16 problems

Let $X(-\infty,0)$ and $X(-\infty,T)$ denote the trajectory-history spaces at times $0$ and $T$, respectively, for a linear time-periodic fractional-order differential equation, an…

Existence conjecture. Assuming only $(C_p^*)$ should suffice to establish existence of a solution to the associated problem.

Let $K=2,3,\dots$ and consider the initial-boundary value problem for a coupled subdiffusion system with $K$ components: … Assume $1\ge\alpha_1\ge\cdots\ge\alpha_K>0$, $\alpha_K<1$…

Consider the three-component coupled subdiffusion system in the case $v_0=w_0\equiv0$ and $u_0\not\equiv0$, with fractional orders $\alpha,\beta,\gamma$. In the numerical examples,…

Consider a coupled subdiffusion system with components having fractional derivative orders, and suppose some initial values may vanish identically. The lowest active fractional ord…

Fractional differential equation systems may be nonlinear and non-autonomous, and an inertial manifold is a reduced finite-dimensional manifold intended to capture their long-time…

Instability conjecture. The regions bounded by the curves $Gamma_1$ and $Gamma_2$ are unstable.

The paper studies fractional partial differential equations and gives a general solution in equation … can be shown to hold for a larger class of fractional partial differential eq…

Regularity conjecture. If $h\in C[0,1]$ or $h\in L^1(0,1)$, then the solution $u$ belongs to $C^1_{2-\alpha}[0,1]$. Moreover, if $h\in C(0,1]$ satisfies $(H)$, then the solution…

Source-type solution conjecture. This function is a source-type solution of the nonlinear time-fractional equation referred to as equation in the source.

The fractional diffusion-wave equation involves an orbit function $\bm\rho(t)\in\mathbb R^3$ and a temporal function $h(t)$ describing the moving source. Moment-theory conjecture.…

Effectiveness conjecture. The proposed method can be equally effective for a very wide range of FDE optimization problems.

Regularity conjecture. If the forcing $\vec{F}$ is nonzero but sufficiently regular, then the same regularity estimate holds as in the case $\vec{F}\equiv\vec{0}$; in particular, o…

Singular-point conjecture. There exist singular points in the trajectory if and only if

The numerical method is applied to the time-fractional porous medium equation, with fractional order $[?]$ and convergence order measured as the number of subdivisions of the inter…

Audounet–Matignon–Montseny conjecture. The local stability of $x^{*}=0$ is governed by the global stability of the linearized system as follows: (i) $x^{*}=0$ is locally asymptotic…