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An analytical solution to the problem of time-fractional heat conduction in a composite sphere

Kukla S., Siedlecka U.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2017

Vol. 65, nr 2

179--186

An analytical solution to the problem of time-fractional heat conduction in a sphere consisting of an inner solid sphere and concentric spherical layers is presented. In the heat conduction equation, the Caputo time-derivative of fractional order and the Robin boundary condition at the outer surface of the sphere are assumed. The spherical layers are characterized by different material properties and perfect thermal contact is assumed between the layers. The analytical solution to the problem of heat conduction in the sphere for time-dependent surrounding temperature and for time-space-dependent volumetric heat source is derived. Numerical examples are presented to show the effect of the harmonically varying intensity of the heat source and the harmonically varying surrounding temperature on the temperature in the sphere for different orders of the Caputo time-derivative.

Controllability criteria for time-delay fractional systems with a retarded state

Sikora B.

International Journal of Applied Mathematics and Computer Science

2016

Vol. 26, no. 3

521--531

The paper is concerned with time-delay linear fractional systems with multiple delays in the state. A formula for the solution of the discussed systems is presented and derived using the Laplace transform. Definitions of relative controllability with and without constraints for linear fractional systems with delays in the state are formulated. Relative controllability, both with and without constraints imposed on control values, is discussed. Various types of necessary and sufficient conditions for relative controllability and relative U-controllability are established and proved. Numerical examples illustrate the obtained theoretical results.

Global existence and stability results for partial fractional random differential equations

Abbas S., Albarakati W., Benchohra M., Hilal E.

Journal of Applied Analysis

2015

Vol. 21, nr 2

79--87

We investigate some existence and stability results for the Darboux problem of partial fractional random differential equations in Banach spaces. Our existence results are based upon some fixed point theorems.