Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels

In this article, we study optimal control problems driven by space-time fractional parabolic equations (STFPEs) of Sturm-Liouville type on metric graphs, considering both distributed and boundary controls. The considered problem involves the fractional time derivative in the Caputo sense, and the spatial fractional derivative is of the Sturm-Liouville type which is the composition of the right-sided Caputo fractional derivative and left-sided Riemann-Liouville fractional derivative. By introducing the appropriate function spaces for the involved fractional operators in both the time and spatial variables, we first establish well-posedness results for the weak solution for the governing STFPEs using the Galerkin approximation method. Then, we prove that the associated optimal control problem possesses a unique optimal solution. Moreover, to characterize the control variables, we develop an adjoint problem involving a right-sided Caputo fractional derivative in time and derive optimality conditions for the considered problem, using the Lagrange multiplier method. Finally, we propose a difference scheme to find the numerical solution of the considered optimal control problem, particularly on a metric star graph by approximating the Caputo time derivative using the L1 method and spatial fractional derivative with the Grünwald-Letnikov formula. The performance and the accuracy of the proposed difference scheme have been illustrated via an example.

In this manuscript, we have proposed a comparison based on newly defined fractional derivative operators which are called as Caputo-Fabrizio (CF) and Atangana-Baleanu (AB). In 2015, Caputo and Fabrizio established a new fractional operator by using exponential kernel. After one year, Atangana and Baleanu recommended a different-type fractional operator that uses the generalized Mittag-Leffler function (MLF). Many real-life problems can be modelled and can be solved by numerical-analytical solution methods which are derived with these operators. In this paper, we suggest an approximate solution method for PDEs of fractional order by using the mentioned operators. We consider the Laplace homotopy transformation method (LHTM) which is the combination of standard homotopy technique (SHT) and Laplace transformation method (LTM). In this study, we aim to demonstrate the effectiveness of the aforementioned method by comparing the solutions we have achieved with the exact solutions. Furthermore, by constructing the error analysis, we test the practicability and usefulness of the method.

This paper focuses on the bounds of the Lyapunov exponents for fractional differential systems, where the fractional derivatives are Riemann–Liouville and Caputo fractional derivatives with the exponential kernel. First, the essential properties of fractional integral and derivatives with the exponential kernel are given. Then the continuous dependence of solutions on the initial value problems of some particular parameters is studied. On these bases, the bounds of Lyapunov exponents are estimated. Finally, the theoretical results are illustrated by numerical simulations.

We present an overview of articles devoted to the analysis of dynamic behavior of viscoelastic rods, whose features are described by rheological models involving fractional derivatives or operators of two different orders. We investigate ten rheological models: the generalized viscoelastic Kelvin-Voigt, Maxwell, Zener models and other models containing fractional derivatives or fractional operators of two different fractional orders. For each of the above-mentioned rheological models, we carry out a comparative analysis between the behavior of its rheological and dynamic characteristics. Vector diagrams are used as the rheological characteristics, but the roots of the characteristic equations in the problems of longitudinal vibrations of rods, the velocities and coefficients of attenuation in the problems of propagation of harmonic waves, and the stress function in the problems of transient wave propagation are considered as the dynamic characteristics. The analysis of the rheological and dynamic characteristics shows that, depending on the relative magnitudes of the orders of the fractional derivatives and fractional operators entering into the rheological model, some of the models considered may describe both the wave and diffusion processes occurring in mechanical systems, but others describe only wave or only diffusion phenomena. The parallels between the behavior of the vector diagrams, the roots of the characteristic equations, the velocities and coefficients of attenuation, and the stress functions are revealed. We also show the analogy in the behavior of viscoelastic rods, whose rheological features are described by the models with fractional time derivatives and ordinary time derivatives.

New Illustrative Applications of Integral Transforms to Financial Models with Different Fractional Derivatives

We derive a time-domain differential equation for modelling seismic wave propagation in constant-<it>Q</it> viscoelastic media based on fractional spatial derivatives, specifically Laplacian differential operators of fractional order. The stress–strain relation is derived from the classical equation expressed in terms of fractional time derivatives. The new formulation has the advantage of not requiring additional field variables that increase the computer time and storage significantly. The spatial derivatives are calculated with a generalization of the Fourier pseudospectral method to the fractional-derivative case. The accuracy of the numerical solution is verified against an analytical solution in a homogeneous medium. An example shows that the proposed wave equation describes the constant-<it>Q</it> attenuation and velocity dispersion behaviour observed in Pierre Shale. Finally, we consider a plane-layer model and the Marmousi model to show how the new formulation applies to inhomogeneous media.

A new time and spatial fractional heat conduction model for Maxwell nanofluid in porous medium

The most important operator in fractional theory that enables the classical theory of integrals to be generalized is the Riemann-Liouville fractional integrals. In this paper, we have introduced new fractional operators in the fuzzy environment known as fuzzy Riemann-Liouville fractional integrals having exponential kernels. All classical fractional integrals that depend upon exponential kernels are exceptional cases of this new one. Moreover, we have defined a new class of convex mappings which is known as exponential trigonometric convex fuzzy-number valued mappings. With the help of this class and the newly proposed fuzzy fractional integral operator, the well-known Hermite-Hadamard type and related inequalities are taken into account in this work. Moreover, some new versions of midpoint Hermite-Hadamard-type inequalities are also established. By applying these definitions, we have amassed some novel and classical exceptional cases that serve as implementations of the key findings. For the purpose of proving the viability of the fuzzy order relations put forth in this research, some nontrivial examples of fuzzy numbered valued convexity are also provided.

The paper considers the initial-boundary value problem for equation Dtρu(x; t) + (−∆)σu(x; t) = 0, ρ; σ 2 (0; 1), in an N-dimensional domain Ω with a homogeneous Dirichlet condition. The fractional derivative is taken in the sense of Caputo. The main goal of the work is to solve the inverse problem of simultaneously determining two parameters: the order of the fractional derivative ρ and the degree of the Laplace operator σ. A new formulation and solution method for this inverse problem are proposed.

The paper considers the initial‐boundary value problem for equation , in an N‐dimensional domain with a homogeneous Dirichlet condition. The fractional derivative is taken in the sense of Caputo. The main goal of the work is to solve the inverse problem of simultaneously determining two parameters: the order of the fractional derivative and the degree of the Laplace operator . A new formulation and solution method for this inverse problem are proposed. It is proved that in the new formulation the solution to the inverse problem exists and is unique for an arbitrary initial function from the class . Note that in previously known works, only the uniqueness of the solution to the inverse problem was proved and the initial function was required to be sufficiently smooth and non‐negative.

In this article, we extend fractional calculus with nonsingular exponential kernels, initiated recently by Caputo and Fabrizio, to higher order. The extension is given to both left and right fractional derivatives and integrals. We prove existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial value problems by using Banach contraction theorem. Then we prove Lyapunov type inequality for the Riemann type fractional boundary value problems within the exponential kernels. Illustrative examples are analyzed and an application about Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.

Fractional diffusion equation with new fractional operator

<abstract> <p>The objective of this study was to investigate the thermodynamic reactions of thermoelastic materials by utilizing a modified mathematical fractional thermoelastic model. This model combines a fractional derivative with Rabotnov's exponential kernel and the idea of a two-phase delay, which makes it possible to show thermoelastic behavior more accurately. The model was utilized to investigate an unbounded material with a spherical cavity subjected to a decreasing and shifting heat flux on its inner surface. The problem was solved using analytical approaches, with a strong focus on the Laplace transform. The transform was numerically inverted to provide time-domain results. The study presented graphs that compared the outcomes of utilizing a single kernel fractional derivative with the results obtained using the Rabotnov kernel and fractional order. These graphs showed how the Rabotnov kernel and fractional order affected the physical fields under investigation. This novel theoretical framework has the potential to be advantageous in diverse domains, including engineering, solid mechanics, and materials science.</p> </abstract>

This work presents an alternative solution for the mathematical analysis of the fractional waves in dielectric media. For the fractional wave equation, the Caputo fractional derivative was considered, the order of the spatial and temporal fractional derivatives are , respectively. In this analysis, we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space and time derivatives into the fractional wave equation. We will consider source free Maxwell equations in isotropic and homogeneous dielectric medium. The general solutions obtained in our research have been expressed in terms of the multivariate Mittag–Leffler functions, these functions depend only on the parameters and preserving the appropriated physical units according to the system studied.

Our motive in this scientific contribution is to deal with nonlinear reaction–diffusion equation having both space and time variable order. The fractional derivatives which are used are non-singular having exponential kernel. These derivatives are also known as Caputo–Fabrizio derivatives. In our model, time fractional derivative is Caputo type while spatial derivative is variable-order Riesz fractional type. To approximate the variable-order time fractional derivative, we used a difference scheme based upon the Taylor series formula. While approximating the variable order spatial derivatives, we apply the quasi-wavelet-based numerical method. Here, double-quasi-wavelet numerical method is used to investigate this type of model. The discretization of boundary conditions with the help of quasi-wavelet is discussed. We have depicted the efficiency and accuracy of this method by solving the some particular cases of our model. The error tables and graphs clearly show that our method has desired accuracy.