scholarly journals Green’s function for the fractional KdV equation on the periodic domain via Mittag–Leffler function

2021 ◽  
Vol 24 (5) ◽  
pp. 1507-1534
Author(s):  
Uyen Le ◽  
Dmitry E. Pelinovsky
Keyword(s):  
Green Function ◽  
Linear Operator ◽  
Fractional Derivatives ◽  
Fractional Laplacian ◽  
Kdv Equation ◽  
Travelling Waves ◽  
Maximum Point ◽  
Periodic Domain ◽  
Caputo Fractional Derivatives ◽  
The Green Function

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.

scholarly journals On the Green function of the killed fractional Laplacian on the periodic domain

2021 ◽  
Vol 24 (5) ◽  
pp. 1629-1635
Author(s):  
Thomas Simon
Keyword(s):  
Green Function ◽  
Simple Proof ◽  
Fractional Laplacian ◽  
Contour Integration ◽  
Triple Product ◽  
Periodic Domain ◽  
Probabilistic Representation ◽  
Positive Part ◽  
Completely Monotone ◽  
The Green Function

Abstract We give a very simple proof of the positivity and unimodality of the Green function for the killed fractional Laplacian on the periodic domain. The argument relies on the Jacobi triple product and a probabilistic representation of the Green function. We also show by a contour integration that the Green function is completely monotone on the positive part of the periodic domain.

scholarly journals Global existence and convergence of a flow to Kazdan–Warner equation with non-negative prescribed function

2021 ◽  
Vol 60 (1) ◽  
Author(s):  
Linlin Sun ◽  
Jingyong Zhu
Keyword(s):  
Riemann Surface ◽  
Green Function ◽  
Global Existence ◽  
Existence Result ◽  
Maximum Point ◽  
Closed Riemann Surface ◽  
Delta G ◽  
The Green Function ◽  
Sigma H ◽  
Sigma G

AbstractWe consider an evolution problem associated to the Kazdan–Warner equation on a closed Riemann surface $$(\Sigma ,g)$$ ( Σ , g ) $$\begin{aligned} -\Delta _{g}u=8\pi \left( \dfrac{he^{u}}{\int _{\Sigma }he^{u}\mathop {}\mathrm {d}\mu _{g}}-\dfrac{1}{\int _{\Sigma }\mathop {}\mathrm {d}\mu _{g}}\right) \end{aligned}$$ - Δ g u = 8 π h e u ∫ Σ h e u d μ g - 1 ∫ Σ d μ g where the prescribed function $$h\ge 0$$ h ≥ 0 and $$\max _{\Sigma }h>0$$ max Σ h > 0 . We prove the global existence and convergence under additional assumptions such as $$\begin{aligned} \Delta _{g}\ln h(p_0)+8\pi -2K(p_0)>0 \end{aligned}$$ Δ g ln h ( p 0 ) + 8 π - 2 K ( p 0 ) > 0 for any maximum point $$p_0$$ p 0 of the sum of $$2\ln h$$ 2 ln h and the regular part of the Green function, where K is the Gaussian curvature of $$\Sigma $$ Σ . In particular, this gives a new proof of the existence result by Yang and Zhu (Pro Am Math Soc 145:3953–3959, 2017) which generalizes existence result of Ding et al. (Asian J Math 1:230–248, 1997) to the non-negative prescribed function case.

Exponentially Convergent Approximation to the Elliptic Solution Operator

2006 ◽  
Vol 6 (4) ◽  
pp. 386-404 ◽  
Author(s):  
Ivan. P. Gavrilyuk ◽  
V.L. Makarov ◽  
V.B. Vasylyk
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Solution Operator ◽  
Hyperbolic Operator ◽  
Elliptic Solution ◽  
Elliptic Boundary Value ◽  
Sine Family ◽  
The Green Function

AbstractWe develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

Solution of inverse coefficient problem for fractional anomalous diffusion equation

2012 ◽  
Vol 9 (2) ◽  
pp. 65-70
Author(s):  
E.V. Karachurina ◽  
S.Yu. Lukashchuk
Keyword(s):  
Anomalous Diffusion ◽  
Fractional Derivatives ◽  
Descent Method ◽  
Extremum Problem ◽  
Steepest Descent Method ◽  
Coefficient Problem ◽  
Caputo Fractional Derivatives ◽  
Inverse Coefficient Problem ◽  
Residual Functional ◽  
Inverse Coefficient

An inverse coefficient problem is considered for time-fractional anomalous diffusion equations with the Riemann-Liouville and Caputo fractional derivatives. A numerical algorithm is proposed for identification of anomalous diffusivity which is considered as a function of concentration. The algorithm is based on transformation of inverse coefficient problem to extremum problem for the residual functional. The steepest descent method is used for numerical solving of this extremum problem. Necessary expressions for calculating gradient of residual functional are presented. The efficiency of the proposed algorithm is illustrated by several test examples.

scholarly journals Stability Analysis and Existence of Solutions for a Modified SIRD Model of COVID-19 with Fractional Derivatives

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1431
Author(s):  
Bilal Basti ◽  
Nacereddine Hammami ◽  
Imadeddine Berrabah ◽  
Farid Nouioua ◽  
Rabah Djemiat ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Biological Models ◽  
Uniqueness Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Single Form ◽  
Stability Results

This paper discusses and provides some analytical studies for a modified fractional-order SIRD mathematical model of the COVID-19 epidemic in the sense of the Caputo–Katugampola fractional derivative that allows treating of the biological models of infectious diseases and unifies the Hadamard and Caputo fractional derivatives into a single form. By considering the vaccine parameter of the suspected population, we compute and derive several stability results based on some symmetrical parameters that satisfy some conditions that prevent the pandemic. The paper also investigates the problem of the existence and uniqueness of solutions for the modified SIRD model. It does so by applying the properties of Schauder’s and Banach’s fixed point theorems.

scholarly journals Ulam–Hyers–Mittag-Leffler stability for tripled system of weighted fractional operator with TIME delay

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed A. Almalahi ◽  
Satish K. Panchal ◽  
Fahd Jarad ◽  
Thabet Abdeljawad
Keyword(s):  
Fixed Point ◽  
Time Delay ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Sufficient Conditions ◽  
Fractional Operator ◽  
Caputo Fractional Derivatives ◽  
Picard Operator

AbstractThis study is aimed to investigate the sufficient conditions of the existence of unique solutions and the Ulam–Hyers–Mittag-Leffler (UHML) stability for a tripled system of weighted generalized Caputo fractional derivatives investigated by Jarad et al. (Fractals 28:2040011 2020) in the frame of Chebyshev and Bielecki norms with time delay. The acquired results are obtained by using Banach fixed point theorems and the Picard operator (PO) method. Finally, a pertinent example of the results obtained is demonstrated.

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