SOME ASPECTS OF SOLUTIONS OF SPACE-TIME FRACTIONAL STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH OSGOOD CONDITION

In this paper we discuss the following problem with additive noise, \[\begin{cases} \frac{\partial^{\beta} u(t,x) }{\partial t}=-(-\triangle)^{\frac{\alpha}{2}} u(t,x)+b(u(t,x))+\sigma\dot{W}(t,x),~~t>0, \\u(0,x)=u_{0}(x),\end{cases},\] where $\alpha \in(0,2) $ and $ \beta \in (0,1)$, the fractional time derivative is in the sense of Caputo, $-(-\Delta)^{\frac{\alpha}{2}}$ is the fractional Laplacian, $\sigma$ is a positive parameter, $\dot{W}$ is a space-time white noise, $u_0(x)$ is assumed to be non-negative, continuous and bounded. We study first the equation on $[0,\,1]$ with homogeneous Drichlet boundary condition and show that the solution of the equation blows up in finite time if and only if $b$ satisfies the Osgood condition, \[ \int_{c}^{\infty} \frac{ds}{b(s)} <\infty \] for some constant $c>0$. We then consider the same equation on the whole line and show that the above Osgood condition is satisfied whenever the solution of the equation blows up.

Steady-State Harmonic Vibrations of Viscoelastic Timoshenko Beams with Fractional Derivative Damping Models

Due to growing demands on newly developed products concerning their weight, sound emission, etc., advanced materials are introduced in the product designs. The modeling of these materials is an important task, and a very promising approach to capture the viscoelastic behavior of a broad class of materials are fractional time derivative operators, since only a small number of parameters is required to fit measurement data. The fractional differential operator in the constitutive equations introduces additional challenges in the solution process of structural models, e.g., beams or plates. Therefore, a highly efficient computational method called Numerical Assembly Technique is proposed in this paper to tackle general beam vibration problems governed by the Timoshenko beam theory and the fractional Zener material model. A general framework is presented, which allows for the modeling of multi-span beams with general linear supports, rigid attachments, and arbitrarily distributed force and moment loading. The efficiency and accuracy of the method is shown in comparison to the Finite Element Method. Additionally, a validation with experimental results for beam systems made of steel and polyvinyl chloride is presented, to illustrate the advantages of the proposed method and the material model.

The Effects of Fractional Time Derivatives in Porothermoelastic Materials Using Finite Element Method

In this work, a new model for porothermoelastic waves under a fractional time derivative and two time delays is utilized to study temperature increments, stress and the displacement components of the solid and fluid phases in porothermoelastic media. The governing equations are presented under Lord–Shulman theory with thermal relaxation time. The finite element method has been adopted to solve these equations due to the complex formulations of this problem. The effects of fractional parameter and porosity in porothermoelastic media have been studied. The numerical outcomes for the temperatures, the stresses and the displacement of the fluid and the solid are presented graphically. These results will allow future studies to gain a detailed insight into non-simple porothermoelasticity with various phases.