Green’s function for the fractional KdV equation on the periodic domain via Mittag–Leffler function
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Uyen Le
Abstract
The linear operator c + (−Δ)α/2, where c > 0 and (−Δ)α/2 is the fractional Laplacian on the periodic domain, arises in the existence of periodic travelling waves in the fractional Korteweg–de Vries equation. We establish a relation of the Green function of this linear operator with the Mittag–Leffler function, which was previously used in the context of the Riemann–Liouville and Caputo fractional derivatives. By using this relation, we prove that the Green function is strictly positive and single-lobe (monotonically decreasing away from the maximum point) for every c > 0 and every α ∈ (0, 2]. On the other hand, we argue from numerical approximations that in the case of α ∈ (2, 4], the Green function is positive and single-lobe for small c and non-positive and non-single lobe for large c.
Acknowledgements
The authors are thankful to G. Alfimov, P.G. Kevrekidis, T. Simon and A. Stefanov for relevant suggestions which helped completing this project.
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© 2021 Diogenes Co., Sofia
Articles in the same Issue
- Frontmatter
- Editorial
- FCAA related news, events and books
- Survey Paper
- Towards a unified theory of fractional and nonlocal vector calculus
- Research Paper
- An adaptive memory method for accurate and efficient computation of the Caputo fractional derivative
- Analysis of solutions of some multi-term fractional Bessel equations
- Existence of solutions for the semilinear abstract Cauchy problem of fractional order
- Summability of formal solutions for a family of generalized moment integro-differential equations
- Analysis and fast approximation of a steady-state spatially-dependent distributed-order space-fractional diffusion equation
- Green’s function for the fractional KdV equation on the periodic domain via Mittag–Leffler function
- First order plus fractional diffusive delay modeling: Interconnected discrete systems
- On a solution of a fractional hyper-Bessel differential equation by means of a multi-index special function
- On the decomposition of solutions: From fractional diffusion to fractional Laplacian
- Output error MISO system identification using fractional models
- Short Paper
- Identification of system with distributed-order derivatives
- Short note
- On the Green function of the killed fractional Laplacian on the periodic domain
Articles in the same Issue
- Frontmatter
- Editorial
- FCAA related news, events and books
- Survey Paper
- Towards a unified theory of fractional and nonlocal vector calculus
- Research Paper
- An adaptive memory method for accurate and efficient computation of the Caputo fractional derivative
- Analysis of solutions of some multi-term fractional Bessel equations
- Existence of solutions for the semilinear abstract Cauchy problem of fractional order
- Summability of formal solutions for a family of generalized moment integro-differential equations
- Analysis and fast approximation of a steady-state spatially-dependent distributed-order space-fractional diffusion equation
- Green’s function for the fractional KdV equation on the periodic domain via Mittag–Leffler function
- First order plus fractional diffusive delay modeling: Interconnected discrete systems
- On a solution of a fractional hyper-Bessel differential equation by means of a multi-index special function
- On the decomposition of solutions: From fractional diffusion to fractional Laplacian
- Output error MISO system identification using fractional models
- Short Paper
- Identification of system with distributed-order derivatives
- Short note
- On the Green function of the killed fractional Laplacian on the periodic domain
