Special Session 67: Fractional Differential Equations: Theory, Methods and Applications

Stability analysis of of Fractional Reaction Diffusion Systems

Sofwah Ahmad Khalifa University United Arab Emirates Co-Author(s):    Szymon Cygan and Grzegorz Karch

Abstract: In the talk, results on the stability of solutions to general evolution equations with the Caputo fractional-in-time derivatives will be presented. Our results can be applied, for example, either to systems of fractional differential equations or to general reaction-diffusion systems on bounded domains with Neumann boundary conditions. In particular, we provide an extended analysis of the so-called linearization principle (i.e. when the linear stability/instability implies the non-linear stability/instability). These results have important biological implications including Turing instability criteria.

Applications of FDE to Real-World Problems

Ricardo Almeida University of Aveiro Portugal Co-Author(s):

Abstract: This talk is devoted to the study of nonlinear fractional differential equations involving Caputo-type fractional derivatives with respect to another function. We establish existence and uniqueness results for these equations using some standard fixed point theorems. Furthermore, we present several applications of these results, specifically in optimal control theory, population growth models, and epidemiological models.

ON A NEW FORMULATION OF THE INVERSE PROBLEM OF DETERMINING THE ORDER OF FRACTIONAL DERIVATIVES IN PARTIAL DERIVATIVE EQUATIONS

Ravshan R. Ashurov Institute of Mathematics, Academy of Scienses of Uzbekistan Uzbekistan Co-Author(s):    Ravshan Ashurov

Abstract: The inverse problem of determining the unknown order of a fractional derivative in differential equations has been actively studied by many specialists. A number of interesting results have been obtained that have a certain applied significance. By now, the authors have investigated various modifications of this inverse problem: determining the order of the derivative or, along with the unknown order, determining some other unknown parameter or function included in the initial-boundary value problem under consideration. Analyzing the known results, we can conclude that in all these works, firstly, only the subdiffusion equation was considered and, secondly, the elliptic part of these equations has a discrete spectrum, and the authors were able to prove only the uniqueness of the solution to the inverse problem under consideration. This report will give a brief overview of the most interesting works in this area, and will propose a new formulation and methods for solving these inverse problems. It will be proved that in the new formulation the solutions of inverse problems are not only unique, but also exist. In this case, not only the subdiffusion equations will be considered, and the elliptic parts of the equation can also have a continuous spectrum.

A fractional Laplacian and its extension problem

Salem Ben Said United Arab Emirates University United Arab Emirates Co-Author(s):

Abstract: In this paper, we establish four equivalent characterizations of the fraction Laplacian operator $(-\Vert x\Vert \Delta_k)^{\sigmaup}$ with $0

On the solvability of some inverse problems for a high-order nonlocal parabolic equation with multiple involution

Meiirkhan Borikhanov Khoja Akhmet Yassawi International Kazakh--Turkish University Kazakhstan Co-Author(s):    Batirkhan Turmetov

Abstract: This work investigated the solvability of a some inverse issues for a high-order parabolic equations nonlocal analogue. Non local equivalent of the biharmonic operator has been developed for this purpose. Transformations of the involution type were used in the definition of this operator. The eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator have been studied in a parallelepiped. For this particular problem, the eigenvalues and eigenfunctions were explicitly constructed, and the proof of the system`s completeness was presented. We examined two different kinds of inverse problems that involved solving the equation and its right-hand side. Using the Fourier variable separation approach or reducing it to an integral equation, the two problems` right hand terms that depended on the spatial and temporal variables were found. The theorems for the existence and uniqueness of the solution were proved.

Blowing-up Solution of a System of Fractional Differential Equations with Variable Order

Muhammad R Fadillah Khalifa University United Arab Emirates Co-Author(s):    Mokhtar Kirane

Abstract: We investigated the necessary condition for the following nonlinear system of fractional differential equations to have a blowing-up solution in finite time $$u'(t) + D_{0|t}^{\alpha(t)} (u(t) - u_0) = |v(t)|^q, t > 0, q > 1 ,$$ and $$v'(t) + D_{0|t}^{\beta(t)} (v(t) - v_0) = |u(t)|^p, t > 0, p > 1$$ where $u(0) = u_0 > 0, v(0) = v_0 > 0, \alpha, \beta \in C^1[0,+\infty)$ such that $0 < \alpha_m \leq \alpha(t) \leq \alpha_M < 1$ and $0 < \beta_m \leq \beta(t) \leq \beta_M < 1$ and $D_{0|t}^{\rho(t)}$ is Riemann-Liouville derivative of order $\rho(t)$. Our method was to use a suitable test function in the weak functions.

Blow-up and lifespan estimate of the solution of the wave equation with critical damping

Mohamed Ali Hamza Imam Abdulrahman Bin Faisal University Saudi Arabia Co-Author(s):

Abstract: In this talk we discuss the wave equation in the critical damping case. In the first part, we study the influence of the damping term in the global dynamics of the solution in the case of time derivative nonlinearity, under the assumption of small initial data. The second part will be devoted for the same problem in the case whith combined nonlinearities.

Positivity properties of discrete time-fractional operators on uniform and nonuniform meshes

Samir Karaa Sultan Qaboos University Oman Co-Author(s):    Samir Karaa

Abstract: The positive definiteness of discrete convolution kernels plays an important role in the stability analysis of time-stepping schemes for nonlocal models. Specifically, when these kernels are generated by convex sequences, their positivity can be verified by applying a classical result due to Zygmund. This talk has two main focuses. First, we improve Zygmund`s result and extend its validity to sequences that are almost convex. Next, we establish a more general inequality applicable to sequences that are nearly convex. Secondly, we consider convolution kernels on nonuniform grids and generalize the previous bounds. Our results are then applied to demonstrate the positivity properties of commonly used approximations for fractional integral and differential operators.

Optimality conditions for control problems involving generalized fractional derivatives

Natalia Martins University of Aveiro Portugal Co-Author(s):    Fatima Cruz and Ricardo Almeida

Abstract: In this talk, we extend fractional optimal control theory by proving a version of Pontryagin`s Maximum Principle and establishing a sufficient optimality condition for an optimal control problem. The dynamical system constraint in this problem is governed by a generalized form of a fractional derivative: the left-sided Caputo distributed-order fractional derivative with an arbitrary kernel. This approach provides a more versatile representation of dynamic processes, accommodating a broader range of memory effects and hereditary properties inherent in diverse physical, biological, and engineering systems.

Distributed order integration and differentiation operators

Arsen Pskhu Institute of Applied Mathematics and Automation KBSC RAS Russia Co-Author(s):

Abstract: The report discusses the properties of distributed order integration and differentiation operators. In particular, representations in terms of convolution are obtained, Sonin pairs for the corresponding kernels are constructed, extremum principles are proved, composition laws are found, including inversion formulas and Newton-Leibniz formulas.

Inverse boundary value problem with integral condition for a hyperbolic equation of fractional order

Makhmud A. Sadybekov Institute of Mathematics and Mathematical Modeling Kazakhstan Co-Author(s):    Danabekova Moldir and Sairam Nurgul

Abstract: In this report we consider initial-boundary value problems for a hyperbolic equation (with a fractional time Caputo derivative): $$D_{0t}^{\alpha} u(x,t) - u_{xx}(x,t) + q(x)u(x,t) = f(x,t), \ 1 0.$$ In the case when the boundary conditions are strongly regular, the system of root vectors of this spectral problem forms a Riesz basis in \(L_2(0,1)\). The Fourier method can be implemented to solve the original problem. However, when the boundary conditions are non strongly regular, the system of root vectors of the spectral problem may not form an unconditional basis, preventing the use of the Fourier method. These are the types of problems that will be presented in this report.

Inverse coefficient problems for fractional heat equations

Durvudkhan Suragan Nazarbayev University Kazakhstan Co-Author(s):    Gulaiym Oralsyn

Abstract: In this talk, we discuss inverse problems related to determining the time-dependent coefficient and unknown source function of fractional heat equations. Our approach shows that having just one set of data at an observation point ensures the existence of a weak solution for the inverse problem. Furthermore, if there is an additional datum at the observation point, it leads to a specific formula for the time-dependent source coefficient. Moreover, we investigate inverse problems involving non-local data and recovering the space-dependent source function of the fractional heat equation. We also discuss extensions of these results to time and space fractional heat equations. The talk is based on our recent results.

Nonexistence of global solutions for an inhomogeneous semiliniar heat equation

Nurdaulet Tobakhanov Nazarbayev University Kazakhstan Co-Author(s):    -

Abstract: This report is devoted to the nonexistence of global weak solutions to the inhomogeneous semilinear heat equations with forcing terms on exterior domains. We investigate the critical behavior of solutions to the semilinear biharmonic heat equation ($|u|^p$ and $I_{0+}^\gamma\left(|u|^p\right)$) with forcing term $f(x)$. By employing a method of test function, we derive blow-up results for the critical case in the sense of Fujita.

On the solvability of some inverse problems for a high-order nonlocal parabolic equation with multiple involution

Batirkhan Turmetov Khoja Akhmet Yassawi International Kazakh-Turkish University Kazakhstan Co-Author(s):    Borikhanov M

Abstract: This work investigated the solvability of a some inverse issues for a high-order parabolic equations nonlocal analogue. Non local equivalent of the biharmonic operator has been developed for this purpose. Transformations of the involution type were used in the definition of this operator. The eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator have been studied in a parallelepiped. For this particular problem, the eigenvalues and eigenfunctions were explicitly constructed, and the proof of the system`s completeness was presented. We examined two different kinds of inverse problems that involved solving the equation and its right-hand side. Using the Fourier variable separation approach or reducing it to an integral equation, the two problems` right hand terms that depended on the spatial and temporal variables were found. The theorems for the existence and uniqueness of the solution were proved.

Determination of the flux terms in a time fractional viscoelastic equation

Suleyman Ulusoy American University of Ras Al Khaimah United Arab Emirates Co-Author(s):    Mohamed BenSalah, Salih Tatar, Masahiro Yamamoto

Abstract: We will introduce our results for the flux identification problem for a nonlinear time-fractional viscoelastic equation with a general source function based on the boundary measurements. We prove that the direct problem is well-posed, i.e., the solution exists, unique and depends continuously on the heat flux. Then the Frechet differentiability of the cost functional is proved. The Conjugate Gradient Algorithm, based on the gradient formula for the cost functional, is proposed for numerical solution of the inverse flux problem. The numerical examples, both with noise-free and noisy data, provide a clear demonstration of the applicability and accuracy of the proposed method. If time permits, we will discuss further directions and ongoing investigations.