A numerical solution for a class of time fractional diffusion equations with delay

Пименов, В. Г.; Hendy, Ahmed S.

doi:10.1515/amcs-2017-0033

Cited by 18 publications

(11 citation statements)

References 33 publications

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“…Consider the following time‐fractional partial differential equations with proportional delay as given in the work of Pimenov et al

D_{t}^{ν} u (x, t) = \frac{\partial^{2}}{\partial x^{2}} u (x, t) - u (x, t - s) + \frac{Γ (3)}{Γ (3 - ν)} (2 x - x^{2}) t^{2 - ν} + 2 t^{2} + x (2 - x) {(t - s)}^{2}, x \in [0, 2], t \in [0, 1],

with the initial and boundary conditions

u (x, 0) = t^{2} (2 x - x^{2}), x \in [0, 2], t \in [- s, 0),

u (0, t) = u (2, t) = 0, t \in [0, 1] .

The exact solution to this problem is u ( x , t )= t 2 (2 x − x 2 ). The numerical calculus is summarized in Table and Figures and to demonstrate the accuracy and the applicability of the proposed method.…”

Section: Numerical Approach Implementationmentioning

confidence: 99%

“…Hosseinpour et al used Müntz‐Legendre polynomials for delay time‐fractional partial differential equations. For additional information, see other works …”

Section: Introductionmentioning

confidence: 99%

“…For additional information, see other works. [36][37][38] During the 1980s, wavelet analysis due to their successful application in signal and image processing became famous in various branches of science. In addition, characteristics such as orthogonality, arbitrary regularity, and good localization of wavelets have attracted many scientists.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay

Dehestani

Ordokhani

Razzaghi

2019

Numerical Linear Algebra App

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Summary A novel collocation method based on Genocchi wavelet is presented for the numerical solution of fractional differential equations and time‐fractional partial differential equations with delay. In this work, to achieve the approximate solution with height accuracy, we employed the operational matrix of integer derivative and the pseudo‐operational matrix of fractional derivative in Caputo sense. Also, based on Genocchi function properties, we presented delay and pantograph operational matrices of Genocchi wavelet functions (GWFs). Due to operational and pseudo‐operational matrices, the equations under this study can be turned into nonlinear algebraic equations with the unknown GWF coefficients. For illustrating the upper bound of error for the proposed method, we estimate the error in the sense of Sobolev space. In addition, to demonstrate the efficacy of the pseudo‐operational matrix of fractional derivative, we investigate the upper bound of error for the mentioned matrix. Finally, the algorithm based on the proposed approach is implemented for some numerical experiments to confirm accuracy and applicability.

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“…Consider the following time‐fractional partial differential equations with proportional delay as given in the work of Pimenov et al

D_{t}^{ν} u (x, t) = \frac{\partial^{2}}{\partial x^{2}} u (x, t) - u (x, t - s) + \frac{Γ (3)}{Γ (3 - ν)} (2 x - x^{2}) t^{2 - ν} + 2 t^{2} + x (2 - x) {(t - s)}^{2}, x \in [0, 2], t \in [0, 1],

with the initial and boundary conditions

u (x, 0) = t^{2} (2 x - x^{2}), x \in [0, 2], t \in [- s, 0),

u (0, t) = u (2, t) = 0, t \in [0, 1] .

Section: Numerical Approach Implementationmentioning

confidence: 99%

“…Hosseinpour et al used Müntz‐Legendre polynomials for delay time‐fractional partial differential equations. For additional information, see other works …”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay

Dehestani

Ordokhani

Razzaghi

2019

Numerical Linear Algebra App

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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

Self Cite

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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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“…The aforementioned solutions, especially QuTIP and QuantumOptics.jl, allow utilizing parallel computing inside the environments of Python and Julia. However, the packages are not intended for High-Performance Computing (HPC) [11], [12] where Message Process Interface, termed as MPI [13], plays a significant role. Using MPI allows us to re-implement the QTM for HPC systems, regardless of their scale (the QTM is scale-free because each trajectory may be calculated separately).…”

Section: Introductionmentioning

confidence: 99%

QTM: Computational package using MPI protocol for Quantum Trajectories Method

Sawerwain¹,

Wiśniewska²

2018

PLoS ONE

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The Quantum Trajectories Method (QTM) is one of the frequently used methods for studying open quantum systems. The main idea of this method is the evolution of wave functions which describe the system (as functions of time). Then, so-called quantum jumps are applied at a randomly selected point in time. The obtained system state is called as a trajectory. After averaging many single trajectories, we obtain the approximation of the behavior of a quantum system. This fact also allows us to use parallel computation methods. In the article, we discuss the QTM package which is supported by the MPI technology. Using MPI allowed utilizing the parallel computing for calculating the trajectories and averaging them—as the effect of these actions, the time taken by calculations is shorter. In spite of using the C++ programming language, the presented solution is easy to utilize and does not need any advanced programming techniques. At the same time, it offers a higher performance than other packages realizing the QTM. It is especially important in the case of harder computational tasks, and the use of MPI allows improving the performance of particular problems which can be solved in the field of open quantum systems.

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A numerical solution for a class of time fractional diffusion equations with delay

Cited by 18 publications

References 33 publications

On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay

On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay

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

QTM: Computational package using MPI protocol for Quantum Trajectories Method

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