Related papers: Variational methods for fractional $q$-Sturm--Liou…
A matrix method for the solution of direct fractional Sturm-Liouville problems on bounded domain is proposed where the fractional derivative is defined in the Riesz sense. The scheme is based on the application of the Galerkin spectral…
We mainly research the Liouville type problem for the stationary Navier-Stokes equations (including the fractional case) in $\mathbb{R}^3$. We first establish a new formula for the Dirichlet integral of solutions and show that the globally…
In many cases, groundwater flow in an unconfined aquifer can be simplified to a one-dimensional Sturm-Liouville model of the form: \begin{equation*} x''(t)+\lambda x(t)=h(t)+\varepsilon f(x(t)),\hspace{.1in}t\in(0,\pi) \end{equation*}…
In this study, we give a regular fractional Sturm Liouville problem for diffusion operator (FSLPDO), research the spectral properties of the eigenfunctions and eigenvalues of the diffusion operator. We show that the eigenvalues and…
We study conformal mappings from the unit disk to circular-arc quadrilaterals with four right angles. The problem is reduced to a Sturm-Liouville boundary value problem on a real interval, with a nonlinear boundary condition, in which the…
In this paper, we consider forward and inverse problems for subdiffusion equations with time-dependent coefficients. The fractional derivative is taken in the sense of Riemann-Liouville. Using the classical Fourier method, the theorem of…
We consider the linear eigenvalue problem \tag{1} -u" = \lambda u, \quad \text{on $(-1,1)$}, where $\lambda \in \mathbb{R}$, together with the general multi-point boundary conditions \tag{2} \alpha_0^\pm u(\pm 1) + \beta_0^\pm u'(\pm 1) =…
(Draft 3) A generalized differential operator on the real line is defined by means of a limiting process. These generalized derivatives include, as a special case, the classical derivative and current studies of fractional differential…
In this paper, we first deal with the general fractional derivatives of arbitrary order defined in the Riemann-Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of…
Paper deals with the singular Sturm-Liouville expressions $$l(y) = -(py')' + qy$$ on a finite interval with coefficients $$q = Q', \quad 1/p, Q/p, Q^2/p \in L_1,$$ where derivative of the function $Q$ is understood in the sense of…
This paper deals with the Sturm-Liouville problem that feature distribution potential, polynomial dependence on the spectral parameter in the first boundary condition, and analytical dependence, in the second one. We study an inverse…
Inverse problems of recovering the coefficients of Sturm-Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: (1) from the sequences of eigenvalues and norming constants; (2)…
In this paper, we study problems of minimization of a functional depending on the fractional Caputo derivative of order $0<\alpha \leq 1$ and the fractional Riemann- Liouville integral of order $\beta > 0$ at fixed endpoints. A fractional…
The author (Bull. Math. Anal. App. 6(4)(2014):1-15), introduced a new fractional derivative, \[{}^\rho \mathcal{D}_a^\alpha f (x) = \frac{\rho^{\alpha-n+1}}{\Gamma({n-\alpha})} \, \bigg(x^{1-\rho} \,\frac{d}{dx}\bigg)^n \int^x_a…
Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. This paper investigates two typical applications:…
We establish a Liouville type result for a backward global solution to the Navier-Stokes equations in the half plane with the no-slip boundary condition. No assumptions on spatial decay for the vorticity nor the velocity field are imposed.…
Summary]{In this paper, we study problems of minimization of a functional depending on the fractional Caputo derivative of order $0<\alpha \leq 1$ and the fractional Riemann- Liouville integral of order $\beta > 0$ at fixed endpoints. A…
This paper is a generalization of the author's previous work [14]. We extend the argument [14] for any uniformly elliptic operator in divergence form $\mathcal{L}u=-div(A(x)\nabla u)$, more precisely, we study a fractional type degenerate…
The present work was primarily motivated by our findings in the literature of some flaws within the proof of the second-order Legendre necessary optimality condition for fractional calculus of variations problems. Therefore we were eager to…
Initialization of fractional differential equations remains an ongoing problem. In recent years, the initialization function approach and the infinite state approach provide two effective ways to deal with this problem. The purpose of this…