An abstract framework in the numerical solution of boundary value problems for neutral functional differential equations

We consider the numerical solution of boundary value problems for general neutral functional differential equations. The problems are restated in an abstract form and, then, a general discretization of the abstract form is introduced and a convergence analysis of this discretization is developed

The Collocation Method in the Numerical Solution of Boundary Value Problems for Neutral Functional Differential Equations. Part I: Convergence Results

We consider the numerical solution of boundary value problems for general neutral functional differential equations by the collocation method. The collocation method can be applied in two versions: the finite element method and the spectral element method. We give convergence results for the collocation method deduced by the convergence theory developed in [S. Maset, Numer. Math., (2015), pp. 1--31] for a general discretization of an abstract reformulation of the problems. Such convergence results are then applied in Part II [S. Maset, SIAM J. Numer. Anal., 53 (2015), pp. 2794--2821] of this paper to boundary values problems for a particular type of neutral functional differential equations, namely, differential equations with deviating arguments

Existence of multiple positive solutions of a nonlinear arbitrary order boundary value problem with advanced arguments

In this paper, we investigate nonlinear fractional differential equations of arbitrary order with advanced arguments \begin{equation*}\left\{\begin {array}{ll} D^\alpha_{0^+} u(t) +a(t)f(u(\theta(t)))=0,&0<t<1,~n-1<\alpha\le n,\\ u^{(i)}(0)=0,&i=0,1,2,\cdots,n-2,\\ ~[D^\beta_{0^+} u(t)]_{t=1}=0,&1\le \beta\le n-2, \end {array}\right.\end{equation*} where $n>3\,\, (n\in\mathbb{N}),~D^\alpha_{0^+}$ is the standard Riemann-Liouville fractional derivative of order $\alpha,$ $f: [0,\infty)\to [0,\infty),$ $a: [0,1]\to (0,\infty)$ and $\theta: (0,1)\to (0,1]$ are continuous functions. By applying fixed point index theory and Leggett-Williams fixed point theorem, sufficient conditions for the existence of multiple positive solutions to the above boundary value problem are established

On a Hadamard-type fractional turbulent flow model with deviating arguments in a porous medium

In this paper, we concern a Hadamard-type fractional-order turbulent flow model with deviating arguments. By using some standard fixed point theorems, the uniqueness, existence and nonexistence of solutions of the fractional turbulent flow model are investigated. Our results are new and are well illustrated with the aid of three examples

Positive solutions of arbitrary order nonlinear fractional differential equations with advanced arguments

In this paper, we investigate the existence and uniqueness of positive solutions to arbitrary order nonlinear fractional differential equations with advanced arguments. By applying some known fixed point theorems, sufficient conditions for the existence and uniqueness of positive solutions are established

A class of BVPS for first order impulsive functional integro-differential equations with a parameter

This paper is concerned with a class of boundary value problems for the nonlinear impulsive functional integro-differential equations with a parameter by establishing new comparison principles and using the method of upper and lower solutions together with monotone iterative technique. Sufficient conditions are established for the existence of extremal system of solutions for the given problem. Finally, we give an example that illustrates our results

Positive solutions of nonlinear fractional differential equations with integral boundary value conditions

AbstractIn this paper, we consider the existence of positive solutions for a class of nonlinear boundary-value problem of fractional differential equations with integral boundary conditions. Our analysis relies on known Guo–Krasnoselskii fixed point theorem