Large solutions for a class of semilinear integro-differential equations with censored jumps

Rossi, Julio D.; Topp, Erwin

doi:10.1016/j.jde.2016.01.016

Cited by 4 publications

(3 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The problem of determining all parameters θ and the corresponding global matrix G θ is called the symmetric classification problem of the fractional differential equation with variable order of the function. If the classification is exhaustive, it is called the complete symmetric classification problem (El-Sayed & Agarwal 2019; Rossi & Topp 2016). For all parameters θ, the symmetry of (8) is called the principal symmetry of (8) (Bauer et al 2015), which is denoted as G θ .…”

Section: Definition Of Fractional Differential With Variable Order Ofmentioning

confidence: 99%

Study on Numerical Solution of a Variable Order Fractional Differential Equation based on Symmetric Algorithm

Liu¹,

Pan²

2019

JSM

View full text Add to dashboard Cite

As the class of fractional differential equations with changing order has attracted more attention and attention in the fields of research and engineering, it is important to study its numerical solutions. Numerical solution algorithm for a class of fractional differential equations with transformed arrays based on the proposed symmetry algorithm. The symmetry classification is used for the class of values of the boundary problem of the fractional differential equation with the order of change. A fully symmetric classification of the boundary value problem for a class of fractional differential equations with variable sequences is determined by using a fully symmetric differential sequence sorting algorithm. The problem of the boundary value of the fractional differential equation with the transformed order is reduced to the initial value of the ordinary differential equation. The Legendre polynomial method is used to solve the numerical solution of the starting value of the differential equation. The common differential equation is transformed into a matrix series product by a different operator matrix. The matrix products are converted to algebraic equations by discrete variables. By solving the equations, the numerical solution of the starting value of the common differential equation is obtained.

show abstract

Section: Definition Of Fractional Differential With Variable Order Ofmentioning

confidence: 99%

Study on Numerical Solution of a Variable Order Fractional Differential Equation based on Symmetric Algorithm

Liu¹,

Pan²

2019

JSM

View full text Add to dashboard Cite

show abstract

“…The integral part of such operator L gives rise to the so-called censored processes in G, cf. [6,72]. If L is local outside G, the abstract Cauchy problem in Y for the evolution equation df dt = L o f can be interpreted as the following Cauchy-Dirichlet problem:…”

Section: Notations and Preliminariesmentioning

confidence: 99%

Chernoff Approximation for Semigroups Generated by Killed Feller Processes and Feynman Formulae for Time-Fractional Fokker–Planck–Kolmogorov Equations

Butko

2018

FCAA

View full text Add to dashboard Cite

Semigroups, generated by Feller processes killed upon leaving a given domain, are considered. These semigroups correspond to Cauchy-Dirichlet type initial-exterior value problems in this domain for a class of evolution equations with (possibly non-local) operators. The considered semigroups are approximated by means of the Chernoff theorem. For a class of killed Feller processes, the constructed Chernoff approximation converts into a Feynman formula, i.e. into a limit of n-fold iterated integrals of certain functions as n → ∞. Representations of solutions of evolution equations by Feynman formulae can be used for direct calculations and simulation of underlying stochasstic processes. Further, a method to approximate solutions of time-fractional (including distributed order time-fractional) evolution equations is suggested. This method is based on connections between time-fractional and time-nonfractional evolution equations as well as on Chernoff approximations for the latter ones. Moreover, this method leads to Feynman formulae for solutions of time-fractional evolution equations. To illustrate the method, a class of distributed order time-fractional diffusion equations is considered; Feynman formulae for solutions of the corresponding Cauchy and Cauchy-Dirichlet problems are obtained.

show abstract

“…They prove that if p ∈ (1 + 2α, p * (α)) there exists a unique large solution, whose precise asymptotic behaviour close to the boundary is given by dist(x, ∂Ω) −γ , with γ = 2α/(p − 1). In [10] it is shown existence and uniqueness of large solutions for the problem…”

Section: Introductionmentioning

confidence: 99%

A Nonlocal Operator Breaking the Keller–Osserman Condition

Ferreira

Pérez‐Llanos

2017

Advanced Nonlinear Studies

View full text Add to dashboard Cite

This work is concerned about the existence of solutions to the nonlocal semilinear problem\left\{\begin{aligned} &\displaystyle{-}\int_{{\mathbb{R}}^{N}}J(x-y)(u(y)-u(x% ))\,dy+h(u(x))=f(x),&&\displaystyle x\in\Omega,\\ &\displaystyle u=g,&&\displaystyle x\in{\mathbb{R}}^{N}\setminus\Omega,\end{% aligned}\right.verifying that {\lim_{x\to\partial\Omega,\,x\in\Omega}u(x)=+\infty}, known in the literature as large solutions. We find out that the relation between the diffusion and the absorption term is not enough to ensure such existence, not even assuming that the boundary datum g blows up close to {\partial\Omega}. On the contrary, the role to obtain large solutions is played only by the interior source f, which gives rise to large solutions even without the presence of the absorption. We determine necessary and sufficient conditions on f providing large solutions and compute the blow-up rates of such solutions in terms of h and f. Finally, we also study the uniqueness of large solutions.

show abstract

Large solutions for a class of semilinear integro-differential equations with censored jumps

Cited by 4 publications

References 28 publications

Study on Numerical Solution of a Variable Order Fractional Differential Equation based on Symmetric Algorithm

Study on Numerical Solution of a Variable Order Fractional Differential Equation based on Symmetric Algorithm

Chernoff Approximation for Semigroups Generated by Killed Feller Processes and Feynman Formulae for Time-Fractional Fokker–Planck–Kolmogorov Equations

A Nonlocal Operator Breaking the Keller–Osserman Condition

Contact Info

Product

Resources

About