Search Results (19)

A class of initial boundary value problems is here considered for a one-dimensional diffusion equation with a general time-fractional derivative with the Sonin kernel. One of the boundary conditions is in a general non-classical form, which includes no-nlocal cases of integral or multi-point boundary conditions. The problem is studied here by applying spectral projection operators to convert it to a system of relaxation equations in generalized eigenspaces. The uniqueness of the solution is established based on the uniqueness property of the spectral expansion. An algorithm is given for constructing the solution in the form of spectral expansion in terms of the generalized eigenfunctions. Estimates for the time-dependent components in this expansion are established and applied to prove the existence of a solution in the classical sense. The obtained results are applied to a particular case in which the specified boundary conditions lead to two sequences of eigenvalues, one of which consists of triple eigenvalues. Full article

The deconvolution of the Mellin convolution is studied for a great variety of functions that are expressed in terms of $\alpha$–log-exponential monomials. It is shown that the generation of pairs of functions satisfying a Sonin-like condition can be worked as a deconvolution process. Applications of deconvolution to scale-invariant linear systems are presented. Full article

Traditional operational calculus, while intuitive and effective in addressing problems in physical fractal spaces, often lacks the rigorous mathematical foundation needed for fractional operations, sometimes resulting in inconsistent outcomes. To address these challenges, we have developed a universal framework for defining the fractional calculus operators using the generalized fractional calculus with the Sonine kernel. In this framework, we prove that the α-th power of a differential operator corresponds precisely to the α-th fractional derivative, ensuring both accuracy and consistency. The relationship between the fractional power operators and fractional calculus is not arbitrary, it must be determined by the specific operator form and the initial conditions. Furthermore, we provide operator representations of commonly used fractional derivatives and illustrate their applications with examples of fractional power operators in physical fractal spaces. A superposition principle is also introduced to simplify fractional differential equations with non-integer exponents by transforming them into zero-initial-condition problems. This framework offers new insights into the commutative properties of fractional calculus operators and their relevance in the study of fractal structures. Full article

This paper explores a general class of singular kernels with the objective of designing new families of uniformly continuous sliding mode controllers. The proposed controller results from filtering a discontinuous switching function by means of a Sonine integral, producing a uniformly continuous control signal, preserving finite-time sliding motion and robustness against continuous but unknown and not necessarily integer-order differentiable disturbances. The principle of dynamic memory resetting is considered to demonstrate finite-time stability. A set of sufficient conditions to design singular kernels, preserving the above characteristics, is presented, and several examples are exposed to propose new families of continuous sliding mode approaches. Simulation results are studied to illustrate the feasibility of some of the proposed schemes. Full article

For the first time, a self-consistent mathematical approach to describe economic processes with a general form of a memory function is proposed. In this approach, power-type memory is a special case of such general memory. The memory is described by pairs of memory functions that satisfy the Sonin and Luchko conditions. We propose using general fractional calculus (GFC) as a mathematical language that allows us to describe a general form of memory in economic processes. The existence of memory (non-locality in time) means that the process depends on the history of changes to this process in the past. Using GFC, exactly solvable economic models of natural growth with a general form of memory are proposed. Equations of natural growth with general memory are equations with general fractional derivatives and general fractional integrals for which the fundamental theorems of GFC are satisfied. Exact solutions for these equations of models of natural growth with general memory are derived. The properties of dynamic maps with a general form of memory are described in the general form and do not depend on the choice of specific types of memory functions. Examples of these solutions for various types of memory functions are suggested. Full article

In this paper, two types of general fractional derivatives of distributed order and a corresponding fractional integral of distributed type are defined, and their basic properties are investigated. The general fractional derivatives of distributed order are constructed for a special class of one-parametric Sonin kernels with power law singularities at the origin. The conventional fractional derivatives of distributed order based on the Riemann–Liouville and Caputo fractional derivatives are particular cases of the general fractional derivatives of distributed order introduced in this paper. Full article

The general delay Hopfield neural network is studied. We consider the case of time-varying delay, continuously distributed delays, time-varying coefficients, and a special type of a Riemann–Liouville fractional derivative (GRLFD) with an exponential kernel. The kernels of the fractional integral and the fractional derivative in this paper are Sonine kernels and satisfy the first and the second fundamental theorems in calculus. The presence of delays and GRLFD in the model require a special type of initial condition. The applied GRLFD also requires a special definition of the equilibrium of the model. A constant equilibrium of the model is defined. An inequality for Lyapunov type of convex functions with the applied GRLFD is proved. It is combined with the Razumikhin method to study stability properties of the equilibrium of the model. As a partial case we apply quadratic Lyapunov functions. We prove some comparison results for Lyapunov function connected deeply with the applied GRLFD and use them to obtain exponential bounds of the solutions. These bounds are satisfied for intervals excluding the initial time. Also, the convergence of any solution of the model to the equilibrium at infinity is proved. An example illustrating the importance of our theoretical results is also included. Full article

This brief report studies conditions to ensure the nonexistence of finite-time stable equilibria in a class of systems that are described by means of nonlinear integral equations, whose kernels are part of some Sonine kernel pairs. It is firstly demonstrated that, under certain criteria, a real-valued function that converges in finite-time to a constant value, different from the initial condition, and remains there afterwards, cannot have a Sonine derivative that also remains at zero after some finite time. Then, the concept of equilibrium is generalized to the case of equivalent equilibrium, and it is demonstrated that a nonlinear integral equation, whose kernel is part of some Sonine kernel pair, cannot possess equivalent finite-time stable equilibria. Finally, illustrative examples are presented. Full article

Integro-differential operators with non-singular kernels have been much discussed among fractional calculus researchers. We present a mathematical study to clearly establish the rigorous foundations of this topic. By considering function spaces and mapping results, we show that operators with non-singular kernels can be defined on larger function spaces than operators with singular kernels, as differentiability conditions can be removed. We also discover an analogue of the Sonine invertibility condition, giving two-sided inversion relations between operators with non-singular kernels that are not possible for operators with singular kernels. Full article

The general fractional integrals and derivatives considered so far in the Fractional Calculus literature have been defined for the functions on the real positive semi-axis. The main contribution of this paper is in introducing the general fractional integrals and derivatives of the functions on a finite interval. As in the case of the Riemann–Liouville fractional integrals and derivatives on a finite interval, we define both the left- and the right-sided operators and investigate their interconnections. The main results presented in the paper are the 1st and the 2nd fundamental theorems of Fractional Calculus formulated for the general fractional integrals and derivatives of the functions on a finite interval as well as the formulas for integration by parts that involve the general fractional integrals and derivatives. Full article

In this paper, the 1st-level general fractional derivatives of arbitrary order are defined and investigated for the first time. We start with a generalization of the Sonin condition for the kernels of the general fractional integrals and derivatives and then specify a set of the kernels that satisfy this condition and possess an integrable singularity of the power law type at the origin. The 1st-level general fractional derivatives of arbitrary order are integro-differential operators of convolution type with the kernels from this set. They contain both the general fractional derivatives of arbitrary order of the Riemann–Liouville type and the regularized general fractional derivatives of arbitrary order considered in the literature so far. For the 1st-level general fractional derivatives of arbitrary order, some important properties, including the 1st and the 2nd fundamental theorems of fractional calculus, are formulated and proved. Full article

Long-term and fault-free operation of power transformers depends on the electrical strength of the insulation system and effective heat dissipation. Forced circulation of the insulating liquid is used to increase the cooling capacity. A negative effect of such a solution is the creation of the phenomenon of streaming electrification, which in unfavorable conditions may lead to damage to the insulating system of the transformer. This paper presents results of research confirming the possibility of using fullerene C₆₀ to reduce the phenomenon of streaming electrification generated by the flow of liquid dielectrics. The volume charge density q_w was used as a material indicator to determine the electrostatic charging tendency (ECT) of nanofluids. This parameter was determined from the Abedian-Sonin electrification model on the basis of electrification current measurements and selected physicochemical and electrical properties of the liquid. The electrification current was measured in a flow system with an aluminum pipe of 4 mm diameter and 400 mm length. All measurements were carried out at a temperature of 20 °C. The influence of flow velocity (from 0.34 m/s to 1.75 m/s) and C₆₀ concentration (25 mg/L, 50 mg/L, 100 mg/L, 200 mg/L and 350 mg/L) was analyzed on the electrification of fresh and aged Trafo En mineral oil, as well as Midel 1204 natural ester and Midel 7131 synthetic ester. The density, kinematic viscosity, dielectric constant, and conductivity were also determined. A negative effect of the C₆₀ doping on the electrostatic properties of fresh mineral oil was demonstrated. For other liquids, fullerene C₆₀ can be used as an inhibitor of the streaming electrification process. Based on the analysis of the q_w parameter, the optimum concentration of C₆₀ (from 100 mg/L to 200 mg/L) resulting in the highest reduction of the electrification phenomenon in nanofluids was identified. Full article

In this paper, we first consider the general fractional derivatives of arbitrary order defined in the Riemann–Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of fractional calculus that leads to a closed form formula for their projector operator. These results allow us to formulate the natural initial conditions for the fractional differential equations with the general fractional derivatives of arbitrary order in the Riemann–Liouville sense. In the second part of the paper, we develop an operational calculus of the Mikusiński type for the general fractional derivatives of arbitrary order in the Riemann–Liouville sense and apply it for derivation of an explicit form of solutions to the Cauchy problems for the single- and multi-term linear fractional differential equations with these derivatives. The solutions are provided in form of the convolution series generated by the kernels of the corresponding general fractional integrals. Full article

In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel $\kappa \left(t\right)=t^{\alpha -1}/\mathsf{\Gamma }\left(\alpha \right),0<\alpha <1$. The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature. Full article

In this paper we present a method of studying a convolution operator under the Sonin conditions imposed on the kernel. The particular case of the Sonin kernel is a kernel of the fractional integral Riemman–Liouville operator, other various types of the Sonin kernels are a Bessel-type function, functions with power-logarithmic singularities at the origin e.t.c. We pay special attention to study kernels close to power type functions. The main our aim is to study the Sonin–Abel equation in the weighted Lebesgue space, the used method allows us to formulate a criterion of existence and uniqueness of the solution and classify a solution, due to the asymptotics of the Jacobi series coefficients of the right-hand side. Full article