Parallel finite-difference algorithms for three-dimensional space-fractional diffusion equation with $$\psi $$-Caputo derivatives

Bohaienko, Vsevolod

doi:10.1007/s40314-020-01191-x

Cited by 7 publications

(3 citation statements)

References 20 publications

(22 reference statements)

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“…12: Update T b and T w . 13: Update the position vectors T s i ¢ using the rule in equation (15). 14: Update T b and T w .…”

Section: Error Analysis Of the Approachmentioning

confidence: 99%

Physics informed neural network based scheme and its error analysis for ψ-Caputo type fractional differential equations

Sivalingam,

Govindaraj

2024

Phys. Scr.

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This paper proposes a scientific machine learning approach based on Deep Physics Informed Neural Network (PINN) to solve ψ-Caputo-type differential equations. The trial solution is constructed based on the Theory of Functional Connection (TFC), and the loss function is built using the L1-based difference and quadrature rule. The learning is handled using the new hybrid average subtraction, standard deviation-based optimizer, and the nonlinear least squares approach. The training error is theoretically obtained, and the generalization error is derived in terms of training error. Numerical experiments are performed to validate the proposed approach. We also validate our scheme on the SIR model.

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“…12: Update T b and T w . 13: Update the position vectors T s i ¢ using the rule in equation (15). 14: Update T b and T w .…”

Section: Error Analysis Of the Approachmentioning

confidence: 99%

Physics informed neural network based scheme and its error analysis for ψ-Caputo type fractional differential equations

Sivalingam,

Govindaraj

2024

Phys. Scr.

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“…Aydi et al examined the existence and uniqueness of positive solutions for a fractional thermostat model for both cases of concave and convex source terms by utilizing ψ-Caputo fractional derivative in [9]. For reference, [10][11][12][13][14][15][16] these are certain articles on differential equations that heavily rely on the ψ-Caputo fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

Controllability of fractional dynamical systems with ψ-Caputo fractional derivative

Selvam¹,

Vellappandi²,

Govindaraj³

2023

Phys. Scr.

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The idea behind this study is to investigate the controllability of dynamical systems in terms of the ψ-Caputo fractional derivative. The Grammian matrix is used to get at necessary and sufficient controllability requirements for linear systems, which are characterized by the Mittag-Leffler functions, while the fixed point approach is used to arrive at adequate controllability criteria for nonlinear systems. The novelty of this research is to inquire into the controllability concepts by utilizing the ψ-Caputo fractional derivative. Since ψ-Caputo fractional derivatives have the advantage of capturing memory effects as well as increasing the accuracy of anticipating real-world scenarios. A few numerical examples are offered to help better understand the theoretical results.

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“…For example, the papers [22,23] investigate the issues of computational acceleration in modeling some diffusion processes that are not based on the fractional Gerasimov-Caputo derivative [24,25], which is non-local to spatial variables. The authors Bogaenko V.A.…”

Section: Introductionmentioning

confidence: 99%

Hybrid GPU–CPU Efficient Implementation of a Parallel Numerical Algorithm for Solving the Cauchy Problem for a Nonlinear Differential Riccati Equation of Fractional Variable Order

Tverdyi

Parovik

2023

Mathematics

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The numerical solution for fractional dynamics problems can create a high computational load, which makes it necessary to implement efficient algorithms for their solution. The main contribution to the computational load of such computations is created by heredity (memory), which is determined by the dependence of the current value of the solution function on previous values in the time interval. In terms of mathematics, the heredity here is described using a fractional differentiation operator in the Gerasimov–Caputo sense of variable order. As an example, we consider the Cauchy problem for the non-linear fractional Riccati equation with non-constant coefficients. An efficient parallel implementation algorithm has been proposed for the known sequential non-local explicit finite-difference numerical solution scheme. This implementation of the algorithm is a hybrid one, since it uses both GPU and CPU computational nodes. The program code of the parallel implementation of the algorithm is described in C and CUDA C languages, and is developed using OpenMP and CUDA hardware, as well as software architectures. This paper presents a study on the computational efficiency of the proposed parallel algorithm based on data from a series of computational experiments that were obtained using a computing server NVIDIA DGX STATION. The average computation time is analyzed in terms of the following: running time, acceleration, efficiency, and the cost of the algorithm. As a result, it is shown on test examples that the hybrid version of the numerical algorithm can give a significant performance increase of 3–5 times in comparison with both the sequential version of the algorithm and OpenMP implementation.

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Parallel finite-difference algorithms for three-dimensional space-fractional diffusion equation with $$\psi $$-Caputo derivatives

Cited by 7 publications

References 20 publications

Physics informed neural network based scheme and its error analysis for ψ-Caputo type fractional differential equations

Physics informed neural network based scheme and its error analysis for ψ-Caputo type fractional differential equations

Controllability of fractional dynamical systems with ψ-Caputo fractional derivative

Hybrid GPU–CPU Efficient Implementation of a Parallel Numerical Algorithm for Solving the Cauchy Problem for a Nonlinear Differential Riccati Equation of Fractional Variable Order

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