Liouville Differential Operators Research Articles

Some Results of R-Matrix Functions and Their Fractional Calculus

In this study, we explore various fractional integral properties of R-matrix functions using the Hilfer fractional derivative operator within the framework of fractional calculus. We introduce the θ integral operator and extend its definition to include the R matrix functions. The composition of Riemann–Liouville fractional integral and differential operators is determined using the θ-integral operator. Additionally, we investigate the compositional properties of θ-integral operators, and we establish their inversion, offering new insights into their structural and functional characteristics.

On recovering Sturm–Liouville operators with two delays

Abstract We study the inverse spectral problems of recovering Sturm–Liouville differential operator with two constant delays a 1 {a_{1}} and a 2 {a_{2}} greater than one third of the interval. It has been proved that the operator can be recovered uniquely from four spectra under the condition 2 ⁢ a 1 + a 2 2 ≥ π {2a_{1}+\frac{a_{2}}{2}\geq\pi} , while it is not possible otherwise.

Separation method of semifixed variables together with integral bifurcation method for solving generalized time‐fractional thin‐film equations

It is well known that investigation on exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work compared with integer‐order nonlinear PDEs. In this paper, based on the separation method of semifixed variables and integral bifurcation method, a combinational method is proposed. By using this new method, a class of generalized time‐fractional thin‐film equations are studied. Under two kinds of definitions of fractional derivatives, exact solutions of two generalized time‐fractional thin‐film equations are investigated respectively. Different kinds of exact solutions are obtained and their dynamic properties are discussed. Compared to the results in the existing references, the types of solutions obtained in this paper are abundant and very different from those in the existing references. Investigation shows that the solutions of the model defined by Riemann–Liouville differential operator converge faster than those defined by Caputo differential operator. It is also found that the profiles of some solutions are very similar to solitons, but they are not true soliton solutions. In order to visually show the dynamic properties of these solutions, the profiles of some representative exact solutions are illustrated by 3D graphs.

COMPOSITIONS OF FRACTIONAL DERIVATIVES AS DZHRBASHYAN - NERSESYAN DERIVATIVE

The issues of unique solvability for a special initial value problem to two classes of a linear inhomogeneous equation with a derivative of Dzhrbashyan - Nersesyan are investigated. In one of the classes, the operator at the unknown function is bounded, in the other it is sectorial. The representation of the composition of any number of Gerasimov - Caputo and/or Riemann - Liouville differential operators in the form of a Dzhrbashyan - Nersesyan derivative is also proved.

Inverse problem for Sturm–Liouville operator with complex-valued weight and eigenparameter dependent boundary conditions

Abstract This paper is concerned with discontinuous inverse problem generated by complex-valued weight Sturm–Liouville differential operator with λ-dependent boundary conditions. We establish some properties of spectral characteristic and prove that the potential on the whole interval can be uniquely determined by the Weyl-type function or two spectra.

Analysis of Volterra Integrodifferential Equations with the Fractal-Fractional Differential Operator

In this paper, a class of integrodifferential equations with the Caputo fractal-fractional derivative is considered. We study the exact and numerical solutions of the said problem with a fractal-fractional differential operator. The abovementioned operator is arising widely in the mathematical modeling of various physical and biological problems. In our scheme, first, the integrodifferential equation with the fractal-fractional differential operator is converted to an integrodifferential equation with the Riemann–Liouville differential operator. Additionally, the obtained integrodifferential equation is then converted to the equivalent integrodifferential equation involving the Caputo differential operator. Then, we convert the integrodifferential equation under the Caputo differential operator using the Laplace transform to an algebraic equation in the Laplace space. Finally, we convert the obtained solution from the Laplace space into the real domain. Moreover, we utilize two techniques which include analytic inversion and numerical inversion. For numerical inversion of the Laplace transforms, we have to evaluate five methods. Extensive results are presented. Furthermore, for numerical illustration of the abovementioned methods, we consider three problems to demonstrate our results.

The Regulator Problem to the Convection–Diffusion Equation

In this paper, from linear operator, semigroup and Sturm–Liouville problem theories, an abstract system model for the convection–diffusion (C–D) equation is proposed. The state operator for this abstract system model is here defined as given in the form of the Sturm–Liouville differential operator (SLDO) plus an integral term of the same SLDO. Our aim is to achieve the trajectory tracking task in the presence of external disturbances to the C–D equation invoking the regulator problem theory, where the state from a finite-dimensional exosystem is the state to the feedback law. In this context, the regulator (Francis) equations, established from the abstract system model for the C–D equation, here are solved; i.e., the state feedback regulator problem (SFRP) for the C–D system has a solution. Our proposal is validated via numerical simulation results.

A partial inverse problem for non-self-adjoint Sturm–Liouville operators with a constant delay

Abstract In this paper we study a partial inverse spectral problem for non-self-adjoint Sturm–Liouville operators with a constant delay and show that subspectra of two boundary value problems with one common boundary condition are sufficient to determine the complex potential. We developed the Horváth’s method in [M. Horváth, On the inverse spectral theory of Schrödinger and Dirac operators, Trans. Amer. Math. Soc. 353 2001, 10, 4155–4171] for the self-adjoint Sturm–Liouville operator without delay into the non-self-adjoint Sturm–Liouville differential operator with a constant delay.

ANALYSIS AND IMPLEMENTATION OF NUMERICAL SCHEME FOR THE VARIABLE-ORDER FRACTIONAL MODIFIED SUB-DIFFUSION EQUATION

This paper addresses the numerical study of variable-order fractional differential equation based on finite-difference method. We utilize the implicit numerical scheme to find out the solution of two-dimensional variable-order fractional modified sub-diffusion equation. The discretized form of the variable-order Riemann–Liouville differential operator is used for the fractional variable-order differential operator. The theoretical analysis including for stability and convergence is made by the von Neumann method. The analysis confirmed that the proposed scheme is unconditionally stable and convergent. Numerical simulation results are given to validate the theoretical analysis as well as demonstrate the accuracy and efficiency of the implicit scheme.

Dynamic Analysis and Control for a Bioreactor in Fractional Order

In this paper, a mathematical model was developed to describe the dynamic behavior of a bioreactor in which a fermentation process takes place. The analysis took into account the bioreactor temperature controlled by the refrigerant fluid flow through the reactor jacket. An optimal LQR control acting in the water flow through a jacket was used in order to maintain the reactor temperature during the process. For the control design, a reduced-order model of the system was considered. Given the heat transfer asymmetry observed in reactors, a model considering the fractional order heat exchange between the reactor and the jacket using the Riemann–Liouville differential operators was proposed. The numerical simulation demonstrated that the proposed control was efficient in maintaining the temperature at the desired levels and was robust for disturbances in the inlet temperature reactor. Additionally, the proposed control proved to be easy to apply in real life, bypassing the singularity problem and the difficulty of initial conditions for real applications that can be observed when considering Riemann–Liouville differential operators.

Riemann–Liouville fractional operators of bicomplex order and its properties

In this paper, we construct the Riemann–Liouville fractional integral and differential operator of bicomplex order and illustrate some examples to calculate the fractional integration and differentiation of bicomplex order of some elementary bicomplex‐valued functions. Also, we discuss some properties of these operators by proving analogues of the semigroup property, Leibniz rule, chain rule, etc.

On Modified Second Paine–de Hoog–Anderssen Boundary Value Problem

This article deals with a special case of the Sturm–Liouville boundary value problem (BVP), an eigenvalue problem characterized by the Sturm–Liouville differential operator with unknown spectra and the associated eigenfunctions. By examining the BVP in the Schrödinger form, we are interested in the problem where the corresponding invariant function takes the form of a reciprocal quadratic form. We call this BVP the modified second Paine–de Hoog–Anderssen (PdHA) problem. We estimate the lowest-order eigenvalue without solving the eigenvalue problem but by utilizing the localized landscape and effective potential functions instead. While for particular combinations of parameter values that the spectrum estimates exhibit a poor quality, the outcomes are generally acceptable although they overestimate the numerical computations. Qualitatively, the eigenvalue estimate is strikingly excellent, and the proposal can be adopted to other BVPs.

Certain Recurrence Relations of Two Parametric Mittag-Leffler Function and Their Application in Fractional Calculus

The purpose of this paper is to develop some new recurrence relations for the two parametric Mittag-Leffler function. Then, we consider some applications of those recurrence relations. Firstly, we express many of the two parametric Mittag-Leffler functions in terms of elementary functions by combining suitable pairings of certain specific instances of those recurrence relations. Secondly, by applying Riemann–Liouville fractional integral and differential operators to one of those recurrence relations, we establish four new relations among the Fox–Wright functions, certain particular cases of which exhibit four relations among the generalized hypergeometric functions. Finally, we raise several relevant issues for further research.

EXISTENCE THEORY TO A CLASS OF FRACTIONAL ORDER HYBRID DIFFERENTIAL EQUATIONS

In this paper, we develop the theory of fractional order hybrid differential equations involving Riemann–Liouville differential operators of order [Formula: see text]. We study the existence theory to a class of boundary value problems for fractional order hybrid differential equations. The sum of three operators is used to prove the key results for a couple of hybrid fixed point theorems. We obtain sufficient conditions for the existence and uniqueness of positive solutions. Moreover, examples are also presented to show the significance of the results.

On Inverse Spectral Problems for Sturm–Liouville Differential Operators on Closed Sets

We study Sturm-Liouville operators on closed sets of a special structure, which are sometimes referred as time scales and often appear in modelling various real processes. Depending on the set structure, such operators unify both differential and difference operators. We obtain properties of their spectral characteristics and study inverse problems with respect to them. We prove that the spectral characteristics uniquely determine the operator.

On Higher Regularized Traces of a Differential Operator with Bounded Operator Coefficient Given in a Finite Interval

In this work, we find a higher regularized trace formula for a regular Sturm–Liouville differential operator with operator coefficient.

On Recovering the Sturm–Liouville Differential Operators on Time Scales

We study Sturm--Liouville differential operators on the time scales consisting of a finite number of isolated points and segments. In a previous paper it was established that such operators are uniquely determined by their spectral characteristics. In the present paper, an algorithm for their recovery based on the method of spectral mappings is obtained. We also prove that the eigenvalues of two Sturm--Liouville boundary value problems with one common boundary condition alternate.

Design of Atangana–Baleanu–Caputo fractional-order digital filter

Atangana–Baleanu–Caputo (ABC) fractional differential operator based upon Mittag-Leffler kernel exhibits all the advantages of conventional Riemann–Liouville and Caputo fractional differential operators; in addition, the kernel associated is non-singular. Therefore, this paper puts forward a closed-form analytical formulation for the design of an ABC-based fractional-order FIR filter for various signal processing and filtering applications. The closed-form expression is derived by utilizing backward finite difference method and fractional sample delay interpolation techniques. Furthermore, several design examples are considered to illustrate the effectiveness of the proposed method. From the analytical and simulation studies done, it is observed that the proposed design efficiently approximates the ideal frequency response of ABC-fractional differential operator. Finally, one-dimensional and two-dimensional applications of the proposed method are validated and compared against state-of-the-art methods for electrocardiogram (ECG) R-peak detection as well as for digital image sharpening.

State Feedback Regulation Problem to the Reaction-Diffusion Equation

In this work, we explore the state feedback regulator problem (SFRP) in order to achieve the goal for trajectory tracking with harmonic disturbance rejection to one-dimensional (1-D) reaction-diffusion (R-D) equation, namely, a partial differential equation of parabolic type, while taking into account bounded input, output, and disturbance operators, a finite-dimensional exosystem (exogenous system), and the state of the exosystem as the state to the feedback law. As is well-known, the SFRP can be solved only if the so-called Francis (regulator) equations have solution. In our work, we try with the solution of the Francis equations from the 1-D R-D equation following given criteria to the eigenvalues from the exosystem and transfer function of the system, but the state operator is here defined in terms of the Sturm–Liouville differential operator (SLDO). Within this framework, the SFRP is then solved for the 1-D R-D equation. The numerical simulation results validate the performance of the regulator.

Reconstruction of the Sturm-Liouville differential operators with discontinuity conditions and a constant delay

In this manuscript, we study second–order differential operators with a constant delay and transmission boundary conditions. We establish properties of the spectral characteristics and investigate the inverse problem of recovering operators from their spectra. Also, we construct the potential function by using the Fourier series and delay point of the Sturm–Liouville differential operator.