Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey

Gong, Chunye; Bao, Weimin; Tang, Guojian; Jiang, Yuewen; Liu, Jie

doi:10.1155/2015/258265

Cited by 20 publications

(11 citation statements)

References 119 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The author has provided many solution techniques for solving three identified FDE problems such as the standard FDEs, the multiorder systems of FDEs, and the multiterm FDEs. One of the studies [13] presented several computational cost evaluations for the numerical solutions of FDEs from the point of view of computer science. Based on that work, the computational complexities for the time-fractional, space-fractional, and space-time FDEs are known to be O(N 2 M), O(NM 2 ), and O(NM(M + N)).…”

Section: Introductionmentioning

confidence: 99%

Iterative method for solving one-dimensional fractional mathematical physics model via quarter-sweep and PAOR

et al. 2021

View full text Add to dashboard Cite

This paper will solve one of the fractional mathematical physics models, a one-dimensional time-fractional differential equation, by utilizing the second-order quarter-sweep finite-difference scheme and the preconditioned accelerated over-relaxation method. The proposed numerical method offers an efficient solution to the time-fractional differential equation by applying the computational complexity reduction approach by the quarter-sweep technique. The finite-difference approximation equation will be formulated based on the Caputo’s time-fractional derivative and quarter-sweep central difference in space. The developed approximation equation generates a linear system on a large scale and has sparse coefficients. With the quarter-sweep technique and the preconditioned iterative method, computing the time-fractional differential equation solutions can be more efficient in terms of the number of iterations and computation time. The quarter-sweep computes a quarter of the total mesh points using the preconditioned iterative method while maintaining the solutions’ accuracy. A numerical example will demonstrate the efficiency of the proposed quarter-sweep preconditioned accelerated over-relaxation method against the half-sweep preconditioned accelerated over-relaxation, and the full-sweep preconditioned accelerated over-relaxation methods. The numerical finding showed that the quarter-sweep finite difference scheme and preconditioned accelerated over-relaxation method can serve as an efficient numerical method to solve fractional differential equations.

show abstract

Section: Introductionmentioning

confidence: 99%

Iterative method for solving one-dimensional fractional mathematical physics model via quarter-sweep and PAOR

et al. 2021

View full text Add to dashboard Cite

show abstract

“…The beauty of the local meshless technique is utilizing just neighbouring collocation points which results in a sparse matrix system and ward off the main deficiency of ill-conditioning. This sparse system of equations can effectively be solved [27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method

Ahmad

Thounthong

et al. 2020

Symmetry

View full text Add to dashboard Cite

Fractional differential equations depict nature sufficiently in light of the symmetry properties which describe biological and physical processes. This article is concerned with the numerical treatment of three-term time fractional-order multi-dimensional diffusion equations by using an efficient local meshless method. The space derivative of the models is discretized by the proposed meshless procedure based on the multiquadric radial basis function though the time-fractional part is discretized by Liouville–Caputo fractional derivative. The numerical results are obtained for one-, two- and three-dimensional cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method.

show abstract

“…In another respect, although simulations of the morphological changes and other hydrodynamic phenomena can be modeled via partial differential equations, this method is implicitly impractical due to the large temporal and spatial grid. Hence, it is not possible to solve within a certain amount of time [40][41][42][43][44][45]. In the 15th Pacific Conference on Computer Graphics and Applications, the United States, Wu and Eftekkarian (2011) emphasized that reducing the calculation speed and improving in-situ visualization will be necessary for integration hydrodynamic models [45][46][47].…”

Section: Introductionmentioning

confidence: 99%

Assessment of the Impact of Sand Mining on Bottom Morphology in the Mekong River in An Giang Province, Vietnam, Using a Hydro-Morphological Model with GPU Computing

Kim¹,

Hương

Huy³

et al. 2020

Water

View full text Add to dashboard Cite

Sand mining, among the many activities that have significant effects on the bed changes of rivers, has increased in many parts of the world in recent decades. Numerical modeling plays a vital role in simulation in the long term; however, computational time remains a challenge. In this paper, we propose a sand mining component integrated into the bedload continuity equation and combine it with high-performance computing using graphics processing units to boost the speed of the simulation. The developed numerical model is applied to the Mekong river segment, flowing through Tan Chau Town, An Giang Province, Vietnam. The 20 years from 1999 to 2019 is examined in this study, both with and without sand mining activities. The results show that the numerical model can simulate the bed change for the period from 1999 to 2019. By adding the sand mining component (2002–2006), the bed change in the river is modeled closely after the actual development. The Tan An sand mine in the area (2002–2006) caused the channel to deviate slightly from that of An Giang and created a slight erosion channel in 2006 (−23 m). From 2006 to 2014, although Tan An mine stopped operating, the riverbed recovered quite slowly with a small accretion rate (0.25 m/year). However, the Tan An sand mine eroded again from 2014–2019 due to a lack of sand. In 2014, in the Vinh Hoa communes, An Giang Province, the Vinh Hoa sand mine began to operate. The results of simulating with sand mining incidents proved that sand mining caused the erosion channel to move towards the sand mines, and the erosion speed was faster when there was no sand mining. Combined with high-performance computing, harnessing the power of accelerators such as graphics processing units (GPUs) can help run numerical simulations up to 23x times faster.

show abstract

Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey

Cited by 20 publications

References 119 publications

Iterative method for solving one-dimensional fractional mathematical physics model via quarter-sweep and PAOR

Iterative method for solving one-dimensional fractional mathematical physics model via quarter-sweep and PAOR

Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method

Assessment of the Impact of Sand Mining on Bottom Morphology in the Mekong River in An Giang Province, Vietnam, Using a Hydro-Morphological Model with GPU Computing

Contact Info

Product

Resources

About