A comparative analysis of two algorithms for the simulation of fractional differential equations

Maamri, Nezha; Trigeassou, Jean‐Claude

doi:10.1007/s40435-019-00547-4

Cited by 6 publications

(4 citation statements)

References 31 publications

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“…Also, to deal with the boundary and initial conditions (3,4), this was conducted through adapting the governing equations. Now, we start to demonstrate the obtained results to know the reliability, stability, convergence, and performance of DQM based on three types of kernels and block marching with Caputo sense as follows:…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Therefore, the results computed via our methods are much more accurate, stable, and efficient. The best values of parameters, which control our methods, are L = 12, R = 3 blocks, δt = 0.1, mesh size > (7 × 7), and σ = 1.45 × ∆ for problem (1) and σ = 1.3 × ∆ for problems (2)(3), and these values were achieved at L ∞ = 10 −7 and computation time = 0.036 s.…”

Section: Discussionmentioning

confidence: 99%

“…Many researchers have been drawn to the study and use of fractional differential equations in recent decades. Classical differential equations are generalized by fractional order differential equations to explain complex problems such as viscoelasticity, biological engineering, fluid mechanics, engineering, stream through porous media [1,2], fractional operators [3], mathematical physics, and other fields. These equations are classified into three kinds based on the presence of fractional order derivatives in space, time, or spacetime [4].…”

Section: Introductionmentioning

confidence: 99%

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Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems

Mustafa,

Ragb,

Salah

et al. 2023

Fractal Fract

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The aim of this study is to utilize a differential quadrature method with various kernels, such as Lagrange interpolation and discrete singular convolution, to tackle problems related to the Riesz fractional diffusion equation and the Riesz fractional advection–dispersion equation. The governing equation for convection and diffusion depends on both spatial and transient factors. By using the block marching technique, we transform these equations into an algebraic system using differential quadrature methods and the Caputo-type fractional operator. Next, we develop a MATLAB program that generates code capable of solving the fractional convection–diffusion equation in (1+2) dimensions for each shape function. Our goal is to ensure that our methods are reliable, accurate, efficient, and capable of convergence. To achieve this, we conduct two experiments, comparing the numerical and graphical results with both analytical and numerical solutions. Additionally, we evaluate the accuracy of our findings using the L∞ error. Our tests show that the differential quadrature method, which relies mainly on the discrete singular convolution shape function, is a highly effective numerical approach for fractional convective diffusion problems. It offers superior accuracy, faster convergence, and greater reliability than other techniques. Furthermore, we study the impact of fractional order derivatives, velocity, and positive diffusion parameters on the results.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems

Mustafa,

Ragb,

Salah

et al. 2023

Fractal Fract

View full text Add to dashboard Cite

show abstract

“…where L is the window size, which limits the memory dependency to a finite window, including a fewer number of the previous iterations [3,14], resulting in another form for the GL operator (2b), which can be implemented on hardware based on the window size L.…”

Section: Introductionmentioning

confidence: 99%

A Unified FPGA Realization for Fractional-Order Integrator and Differentiator

et al. 2022

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This paper proposes a generic FPGA realization of an IP core for fractional-order integration and differentiation based on the Grünwald–Letnikov approximation. All fractional-order dependent terms are approximated to simpler relations using curve fitting to enable an efficient hardware realization. Compared to previous works, the proposed design introduces enhancements in the fractional-order range covering both integration and differentiation. An error analysis between software and hardware results is presented for sine, triangle and sawtooth signals. The proposed generic design is realized on XC7A100T FPGA achieving frequency of 9.328 MHz and validated experimentally for a sine input signal on the oscilloscope. The proposed unified generic design is suitable for biomedical signal processing applications. In addition, it can be employed as a laboratory tool for fractional calculus education.

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The Infinite State Representation of Fractional Order Differential Systems: A Survey - Part 2

Trigeassou,

Maamri

2024

IFAC-PapersOnLine

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A comparative analysis of two algorithms for the simulation of fractional differential equations

Cited by 6 publications

References 31 publications

Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems

Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems

A Unified FPGA Realization for Fractional-Order Integrator and Differentiator

The Infinite State Representation of Fractional Order Differential Systems: A Survey - Part 2

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