Solutions for a category of singular nonlinear fractional differential equations subject to integral boundary conditions

We concentrate on a category of singular boundary value problems of fractional differential equations with integral boundary conditions, in which the nonlinear function f is singular at $t=0$ t = 0 , 1. We use Banach’s fixed-point theorem and Hölder’s inequality to verify the existence and uniqueness of a solution. Moreover, also we prove the existence of solutions by Krasnoselskii’s and Schaefer’s fixed point theorems.

Existence of positive solutions of Hadamard fractional defferential equations with integral boundary conditions

In this paper, We study the existence of positive solutions for Hadamard fractional differential equations with integral conditions. We employ Avery-Peterson fixed point theorem and properties of Green's function to show the existence of positive solutions of our problem. Furthermore, we present an example to illustrate our main result.

On a class of differential inclusions in the frame of generalized Hilfer fractional derivative

<abstract><p>In the present paper, we extend and develop a qualitative analysis for a class of nonlinear fractional inclusion problems subjected to nonlocal integral boundary conditions (nonlocal IBC) under the $ \varphi $-Hilfer operator. Both claims of convex valued and nonconvex valued right-hand sides are investigated. The obtained existence results of the proposed problem are new in the frame of a $ \varphi $-Hilfer fractional derivative with nonlocal IBC, which are derived via the fixed point theorems (FPT's) for set-valued analysis. Eventually, we give some illustrative examples for the acquired results.</p></abstract>

Rothe–Legendre pseudospectral method for a semilinear pseudoparabolic equation with nonclassical boundary condition

A semilinear pseudoparabolic equation with nonlocal integral boundary conditions is studied in the present paper. Using Rothe method, which is based on backward Euler finitedifference schema, we designed a suitable semidiscretization in time to approximate the original problem by a sequence of standard elliptic problems. The questions of convergence of the approximation scheme as well as the existence and uniqueness of the solution are investigated. Moreover, the Legendre pseudospectral method is employed to discretize the time-discrete approximation scheme in the space direction. The main advantage of the proposed approach lies in the fact that the full-discretization schema leads to a symmetric linear algebraic system, which may be useful for theoretical and practical reasons. Finally, numerical experiments are included to illustrate the effectiveness and robustness of the presented algorithm.

Modeling the Effect of Low Pt loading Cathode Catalyst Layer in Polymer Electrolyte Fuel Cells: Part I. Model Formulation and Validation

A model for the cathode catalyst layer (CL) is presented, which is validated with previous experimental data in terms of both performance and oxygen transport resistance. The model includes a 1D macroscopic description of proton, electron and oxygen transport across the thickness, which is locally coupled to a 1D microscopic model that describes oxygen transport toward Pt sites. Oxygen transport from the channel to the CL and ionic transport across the membrane are incorporated through integral boundary conditions. The model is complemented with data of effective transport and electrochemical properties extracted from multiple experimental works. The results show that the contribution of the thin ionomer film and Pt/ionomer interface increases with the inverse of the roughness factor. Whereas the contribution of the water film and the water/ionomer interface increases with the ratio between the geometric area and the surface area of active ionomer. Moreover, it is found that CLs diluted with bare carbon provide lower performance than non-diluted samples due to their lower electrochemical surface area and larger local oxygen transport resistance. Optimized design of non-diluted samples, with a good distribution of the overall oxygen flux among Pt sites, is critical to reduce mass transport losses at low Pt loading.

Positive solution of Hilfer fractional differential equations with integral boundary conditions

In this article, we have interested the study of the existence and uniqueness of positive solutions of the first-order nonlinear Hilfer fractional differential equation $$D_{0^{+}}^{\alpha ,\beta }y(t)=f(t,y(t)),\text{ }0<t\leq 1,$$ with the integral boundary condition $$I_{0^{+}}^{1-\gamma }y(0)=\lambda \int_{0}^{1}y(s)ds+d,$$ where $0<\alpha \leq 1,$ $0\leq \beta \leq 1,$ $\lambda \geq 0,$ $d\in \mathbb{R}^{+},$ and $D_{0^{+}}^{\alpha ,\beta }$, $I_{0^{+}}^{1-\gamma }$ are fractional ope\-rators in the Hilfer, Riemann-Liouville concepts, respectively. In this approach, we transform the given fractional differential equation into an equivalent integral equation. Then we establish sufficient conditions and employ the Schauder fixed point theorem and the method of upper and lower solutions to obtain the existence of a positive solution of a given problem. We also use the Banach contraction principle theorem to show the existence of a unique positive solution. The result of existence obtained by structure the upper and lower control functions of the nonlinear term is without any monotonous conditions. Finally, an example is presented to show the effectiveness of our main results.