Qualitative Analysis of Fractional Deterministic and Stochastic Systems

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Probability and Statistics".

Deadline for manuscript submissions: closed (30 September 2021) | Viewed by 7679

Special Issue Editors

Department of Mathematics, Eastern Mediterranean University, Gazimagusa 99628, TRNC, via Mersin 10, Turkey
Interests: fractional deterministic and stochastic differential/difference equations; stochastic calculus; fractional calculus; stability analysis; control theory; numerical methods for fractional differential equations, discrete fractional calculus
Special Issues, Collections and Topics in MDPI journals
Department of Applied Mathematics, Bharathiar University, Coimbatore 641046, India
Interests: dynamical systems; fractional differential equations; systems and control theory; differential/integral equations; numerical methods for fractional models; stability analysis
Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
Interests: wavelets; fractals; fractional and stochastic equations; numerical and computational methods; mathematical physics; nonlinear systems; artificial intelligence
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Over the years, many results have emerged on the theory and applications of stochastic differential equations. In particular, fractional stochastic differential equations, which are a generalization of differential equations by the use of fractional and stochastic calculus, are more popular due to their applications in mathematical finance, biology, biomedicine, and so on. The research area of stochastic differential equations has occupied one of the primary areas of numerical and applied mathematics for the last three decades, providing new techniques for analyzing complex systems. Thus, it is of great importance to design stochastic effects in the study of fractional-order dynamical systems.

This Special Issue invites original contributions that cover recent advances in the theory and applications of fractional deterministic and stochastic differential equations. The main focus of this Special Issue is to describe and analyze new methods and techniques for solving nonlinear dynamical systems described by fractional deterministic and stochastic differential equations that examine important applications. We kindly invite strong and interesting contributions that provide original results obtained from modern computational techniques of theoretical, experimental, and applied aspects of both deterministic and stochastic fractional dynamic systems. We also strongly encourage young researchers/PhD students who have achieved exciting results under the supervision and guidance of their scientific advisors to submit their works to this Special Issue.

Potential topics include, but are not limited to:

  • Stochastic fractional differential equations;
  • Stochastic differential and partial differential equations (SPDEs);
  • A class of backward stochastic differential equations with a conformable derivative;
  • Stochastic differential equations with fractional Brownian motion;
  • Mean-field stochastic equations;
  • Instantaneous and non-instantaneous impulsive deterministic and fractional differential equations;
  • Fractional reaction–diffusion and Navier–Stokes equation;
  • Continuous and discrete fractional systems with randomness;
  • Control and optimization for fractional deterministic and stochastic systems;
  • Numerical methods for fractional-order stochastic systems;
  • Stability analysis of fractional deterministic and stochastic systems;
  • Optimal control;
  • Fractional q-calculus;
  • Fractional time-delay systems;
  • Fractional Brownian process;
  • Applications of fractional stochastic differential equations in finance and engineering;
  • Stochastic analysis in finance;
  • Stochastic analysis in biology and biomedicine.

Prof. Dr. Nazim Mahmudov
Prof. Dr. Rathinasamy Sakthivel
Prof. Dr. Carlo Cattani
Guest Editors

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Keywords

  • fractional stochastic equations
  • fractional calculus
  • stability, optimal control
  • numerical methods

Published Papers (4 papers)

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Research

16 pages, 314 KiB  
Article
Ulam–Hyers Stability and Uniqueness for Nonlinear Sequential Fractional Differential Equations Involving Integral Boundary Conditions
Fractal Fract. 2021, 5(4), 235; https://doi.org/10.3390/fractalfract5040235 - 19 Nov 2021
Cited by 11 | Viewed by 1231
Abstract
Fractional-order boundary value problems are used to model certain phenomena in chemistry, physics, biology, and engineering. However, some of these models do not meet the existence and uniqueness required in the mainstream of mathematical processes. Therefore, in this paper, the existence, stability, and [...] Read more.
Fractional-order boundary value problems are used to model certain phenomena in chemistry, physics, biology, and engineering. However, some of these models do not meet the existence and uniqueness required in the mainstream of mathematical processes. Therefore, in this paper, the existence, stability, and uniqueness for the solution of the coupled system of the Caputo-type sequential fractional differential equation, involving integral boundary conditions, was discussed, and investigated. Leray–Schauder’s alternative was applied to derive the existence of the solution, while Banach’s contraction principle was used to examine the uniqueness of the solution. Moreover, Ulam–Hyers stability of the presented system was investigated. It was found that the theoretical-related aspects (existence, uniqueness, and stability) that were examined for the governing system were satisfactory. Finally, an example was given to illustrate and examine certain related aspects. Full article
34 pages, 2067 KiB  
Article
Hexagonal Grid Computation of the Derivatives of the Solution to the Heat Equation by Using Fourth-Order Accurate Two-Stage Implicit Methods
Fractal Fract. 2021, 5(4), 203; https://doi.org/10.3390/fractalfract5040203 - 07 Nov 2021
Cited by 7 | Viewed by 1448
Abstract
We give fourth-order accurate implicit methods for the computation of the first-order spatial derivatives and second-order mixed derivatives involving the time derivative of the solution of first type boundary value problem of two dimensional heat equation. The methods are constructed based on two [...] Read more.
We give fourth-order accurate implicit methods for the computation of the first-order spatial derivatives and second-order mixed derivatives involving the time derivative of the solution of first type boundary value problem of two dimensional heat equation. The methods are constructed based on two stages: At the first stage of the methods, the solution and its derivative with respect to time variable are approximated by using the implicit scheme in Buranay and Arshad in 2020. Therefore, Oh4+τ of convergence on constructed hexagonal grids is obtained that the step sizes in the space variables x1, x2 and in time variable are indicated by h, 32h and τ, respectively. Special difference boundary value problems on hexagonal grids are constructed at the second stages to approximate the first order spatial derivatives and the second order mixed derivatives of the solution. Further, Oh4+τ order of uniform convergence of these schemes are shown for r=ωτh2116, ω>0. Additionally, the methods are applied on two sample problems. Full article
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24 pages, 386 KiB  
Article
Finite-Approximate Controllability of Riemann–Liouville Fractional Evolution Systems via Resolvent-Like Operators
Fractal Fract. 2021, 5(4), 199; https://doi.org/10.3390/fractalfract5040199 - 04 Nov 2021
Cited by 7 | Viewed by 1316
Abstract
This paper presents a variational method for studying approximate controllability and infinite-dimensional exact controllability (finite-approximate controllability) for Riemann–Liouville fractional linear/semilinear evolution equations in Hilbert spaces. A useful criterion for finite-approximate controllability of Riemann–Liouville fractional linear evolution equations is formulated in terms of resolvent-like [...] Read more.
This paper presents a variational method for studying approximate controllability and infinite-dimensional exact controllability (finite-approximate controllability) for Riemann–Liouville fractional linear/semilinear evolution equations in Hilbert spaces. A useful criterion for finite-approximate controllability of Riemann–Liouville fractional linear evolution equations is formulated in terms of resolvent-like operators. We also find that such a control provides finite-dimensional exact controllability in addition to the approximate controllability requirement. Assuming the finite-approximate controllability of the corresponding linearized RL fractional evolution equation, we obtain sufficient conditions for finite-approximate controllability of the semilinear RL fractional evolution equation under natural conditions. The results are a generalization and continuation of recent results on this subject. Applications to fractional heat equations are considered. Full article
26 pages, 1014 KiB  
Article
Two Stage Implicit Method on Hexagonal Grids for Approximating the First Derivatives of the Solution to the Heat Equation
Fractal Fract. 2021, 5(1), 19; https://doi.org/10.3390/fractalfract5010019 - 26 Feb 2021
Cited by 6 | Viewed by 2541
Abstract
The first type of boundary value problem for the heat equation on a rectangle is considered. We propose a two stage implicit method for the approximation of the first order derivatives of the solution with respect to the spatial variables. To approximate the [...] Read more.
The first type of boundary value problem for the heat equation on a rectangle is considered. We propose a two stage implicit method for the approximation of the first order derivatives of the solution with respect to the spatial variables. To approximate the solution at the first stage, the unconditionally stable two layer implicit method on hexagonal grids given by Buranay and Arshad in 2020 is used which converges with Oh2+τ2 of accuracy on the grids. Here, h and 32h are the step sizes in space variables x1 and x2, respectively and τ is the step size in time. At the second stage, we propose special difference boundary value problems on hexagonal grids for the approximation of first derivatives with respect to spatial variables of which the boundary conditions are defined by using the obtained solution from the first stage. It is proved that the given schemes in the difference problems are unconditionally stable. Further, for r=ωτh237, uniform convergence of the solution of the constructed special difference boundary value problems to the corresponding exact derivatives on hexagonal grids with order Oh2+τ2 is shown. Finally, the method is applied on a test problem and the numerical results are presented through tables and figures. Full article
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