Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 07/2018, Volume 41, Issue 10, pp. 3841 - 3855

In this paper, we construct a new fractional weighted reproducing kernel space, which is the minimum space containing the exact solution. The closed form of...

weakly singular kernel | fractional reproducing kernel space | fractional integro‐differential equation | fractional integro-differential equation | ORDER | MATHEMATICS, APPLIED | COLLOCATION METHOD | Differential equations | Error correction | Mathematical analysis | Exact solutions

weakly singular kernel | fractional reproducing kernel space | fractional integro‐differential equation | fractional integro-differential equation | ORDER | MATHEMATICS, APPLIED | COLLOCATION METHOD | Differential equations | Error correction | Mathematical analysis | Exact solutions

Journal Article

Numerical Methods for Partial Differential Equations, ISSN 0749-159X, 09/2017, Volume 33, Issue 5, pp. 1565 - 1581

The main objective of the paper is to find the approximate solution of fractional integro partial differential equation with a weakly singular kernel. Integro...

collocation method | cubic B‐spline | weakly singular kernel | finite differences | integro partial differential equation | cubic B-spline | SCHEME | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | PARTIAL INTEGRODIFFERENTIAL EQUATIONS | DIFFUSION EQUATION | Differential equations | Viscoelasticity | Partial differential equations | Discretization | Collocation | Finite difference method

collocation method | cubic B‐spline | weakly singular kernel | finite differences | integro partial differential equation | cubic B-spline | SCHEME | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | PARTIAL INTEGRODIFFERENTIAL EQUATIONS | DIFFUSION EQUATION | Differential equations | Viscoelasticity | Partial differential equations | Discretization | Collocation | Finite difference method

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 08/2019, Volume 354, pp. 103 - 114

In this paper, we propose a fast and efficient numerical method to solve the two-dimensional integro-differential equation with a weakly singular kernel. The...

Integro-differential equation with weakly singular kernel | Stability | Convolution quadrature rule | ADI difference scheme | Convergence | ORDER | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | CONVOLUTION QUADRATURE | APPROXIMATIONS | EVOLUTION EQUATION | TIME DISCRETIZATION | Analysis | Differential equations

Integro-differential equation with weakly singular kernel | Stability | Convolution quadrature rule | ADI difference scheme | Convergence | ORDER | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | CONVOLUTION QUADRATURE | APPROXIMATIONS | EVOLUTION EQUATION | TIME DISCRETIZATION | Analysis | Differential equations

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 07/2018, Volume 328, pp. 353 - 364

In this paper, we consider the hybrid collocation methods to solve the eigenvalue problem of a compact integral operator with weakly singular kernels of...

Hybrid collocation methods | Eigenvalue problem | Convergence rates | Weakly singular kernel | MATHEMATICS, APPLIED | APPROXIMATION | EQUATIONS | 2ND KIND | PRODUCT INTEGRATION

Hybrid collocation methods | Eigenvalue problem | Convergence rates | Weakly singular kernel | MATHEMATICS, APPLIED | APPROXIMATION | EQUATIONS | 2ND KIND | PRODUCT INTEGRATION

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 02/2013, Volume 234, Issue 1, pp. 317 - 329

In this paper, we study a novel numerical scheme for the fourth order partial integro-differential equation with a weakly singular kernel. In the time...

Weakly singular | Integro-differential equation | Quasi-wavelets | Crank–Nicolson scheme | Crank-nicolson scheme | FOKKER-PLANCK EQUATION | APPROXIMATIONS | FINITE-ELEMENT METHODS | PLATES | PHYSICS, MATHEMATICAL | CONVOLUTION | DISCRETIZATION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Crank-Nicolson scheme | FLOWS | MONOTONIC KERNELS | Kernels | Approximation | Discretization | Integrals | Mathematical analysis | Oscillations | Mathematical models | Standards

Weakly singular | Integro-differential equation | Quasi-wavelets | Crank–Nicolson scheme | Crank-nicolson scheme | FOKKER-PLANCK EQUATION | APPROXIMATIONS | FINITE-ELEMENT METHODS | PLATES | PHYSICS, MATHEMATICAL | CONVOLUTION | DISCRETIZATION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Crank-Nicolson scheme | FLOWS | MONOTONIC KERNELS | Kernels | Approximation | Discretization | Integrals | Mathematical analysis | Oscillations | Mathematical models | Standards

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 02/2016, Volume 275, pp. 72 - 80

In this paper, based on the second Chebyshev wavelets (SCW) operational matrix of fractional order integration, a numerical method for solving a class of...

Operational matrix | Weakly singular integro-differential equations | Block pulse functions | SCW | Fractional calculus | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | INTEGRATION | CALCULUS | MATRICES | Methods | Differential equations

Operational matrix | Weakly singular integro-differential equations | Block pulse functions | SCW | Fractional calculus | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | INTEGRATION | CALCULUS | MATRICES | Methods | Differential equations

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 01/2019, Volume 346, pp. 224 - 236

In this paper, we consider Legendre multi-Galerkin methods to solve Fredholm integral equations of the second kind with weakly singular kernel and the...

Eigenvalue problem | Legendre spectral methods | Multi-Galerkin methods | Fredholm integral equations | Weakly singular kernel | MATHEMATICS, APPLIED | APPROXIMATION | PROJECTION METHODS | PRODUCT INTEGRATION

Eigenvalue problem | Legendre spectral methods | Multi-Galerkin methods | Fredholm integral equations | Weakly singular kernel | MATHEMATICS, APPLIED | APPROXIMATION | PROJECTION METHODS | PRODUCT INTEGRATION

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 12/2017, Volume 40, Issue 18, pp. 7627 - 7639

In this paper, we develop a high‐order finite difference scheme for the solution of a time fractional partial integro‐differential equation with a weakly...

weakly singular kernel | compact finite difference | convergence | stability | fractional integro‐differential equation | Fractional integro-differential equation | Stability | Weakly singular kernel | Compact finite difference | Convergence | MATHEMATICS, APPLIED | fractional integro-differential equation | APPROXIMATIONS | SUB-DIFFUSION EQUATION | Differential equations | Viscosity | Energy consumption | Finite difference method

weakly singular kernel | compact finite difference | convergence | stability | fractional integro‐differential equation | Fractional integro-differential equation | Stability | Weakly singular kernel | Compact finite difference | Convergence | MATHEMATICS, APPLIED | fractional integro-differential equation | APPROXIMATIONS | SUB-DIFFUSION EQUATION | Differential equations | Viscosity | Energy consumption | Finite difference method

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 08/2016, Volume 302, pp. 71 - 80

Two reliable methods, namely the Adomian decomposition method (ADM) and the variational iteration method (VIM), are used for solving the Volterra integral...

Variational iteration method | Adomian decomposition method | Weakly singular Volterra equation | TRANSFORMATION | MATHEMATICS, APPLIED | ALGORITHM | Kernels | Sequences | Approximation | Mathematical analysis | Mathematical models | Decomposition | Iterative methods | Volterra integral equations

Variational iteration method | Adomian decomposition method | Weakly singular Volterra equation | TRANSFORMATION | MATHEMATICS, APPLIED | ALGORITHM | Kernels | Sequences | Approximation | Mathematical analysis | Mathematical models | Decomposition | Iterative methods | Volterra integral equations

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 09/2019, Volume 42, Issue 13, pp. 4427 - 4443

In this paper, a fast numerical algorithm based on the Taylor wavelets is proposed for finding the numerical solutions of the fractional integro‐differential...

numerical solution | operational matrix of fractional integration | weakly singular integro‐differential equations | convergence | Taylor wavelets | ORDER | MATHEMATICS, APPLIED | VOLTERRA INTEGRAL-EQUATIONS | SERIES | weakly singular integro-differential equations | COLLOCATION METHODS | Algorithms | Differential equations | Kernels | Wavelet analysis | Numerical analysis

numerical solution | operational matrix of fractional integration | weakly singular integro‐differential equations | convergence | Taylor wavelets | ORDER | MATHEMATICS, APPLIED | VOLTERRA INTEGRAL-EQUATIONS | SERIES | weakly singular integro-differential equations | COLLOCATION METHODS | Algorithms | Differential equations | Kernels | Wavelet analysis | Numerical analysis

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 08/2015, Volume 39, Issue 15, pp. 4421 - 4431

The Jacobi spectral Galerkin method for Volterra integral equations of the second kind with a weakly singular kernel is proposed in this paper. We provide a...

Volterra integral equation | Jacobi spectral Galerkin method | Weakly singular kernel | Convergence | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | POLYNOMIAL-APPROXIMATION | ENGINEERING, MULTIDISCIPLINARY

Volterra integral equation | Jacobi spectral Galerkin method | Weakly singular kernel | Convergence | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | POLYNOMIAL-APPROXIMATION | ENGINEERING, MULTIDISCIPLINARY

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 06/2015, Volume 260, pp. 63 - 70

Numerical methods for weakly singular Volterra integral equations are rarely considered in the literature. The solutions of such equations may exhibit a...

Operational matrix | Weakly singular Volterra integral equations | Block pulse functions | SCW | Fractional calculus | Weakly singular | Volterra integral equations | KIND CHEBYSHEV WAVELET | MATHEMATICS, APPLIED | CALCULUS | INTEGRODIFFERENTIAL EQUATIONS | COLLOCATION METHODS

Operational matrix | Weakly singular Volterra integral equations | Block pulse functions | SCW | Fractional calculus | Weakly singular | Volterra integral equations | KIND CHEBYSHEV WAVELET | MATHEMATICS, APPLIED | CALCULUS | INTEGRODIFFERENTIAL EQUATIONS | COLLOCATION METHODS

Journal Article

Journal of Scientific Computing, ISSN 0885-7474, 11/2016, Volume 69, Issue 2, pp. 673 - 695

A Jacobi spectral collocation method is proposed for the solution of a class of nonlinear Volterra integral equations with a kernel of the general form $$...

Computational Mathematics and Numerical Analysis | Algorithms | 65R20 | Jacobi spectral collocation method | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Nonlinear Volterra integral equation | Weakly singular kernel | Mathematics | 45J05 | Convergence analysis | MATHEMATICS, APPLIED | POLYNOMIAL-APPROXIMATION

Computational Mathematics and Numerical Analysis | Algorithms | 65R20 | Jacobi spectral collocation method | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Nonlinear Volterra integral equation | Weakly singular kernel | Mathematics | 45J05 | Convergence analysis | MATHEMATICS, APPLIED | POLYNOMIAL-APPROXIMATION

Journal Article

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 08/2017, Volume 20, Issue 4, pp. 1023 - 1042

A new computational approach for approximating of variable-order fractional derivatives is proposed. The technique is based on piecewise cubic spline...

65C20 | weakly singular integro-differential equation | convergence order | Primary 26A33 | finite difference approximation | spline approximation | 45J05 | Secondary 33F05 | 41A15 | 65L12 | variable-order fractional calculus | 45G05 | MATHEMATICS, APPLIED | DIFFERENTIAL-EQUATION | CALCULUS | SCW | ALGORITHMS | MATHEMATICS | NUMERICAL APPROXIMATIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | SYSTEMS | DERIVATIVES | Approximation theory | Research | Kernel functions | Mathematical research | Differential equations | Kernels | Computation | Mathematical analysis

65C20 | weakly singular integro-differential equation | convergence order | Primary 26A33 | finite difference approximation | spline approximation | 45J05 | Secondary 33F05 | 41A15 | 65L12 | variable-order fractional calculus | 45G05 | MATHEMATICS, APPLIED | DIFFERENTIAL-EQUATION | CALCULUS | SCW | ALGORITHMS | MATHEMATICS | NUMERICAL APPROXIMATIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | SYSTEMS | DERIVATIVES | Approximation theory | Research | Kernel functions | Mathematical research | Differential equations | Kernels | Computation | Mathematical analysis

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 01/2015, Volume 39, Issue 2, pp. 947 - 954

A compact difference scheme is presented for a partial integro-differential equation. The integral term is treated by means of the product trapezoidal method....

Compact difference scheme | Integro-differential equation | Product trapezoidal method | Singular weakly kernel | DISCRETIZATION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | TIME | INITIAL DATA | Differential equations | Kernels | Stability | Integrals | Mathematical analysis | Mathematical models | Energy methods | Models | Convergence

Compact difference scheme | Integro-differential equation | Product trapezoidal method | Singular weakly kernel | DISCRETIZATION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | TIME | INITIAL DATA | Differential equations | Kernels | Stability | Integrals | Mathematical analysis | Mathematical models | Energy methods | Models | Convergence

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 04/2015, Volume 278, pp. 1 - 11

In this paper, the efficient methods are proposed for solving nonlinear Fredholm integral equations with a weakly singular kernel. By using Sinc approximation...

Nonlinear Fredholm integral equation | Sinc approximation | Collocation method | Weakly singular kernel | Convergence analysis | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | APPROXIMATION | 2ND KIND

Nonlinear Fredholm integral equation | Sinc approximation | Collocation method | Weakly singular kernel | Convergence analysis | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | APPROXIMATION | 2ND KIND

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 04/2019, Volume 347, pp. 149 - 161

This paper proposes a quadrature method based on multi-variate Bernstein polynomials. The method is used to solve multidimensional Volterra integral equations...

Quadrature method | Error analysis | Weakly singular integral equation | Multi-variate Bernstein polynomial | MESHLESS METHOD | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | PARTIAL INTEGRODIFFERENTIAL EQUATION | SOLVE | RADIAL BASIS FUNCTIONS | 2ND KIND | COLLOCATION METHODS

Quadrature method | Error analysis | Weakly singular integral equation | Multi-variate Bernstein polynomial | MESHLESS METHOD | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | PARTIAL INTEGRODIFFERENTIAL EQUATION | SOLVE | RADIAL BASIS FUNCTIONS | 2ND KIND | COLLOCATION METHODS

Journal Article

Advances in Difference Equations, ISSN 1687-1839, 12/2018, Volume 2018, Issue 1, pp. 1 - 13

In this paper, a new set of functions called fractional-order Euler functions (FEFs) is constructed to obtain the solution of fractional integro-differential...

Fractional integro-differential equation | Ordinary Differential Equations | Functional Analysis | Analysis | Weakly singular kernel | Difference and Functional Equations | Operational matrix | Mathematics, general | Mathematics | Fractional-order Euler functions | Partial Differential Equations | Fractional calculus | MATHEMATICS | MATHEMATICS, APPLIED | COLLOCATION METHOD | DIFFERENTIAL-EQUATIONS | BERNOULLI

Fractional integro-differential equation | Ordinary Differential Equations | Functional Analysis | Analysis | Weakly singular kernel | Difference and Functional Equations | Operational matrix | Mathematics, general | Mathematics | Fractional-order Euler functions | Partial Differential Equations | Fractional calculus | MATHEMATICS | MATHEMATICS, APPLIED | COLLOCATION METHOD | DIFFERENTIAL-EQUATIONS | BERNOULLI

Journal Article

Journal of Scientific Computing, ISSN 0885-7474, 5/2018, Volume 75, Issue 2, pp. 970 - 992

In this paper, we develop a fractional order spectral collocation method for solving second kind Volterra integral equations with weakly singular kernels. It...

Computational Mathematics and Numerical Analysis | Second kind Volterra integral equations with weakly singular kernels | Algorithms | 65R20 | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Condition number | Mathematics | A fractional order collocations spectral method | Stability analysis | 45E05 | Convergence analysis | MATHEMATICS, APPLIED | SPLINE COLLOCATION | SPECTRAL METHODS | GENERALIZED JACOBI FUNCTIONS | DIFFERENTIAL-EQUATIONS | GALERKIN METHODS | INTEGRODIFFERENTIAL EQUATIONS | SMOOTH | CONVERGENCE

Computational Mathematics and Numerical Analysis | Second kind Volterra integral equations with weakly singular kernels | Algorithms | 65R20 | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Condition number | Mathematics | A fractional order collocations spectral method | Stability analysis | 45E05 | Convergence analysis | MATHEMATICS, APPLIED | SPLINE COLLOCATION | SPECTRAL METHODS | GENERALIZED JACOBI FUNCTIONS | DIFFERENTIAL-EQUATIONS | GALERKIN METHODS | INTEGRODIFFERENTIAL EQUATIONS | SMOOTH | CONVERGENCE

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 10/2017, Volume 323, pp. 136 - 146

Numerical methods for solving nonlinear systems of weakly singular Volterra integral equations (VIEs) possessing weakly singular solutions appear almost...

Adaptive methods | Weakly singular solutions | Computational electrochemistry | Weakly singular kernels | Volterra integral equations | Product-integration | DESCRIBING ELECTROCHEMICAL TRANSIENTS | CYLINDRICAL WIRE ELECTRODES | MATHEMATICS, APPLIED | SIMULATION | NUMERICAL-SOLUTION | ABEL-TYPE | AUTOMATIC SOLUTION | PACKAGE | KIND | DIFFUSION | CYCLIC VOLTAMMOGRAMS | Electrochemistry | Electrochemical reactions | Methods

Adaptive methods | Weakly singular solutions | Computational electrochemistry | Weakly singular kernels | Volterra integral equations | Product-integration | DESCRIBING ELECTROCHEMICAL TRANSIENTS | CYLINDRICAL WIRE ELECTRODES | MATHEMATICS, APPLIED | SIMULATION | NUMERICAL-SOLUTION | ABEL-TYPE | AUTOMATIC SOLUTION | PACKAGE | KIND | DIFFUSION | CYCLIC VOLTAMMOGRAMS | Electrochemistry | Electrochemical reactions | Methods

Journal Article

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