Abstract
In many engineering problems associated with various physical phenomena, there occurs a necessity of analysis of signals that are described by multidimensional functions of more than one variable such as time t or space coordinates x, y, z. Therefore, in such cases, we should consider dynamical models of two or more dimensions. In this paper, a new two-dimensional (2D) model described by the Roesser type of state-space equations will be considered. In the introduced model, partial differential operators described by the Conformable Fractional Derivative (CFD) definition with respect to the first (horizontal) and second (vertical) variables will be applied. For the model under consideration, the general response formula is derived using the inverse fractional Laplace method. Next, the properties of the solution will be considered. Usefulness of the general response formula will be discussed and illustrated by a numerical example.
Highlights
In the analysis of many real-world problems that arise in modern engineering, considerations cannot be limited to only one dimension, predominately the time variable
A non-integer order Conformable Fractional Derivative (CFD) derivative is more comfortable in theoretical analysis and more efficient in numerical computations; it may be useful in the modelling of multidimensional problems in control theory
We will show that the well-known Cayley–Hamilton theorem is satisfied for transition matrices (25) of CFD pseudo-fractional 2D systems described by the Roesser model (12)
Summary
In the analysis of many real-world problems that arise in modern engineering, considerations cannot be limited to only one dimension, predominately the time variable. In the analysis of many physical phenomena, researchers consider partial differential equations in two dimensions using fractional (non-integer) order partial differential operators. Gazizov et al investigated Lie point symmetries of nonlinear anomalous diffusion equations with time-domain Riemann–Liouville and Caputo fractional derivatives in [23]. A new two-dimensional continuous non-integer order Roesser-type model with partial derivatives described by the Conformable Fractional Derivative definition will be introduced. A non-integer order CFD derivative is more comfortable in theoretical analysis and more efficient in numerical computations; it may be useful in the modelling of multidimensional problems in control theory. To the best knowledge of the author, the two-dimensional continuous CFD pseudo-fractional systems described by the Roesser model have not been considered yet
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