On a class of non-linear delay distributed order fractional diffusion equations

Пименов, В. Г.; Hendy, Ahmed S.; Staelen, Rob De

doi:10.1016/j.cam.2016.02.039

Cited by 56 publications

(18 citation statements)

References 30 publications

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“…This novel form of Gronwall's inequality helps in obtaining an optimal error estimate for fixed time‐delay fractional parabolic equations. The task is achieved employing arguments from other works which employ local assumptions in time. The present work aims at extending the results in to study time‐fractional differential equations with functional delay.…”

Section: Introductionmentioning

confidence: 99%

Convergence and stability estimates in difference setting for time‐fractional parabolic equations with functional delay

Hendy

Пименов

Macías‐Díaz

2019

Numerical Methods Partial

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A class of one‐dimensional time‐fractional parabolic differential equations with delay effects of functional type in the time component is numerically investigated in this work. To that end, a compact difference scheme is constructed for the numerical solution of those equations based on the idea of separating the current state and the prehistory function. In these terms, the prehistory function is approximated by means of an appropriate interpolation–extrapolation operator. A discrete form of the fractional Gronwall inequality is employed to provide an optimal error estimate. The existence and uniqueness of the numerical solutions, the order of approximation error for the constructed scheme, the stability and the order of convergence are mathematically investigated in this work.

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Section: Introductionmentioning

confidence: 99%

Convergence and stability estimates in difference setting for time‐fractional parabolic equations with functional delay

Hendy

Пименов

Macías‐Díaz

2019

Numerical Methods Partial

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“…On the other hand, for time-fractional diffusion-wave equations, analytic and week solutions are respectively studied in [1,12] and [28], the unique solvability of semilinear time-fractional wave equations in [16], and approximation methods in [4,6,7]. Hendy et al [11] and Pimenov et al [26] extended the Crank-Nicolson scheme to non-linearly distributed orders in time fractional diffusion-wave equation.…”

Section: Introductionmentioning

confidence: 99%

The Identification of the Time-Dependent Source Term in Time-Fractional Diffusion-Wave Equations

Liao¹

2019

EAJAM

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An inverse time-dependent source problem for a multi-dimensional fractional diffusion wave equation is considered. The regularity of the weak solution for the direct problem under strong conditions is studied and the unique solvability of the inverse problem is proved. The regularised variational problem is solved by the conjugate gradient method combined with Morozov's discrepancy principle. Numerical examples show the stability and efficiency of the method.

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“…Since the analytical solutions for fractional partial differential equations with delay are not so easy to obtain, therefore, it becomes necessary to look for efficient numerical schemes. Various numerical methods and existence and uniqueness theory of Fractional Differential Equations (FDEs) have been studied extensively by researchers in last few years and their study comprises of numerical methods such as Finite Difference, Finite Volume, Finite Element, Weighted residual method, Spectral methods, Hybrid methods, discontinuous Galerkin method and so on [7]- [13] and the reader can also refer to the references therein. Nevertheless, relatively less work has been done using neutral delay partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of time fractional non-linear neutral delay differential equations of fourth-order

Nandal¹,

Pandey²

2019

Malaya J. Mat

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In this paper, we present a numerical technique for the solution of a class of time fractional nonlinear neutral delay sub-diffusion differential equation of fourth order with variable coefficients. We constructed a numerical scheme which is of second-order convergence in time and is based on L2-1σ formula for the temporal variable. The stability of the scheme is proved using discrete energy method considering several auxiliary assumptions and then we showed that our scheme is convergent in L 2 norm with convergence order O(τ 2 + h 4), where τ and h are temporal and space mesh sizes respectively. In the end, we provide some numerical experiments to validate the theoretical results.

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On a class of non-linear delay distributed order fractional diffusion equations

Cited by 56 publications

References 30 publications

Convergence and stability estimates in difference setting for time‐fractional parabolic equations with functional delay

Convergence and stability estimates in difference setting for time‐fractional parabolic equations with functional delay

The Identification of the Time-Dependent Source Term in Time-Fractional Diffusion-Wave Equations

Numerical solution of time fractional non-linear neutral delay differential equations of fourth-order

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