The Analytical Solution of Parabolic Volterra Integro-Differential Equations in the Infinite Domain

Zhao, Yun; Zhao, Fengqun

doi:10.3390/e18100344

Cited by 7 publications

(4 citation statements)

References 24 publications

(26 reference statements)

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“…Taking into account a Hankel transform and the properties Fox-H function [37,41], last two equations can be written as…”

Section: Preliminariesmentioning

confidence: 99%

The explicit formula for solution of anomalous diffusion equation in the multi-dimensional space

Durdiev¹,

Shishkina²,

Sitnik³

2020

Preprint

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This paper intends on obtaining the explicit solution of n-dimensional anomalous diffusion equation in the infinite domain with non-zero initial condition and vanishing condition at infinity. It is shown that this equation can be derived from the parabolic integro-differential equation with memory in which the kernel is t, where E α,β is the Mittag-Liffler function. Based on Laplace and Fourier transforms the properties of the Fox Hfunction and convolution theorem, explicit solution for anomalous diffusion equation is obtained.

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“…Taking into account a Hankel transform and the properties Fox-H function [37,41], last two equations can be written as…”

Section: Preliminariesmentioning

confidence: 99%

The explicit formula for solution of anomalous diffusion equation in the multi-dimensional space

Durdiev¹,

Shishkina²,

Sitnik³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…The unique solvability of Cauchy and initial-boundary value problems for different types of FDE (fractional differential equations) were analyzed in the works [25][26][27][28][29][30][31][32][33][34][35][36][37][38].…”

Section: Introduction To the Problem And Its Settingmentioning

confidence: 99%

Construction of an Explicit Solution of a Time-Fractional Multidimensional Differential Equation

2021

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In this work, an explicit solution of the initial-boundary value problem for a multidimensional time-fractional differential equation is constructed. The possibility of obtaining this equation from an integro-differential wave equation with a Mittag–Leffler–type memory kernel is shown. An explicit solution to the problem under consideration is obtained using the Laplace and Fourier transforms, the properties of the Fox H-functions and the convolution theorem.

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“…Convection-diffusion integro-differential equation has been used to model several important physical phenomena such as heat and mass transfer, flows in porous media, current density in fluids, and pollutant transport in atmosphere, streams, rivers and oceans (see [8,10,11] and the references therein). PIDE models have been rarely solved analytically and their general solution is only obtained under restrictive conditions [12]. Due to such restrictive conditions the models become over simplified and also limit their physical relevance.…”

Section: Introductionmentioning

confidence: 99%

A Comparative Numerical Study of Parabolic Partial Integro-Differential Equation Arising from Convection-Diffusion

Khan¹,

Ali²,

Fazal-i-Haq³

et al. 2021

Computer Modeling in Engineering &Amp; Sciences

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This article studies the development of two numerical techniques for solving convection-diffusion type partial integro-differential equation (PIDE) with a weakly singular kernel. Cubic trigonometric B-spline (CTBS) functions are used for interpolation in both methods. The first method is CTBS based collocation method which reduces the PIDE to an algebraic tridiagonal system of linear equations. The other method is CTBS based differential quadrature method which converts the PIDE to a system of ODEs by computing spatial derivatives as weighted sum of function values. An efficient tridiagonal solver is used for the solution of the linear system obtained in the first method as well as for determination of weighting coefficients in the second method. An explicit scheme is employed as time integrator to solve the system of ODEs obtained in the second method. The methods are tested with three nonhomogeneous problems for their validation. Stability, computational efficiency and numerical convergence of the methods are analyzed. Comparison of errors in approximations produced by the present methods versus different values of discretization parameters and convection-diffusion coefficients are made. Convection and diffusion dominant cases are discussed in terms of Peclet number. The results are also compared with cubic B-spline collocation method.

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The Analytical Solution of Parabolic Volterra Integro-Differential Equations in the Infinite Domain

Cited by 7 publications

References 24 publications

The explicit formula for solution of anomalous diffusion equation in the multi-dimensional space

The explicit formula for solution of anomalous diffusion equation in the multi-dimensional space

Construction of an Explicit Solution of a Time-Fractional Multidimensional Differential Equation

A Comparative Numerical Study of Parabolic Partial Integro-Differential Equation Arising from Convection-Diffusion

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