudo-differential operator | theory | Non-stationary increment | MATHEMATICS | pseudo-differential operator | INTEGRODIFFERENTIAL EQUATIONS | non-stationary increment | OPERATORS | L-p-theory

Journal Article

Archive for Rational Mechanics and Analysis, ISSN 0003-9527, 4/2011, Volume 200, Issue 1, pp. 59 - 88

We obtain C 1,α regularity estimates for nonlocal elliptic equations that are not necessarily translation-invariant using compactness and perturbative methods...

Mechanics | Physics, general | Fluid- and Aerodynamics | Statistical Physics, Dynamical Systems and Complexity | Theoretical, Mathematical and Computational Physics | Physics | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | INTEGRODIFFERENTIAL EQUATIONS

Mechanics | Physics, general | Fluid- and Aerodynamics | Statistical Physics, Dynamical Systems and Complexity | Theoretical, Mathematical and Computational Physics | Physics | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | INTEGRODIFFERENTIAL EQUATIONS

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 08/2014, Volume 270, pp. 402 - 415

In this paper, a new computational method based on the generalized hat basis functions is proposed for solving stochastic Itô–Volterra integral equations. In...

Brownian motion process | Stochastic Itô–Volterra integral equations | Generalized hat basis functions | Itô integral | Stochastic operational matrix | ItÔ integral | Stochastic ItÔ-Volterra integral equations | Ito integral | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | RANDOM DIFFERENTIAL-EQUATIONS | INTEGRODIFFERENTIAL EQUATIONS | PHYSICS, MATHEMATICAL | Stochastic Ito Volterra integral equations | Basis functions | Computation | Integral equations | Mathematical analysis | Blocking | Texts | Mathematical models | Stochasticity | INTEGRALS | STOCHASTIC PROCESSES | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | MATRICES | VOLTERRA INTEGRAL EQUATIONS | CONVERGENCE | BROWNIAN MOVEMENT | RELIABILITY | COMPARATIVE EVALUATIONS | ACCURACY | PULSES

Brownian motion process | Stochastic Itô–Volterra integral equations | Generalized hat basis functions | Itô integral | Stochastic operational matrix | ItÔ integral | Stochastic ItÔ-Volterra integral equations | Ito integral | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | RANDOM DIFFERENTIAL-EQUATIONS | INTEGRODIFFERENTIAL EQUATIONS | PHYSICS, MATHEMATICAL | Stochastic Ito Volterra integral equations | Basis functions | Computation | Integral equations | Mathematical analysis | Blocking | Texts | Mathematical models | Stochasticity | INTEGRALS | STOCHASTIC PROCESSES | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | MATRICES | VOLTERRA INTEGRAL EQUATIONS | CONVERGENCE | BROWNIAN MOVEMENT | RELIABILITY | COMPARATIVE EVALUATIONS | ACCURACY | PULSES

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 2011, Volume 382, Issue 1, pp. 426 - 447

We consider initial value/boundary value problems for fractional diffusion-wave equation: ∂ t α u ( x , t ) = L u ( x , t ) , where 0 < α ⩽ 2 , where L is a...

Fractional diffusion equation | Initial value/boundary value problem | Well-posedness | Inverse problem | MATHEMATICS | MATHEMATICS, APPLIED | INTEGRODIFFERENTIAL EQUATION | HEAT-EQUATION | Universities and colleges

Fractional diffusion equation | Initial value/boundary value problem | Well-posedness | Inverse problem | MATHEMATICS | MATHEMATICS, APPLIED | INTEGRODIFFERENTIAL EQUATION | HEAT-EQUATION | Universities and colleges

Journal Article

IEEE Transactions on Automatic Control, ISSN 0018-9286, 05/2018, Volume 63, Issue 5, pp. 1517 - 1522

In this paper, we study the quadratic regulator problem for a process governed by a Volterra integrodifferential equation in ℝ n . Our main goal is the proof...

Regulators | Integrodifferential equations | Riccati equations | Integral equations | Optimal control | Process control | Differential equations | Riccati equation | optimal control | STRUCTURAL OPERATOR-F | RETARDED-SYSTEMS | BOUNDARY CONTROL-SYSTEMS | THEOREM | AUTOMATION & CONTROL SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC

Regulators | Integrodifferential equations | Riccati equations | Integral equations | Optimal control | Process control | Differential equations | Riccati equation | optimal control | STRUCTURAL OPERATOR-F | RETARDED-SYSTEMS | BOUNDARY CONTROL-SYSTEMS | THEOREM | AUTOMATION & CONTROL SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 11/2018, Volume 337, pp. 452 - 460

In this work we prove that a family of explicit numerical methods is convergent when applied to a nonlinear Volterra equation with a power-type nonlinearity....

Nonlinearity | Numerical method | Power-type | Volterra equation | GRONWALL INEQUALITY | EXISTENCE | MATHEMATICS, APPLIED | APPROXIMATION | NONTRIVIAL SOLUTIONS | INTEGRAL-EQUATIONS | INTEGRODIFFERENTIAL EQUATIONS | UNIQUENESS | OPERATOR

Nonlinearity | Numerical method | Power-type | Volterra equation | GRONWALL INEQUALITY | EXISTENCE | MATHEMATICS, APPLIED | APPROXIMATION | NONTRIVIAL SOLUTIONS | INTEGRAL-EQUATIONS | INTEGRODIFFERENTIAL EQUATIONS | UNIQUENESS | OPERATOR

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 09/2015, Volume 39, Issue 17, pp. 5121 - 5130

•A method based on Chebyshev wavelet expansion has been proposed.•The proposed method is well suited for fractional Sawada–Kotera equation.•The results of this...

Caputo derivative | Fractional Sawada–Kotera equation | Chebyshev wavelet method | Homotopy analysis method | Fractional sawada-kotera equation | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | INTEGRODIFFERENTIAL EQUATIONS | TIME | Fractional Sawada-Kotera equation

Caputo derivative | Fractional Sawada–Kotera equation | Chebyshev wavelet method | Homotopy analysis method | Fractional sawada-kotera equation | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | INTEGRODIFFERENTIAL EQUATIONS | TIME | Fractional Sawada-Kotera equation

Journal Article

Soft Computing, ISSN 1432-7643, 8/2016, Volume 20, Issue 8, pp. 3283 - 3302

Modeling of uncertainty differential equations is very important issue in applied sciences and engineering, while the natural way to model such dynamical...

Engineering | Computational Intelligence | Gram–Schmidt process | Control, Robotics, Mechatronics | Reproducing kernel Hilbert space method | Artificial Intelligence (incl. Robotics) | Mathematical Logic and Foundations | Strongly generalized differentiability | Fuzzy differential equations | SYSTEM | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Gram-Schmidt process | TURNING-POINT PROBLEMS | BOUNDARY-VALUE-PROBLEMS | INTEGRODIFFERENTIAL EQUATIONS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | Methods | Differential equations | Resveratrol

Engineering | Computational Intelligence | Gram–Schmidt process | Control, Robotics, Mechatronics | Reproducing kernel Hilbert space method | Artificial Intelligence (incl. Robotics) | Mathematical Logic and Foundations | Strongly generalized differentiability | Fuzzy differential equations | SYSTEM | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Gram-Schmidt process | TURNING-POINT PROBLEMS | BOUNDARY-VALUE-PROBLEMS | INTEGRODIFFERENTIAL EQUATIONS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | Methods | Differential equations | Resveratrol

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

SIAM JOURNAL ON MATHEMATICAL ANALYSIS, ISSN 0036-1410, 2015, Volume 47, Issue 1, pp. 210 - 239

We prove sharp estimates for the decay in time of solutions to a rather general class of nonlocal in time subdiffusion equations on a bounded domain subject to...

MATHEMATICS, APPLIED | temporal decay estimates | p-Laplacian | NONLINEAR VOLTERRA-EQUATIONS | porous medium equation | DIFFERENTIAL-EQUATIONS | INTEGRODIFFERENTIAL EQUATIONS | EVOLUTION-EQUATIONS | quasilinear equation | STOCHASTIC HEAT-EQUATIONS | comparison principle | ASYMPTOTIC-BEHAVIOR | energy estimates | Tauberian theorem | ANOMALOUS DIFFUSION | weak solution | subdiffusion equations | MEMORY | heat conduction with memory | time-fractional diffusion | CONDUCTION | ultraslow diffusion | WEAK SOLUTIONS | Porous media | Mathematical analysis | Inequalities | Decay | Dirichlet problem | Mathematical models | Energy methods | Diffusion | Estimates

MATHEMATICS, APPLIED | temporal decay estimates | p-Laplacian | NONLINEAR VOLTERRA-EQUATIONS | porous medium equation | DIFFERENTIAL-EQUATIONS | INTEGRODIFFERENTIAL EQUATIONS | EVOLUTION-EQUATIONS | quasilinear equation | STOCHASTIC HEAT-EQUATIONS | comparison principle | ASYMPTOTIC-BEHAVIOR | energy estimates | Tauberian theorem | ANOMALOUS DIFFUSION | weak solution | subdiffusion equations | MEMORY | heat conduction with memory | time-fractional diffusion | CONDUCTION | ultraslow diffusion | WEAK SOLUTIONS | Porous media | Mathematical analysis | Inequalities | Decay | Dirichlet problem | Mathematical models | Energy methods | Diffusion | Estimates

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