Computation of Fractional Derivatives of Analytic Functions

Fornberg, Bengt; Piret, Cécile

doi:10.1007/s10915-023-02293-4

Cited by 3 publications

(6 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These include adaptive finite element methods that use much finer grids close to the boundary of integration domains, to address the lower regularity of fractional derivatives close to said boundaries 2 and finite difference methods on stencils in the complex plane. 15 Walking on spheres is a classical Monte Carlo method, recently applied to computing the fractional Poisson equation, 20 memory-efficient approaches based on convolution quadrature via Runge-Kutta schemes 3 ; an excellent review can be found in Ref. [31].…”

Section: Numerical Approachesmentioning

confidence: 99%

“…A dependence of the numerical error can be seen in Table 1 on the left. For values of 𝑁 𝐹𝐹𝑇 = 2 16 , 2 17 , 2 18 , 2 19 , the maximal difference (err) between DFT computation and the exact expression is as for 𝑁 𝐹𝐹𝑇 = 2 15 . The dependence on the size of the torus is shown in Table 1 on the right where we have used 𝑁 𝐹𝐹𝑇 = 2 19 .…”

Section: Comparison With Dftmentioning

confidence: 99%

See 1 more Smart Citation

Multidomain spectral approach to rational‐order fractional derivatives

Klein,

Stoilov

2024

Stud Appl Math

View full text Add to dashboard Cite

We propose a method to numerically compute fractional derivatives (or the fractional Laplacian) on the whole real line via Riesz fractional integrals. The compactified real line is divided into a number of intervals, thus amounting to a multidomain approach; after transformations in accordance with the underlying curve ensuring analyticity of the respective integrands, the integrals over the different domains are computed with a Clenshaw–Curtis algorithm. As an example, we consider solitary waves for fractional Korteweg‐de Vries equations and compare these to results obtained with a discrete Fourier transform.

show abstract

Section: Numerical Approachesmentioning

confidence: 99%

Section: Comparison With Dftmentioning

confidence: 99%

Multidomain spectral approach to rational‐order fractional derivatives

Klein,

Stoilov

2024

Stud Appl Math

View full text Add to dashboard Cite

show abstract

“…In another example, for a second-order centered partial derivative with second-order error (18), one has:…”

Section: Finite Difference Equations For Derivative Approximations Wi...mentioning

confidence: 99%

Section: Finite Difference Coefficients For Complex Variablesmentioning

confidence: 99%

“…is uniquely defined, no matter from which direction in the complex plane ∆x 1 approaches zero [17,18]. Any function u(x) that possesses a Taylor expansion at some x-location can be extended to an analytic function, with a vast range of further consequences [17][18][19][20]. Now, instead of having forward, backward, and central difference approximations on the real line, we have first quadrant (Q1), second quadrant (Q2), third quadrant (Q3), fourth quadrant (Q4), and central (C) on the complex plane.…”

Section: Finite Difference Coefficients For Complex Variablesmentioning

confidence: 99%

See 1 more Smart Citation

Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain

Almeida Magalhães,

Brito,

Lamon

et al. 2024

Mathematics

View full text Add to dashboard Cite

The paper expands the finite difference method to the complex plane, and thus obtains an improvement in the resolution of differential equations with an increase in numerical precision and a generalization in the mathematical modeling of problems. The article begins with a selection of the best techniques for obtaining finite difference coefficients for approximating derivatives in the real domain. Then, the calculation is expanded to the complex domain. The research expands forward, backward, and central difference approximations of the real case by a quadrant approximation in the complex plane, which facilitates the use in boundary conditions of differential equations. The article shows many real and complex finite difference equations with their respective order of error, intended to serve as a basis and reference, which have been tested in practical examples of solving differential equations used in engineering. Finally, a comparison is made between the real and complex techniques of finite difference methods applied in the Theory of Elasticity. As a surprising result, the article shows that the finite difference method has great advantages in numerical precision, diversity of formulas, and modeling generalities in the complex domain when compared to the real domain.

show abstract

Numerical Methods for Fractional PDEs

Klein,

Stoilov

2024

Nonlinear Systems and Complexity

View full text Add to dashboard Cite

Computation of Fractional Derivatives of Analytic Functions

Cited by 3 publications

References 18 publications

Multidomain spectral approach to rational‐order fractional derivatives

Multidomain spectral approach to rational‐order fractional derivatives

Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain

Numerical Methods for Fractional PDEs

Contact Info

Product

Resources

About