Operational Calculus Research Articles

Theoretical studies of filtration movement of moisture in capillaries of woody plants

The article presents theoretical studies of filtration transport of moisture through capillaries of woody plants that is determined by humidity gradients. The emerging humidity fields and the magnitude of the gradients significantly affect the growth and development of woody plants biomass. The aim of theoretical research was to use mathematical tools of operational calculus that provide an opportunity to consider the filtration of moisture through the capillaries of wood materials using a third-kind boundary condition, when the substance is exchanged with the environment at the considered boundary. As a result of the use of Laplace transform a mathematical model for studying the movement of moisture in the capillaries of woody vegetation through filtration transport has been obtained. Modelling of a moisture transport process in wood materials is an important tool in theoretical and experimental studies. This is particularly important for conducting long-term experimental studies of the site quality of stand of trees from the technogenic impact of vehicles on roadside ecological systems.

Design of Fractional-Order PID for Stabilized Sight System via Internal Model Control Approach

This paper aims at the problem of accuracy reduction of stabilized sight system caused by uncertain factors and disturbances under complex working conditions. Here, a fractional-order PID (FOPID) controller design method based on internal model control (IMC) approach is proposed for stabilized sight system. First, the IMC-PID is constructed by IMC approach, and then the fractional order is added to its differential and integral terms to obtain the FOPID. The traditional FOPID requires five parameters for tuned, while the proposed FOPID has only three tuning parameters, which can be obtained through the CRONE control technology. The optimal rational approximation algorithm (ORAA) with a high degree of fit is used to approximate the fractional-order calculus operator. Matlab simulation experiments show that the proposed controller outperforms other controllers and can obtain good robustness, disturbance rejection and tracking performance.

Hardy–Littlewood–Sobolev inequalities for a class of non-symmetric and non-doubling hypoelliptic semigroups

In his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order evolution equation which displays some challenging features. In the opening of his 1967 hypoellipticity paper Hörmander discussed a general class of degenerate Ornstein–Uhlenbeck operators that includes Kolmogorov’s as a special case. In this note we combine semigroup theory with a nonlocal calculus for these hypoelliptic operators to establish new inequalities of Hardy–Littlewood–Sobolev type in the situation when the drift matrix has nonnegative trace. Our work has been influenced by ideas of E. Stein and Varopoulos in the framework of symmetric semigroups. One of our objectives is to show that such ideas can be pushed to successfully handle the present degenerate non-symmetric setting.

On an extension of the Mikusiński type operational calculus for the Caputo fractional derivative

In this paper, a two-parameter extension of the operational calculus of Mikusiński's type for the Caputo fractional derivative is presented. The first parameter is connected with the rings of functions that are used as a basis for construction of the convolution quotients fields. The convolutions by themselves are characterized by another parameter. The obtained two-parameter operational calculi are compared each to other and some homomorphisms between the fields of convolution quotients are established.

On certain fractional calculus operators and applications in mathematical physics

In this paper, we modify the (k, s) fractional integral operator involving k-Mittag-Leffler function and discuss its properties. We originate a new fractional operator named (k, s)-Prabhakar derivative and obtained some classical fractional operators as a special case of the newly proposed derivative. Some properties of the introduced operator are also part of the present work. The generalized Laplace transform is employed to study the characteristics of fractional operators. We modeled the free-electron laser (FEL) equation by involving the proposed derivative and can find the solution by using the said Laplace transform.

A new approach on approximate controllability of fractional evolution inclusions of order 1 < r < 2 with infinite delay

This manuscript is mainly focusing on the approximate controllability of fractional differential evolution inclusions of order 1 < r < 2 with infinite delay. We study our primary outcomes by using the theoretical concepts about fractional calculus, cosine, and sine function of operators and Dhage’s fixed point theorem. Initially, we prove the approximate controllability for the fractional evolution system. The results are established under the assumption that the associated linear system is approximately controllable. Then, we develop our conclusions to the ideas of nonlocal conditions. Finally, we present theoretical and practical applications to support the validity of the study.

A New Operation Method of Relay Protection Hardwarewith VFC Pulse Train as Variable

Introduction In the basic theory of digital signal processing， the calculus operation of continuous time variable function can be transformed into the difference and accumulation operation of corresponding discrete signal.At present， the application of this principle in microcomputer type relay protection is based on the software operation of fixed interval sampling value.In this paper， A new application type of the above theory is proposed， which integrates the sampling and operation process into the discrete， quantization and coding links of A/D conversion， so as to simplify the computational overhead and improve the protection performance. Methods The real-time sequence and delay sequence of VFC serial pulse signal were taken as variables， the counter was taken as the core computing hardware， and the single and multiple floating operation Windows were constructed with pulse delay devices to realize the hardware synchronization of calculus and phasor addition and subtraction.Based on the mature theory of software protection algorithm， the new algorithm formula is obtained by interval transformation and the loop construction mode is proposed. Result After discussion and calculation， the following results are obtained. The new method can be used to construct the instantaneous integral value of the electrical quantity， the instantaneous sampling value and the effective value of the definite integral. Conclusion Although the operation is based on the same basic principle as software operation， the physical process of operation is very simple， which can complete high-precision synchronous operation， and form a variety of operators， which can construct a protection scheme of multiple principles.

Meromorphic harmonic univalent functions related with generalized (p,q)-post quantum calculus operators

In this paper, we introduce certain subclasses of meromorphic harmonic univalent functions, which are defined by using generalized (p, q)-post quantum calculus operators as well as subordination relationship. Sufficient coefficient conditions, extreme points, distortion bounds and convolution properties for functions belonging to the subclasses are obtained.

Operator calculus: the lost formulation of quantum mechanics

Traditionally, “the operator calculus of Born and Wiener” has been considered one of the four formulations of quantum mechanics that existed in 1926. The present paper reviews the operator calculus as applied by Max Born and Norbert Wiener during the last months of 1925 and the early months of 1926 and its connections with the rise of the new quantum theory. Despite the relevance of this operator calculus, Born–Wiener’s joint contribution to the topic is generally bypassed in historical accounts of quantum mechanics. In this study, we analyse the paper that epitomises the contribution, and we explain the main reasons for the apparent lack of interest in Born and Wiener’s work. We argue that they did not solve the main problem for which the tool was intended, that of linear motion, because of their reluctance to use Dirac delta functions.

A new study on existence and uniqueness of nonlocal fractional delay differential systems of order 1 &lt; r &lt; 2 in Banach spaces

AbstractThis article is mainly focusing on the existence and uniqueness of nonlocal fractional delay differential systems of order 1 &lt; r &lt; 2 in Banach spaces. By using the theoretical concepts related to the fractional calculus, cosine, and sine functions of operators and fixed point approach, we prove our main results. By using Kranoselskii's fixed point theorem, we discuss the existence of the mild solution and by applying the Banach contraction principle, we prove the existence and uniqueness of the mild solution of nonlocal fractional delay differential system. Finally, we provide an example to illustrate the obtained theoretical results.

Intelligent optimal controller design applied to quarter car model based on non-asymptotic observer for improved vehicle dynamics

A novel active suspension control strategy is introduced to improve dynamic response of vehicle suspension systems. The proposed algorithm is a fusion of classical controller design methods together with an online observer and is based on the cancelation of system disturbances. The operational calculus method and the differential algebraic theory are applied to build the observer/compensator that is appended to the classical linear quadratic regulator. An ultra-local model based on linear algebraic rules is presented avoiding the use of a precise mathematical model while guaranteeing the stability of the overall system. Simplicity of implementation, low power demand and significant enhancement of active suspension performance are the observed features of the proposed controller. The numerical simulations illustrate the effectiveness and the robustness against sprung mass variation of the proposed control method compared to proportional–integral–derivative controller, intelligent proportional–derivative controller, linear quadratic regulator and active disturbance rejection—linear quadratic regulator.

English

English

Research and Analysis of Tunable Differential N-Path Band-Pass Filter

Traditional filters usually have low Q and gain values and it is difficult to adjust their center frequencies. Moreover, it is very complicated to analyze their transmission charateristics through conventional methods. Therefore, in this paper, a tunable differential N-path bandpass filter that uses a new adjoint network method to analyze the transmission characteristics of the differential N-path structure is proposed. The filter circuit adopts a novel circuit structure consisting of two differential N-path structures, two transconductance amplifiers and an off-chip transformer. The differential structure eliminates even harmonics, the transconductance amplifier increases the circuit gain and the off-chip transformer acts as a balun, improving the filter’s Q value and achieving impedance matching. Unlike the traditional switching capacitance method used for analyzing the differential circuit structure, the method proposed in this paper does not involve complicated calculus operations. In fact, the method greatly simplifies these complex operations, and the transmission function of the circuit can be obtained through simple algebraic operations. The proposed filter was designed using TSMC 180[Formula: see text]nm CMOS process. Simulation results for a differential four-path bandpass filter formed under 1.2[Formula: see text]V supply voltage show that the gain of the filter is greater than 8.5 dB, the center frequency can be adjusted from 0.1[Formula: see text]GHz to 1[Formula: see text]GHz, the in-band insertion loss S11 is greater than 10 dB, the out-of-band IIP3 is greater than 10 dBm, the out-of-band rejection is 28 dB and the noise figure is less than 2.2 dB at [Formula: see text][Formula: see text]MHz.

Analytical stochastic responses of thin cylindrical shells under various stationary excitations

This study proposed an efficient analytical framework to achieve exact stochastic responses of circular cylindrical shells under various stationary excitations. By introducing the exact modal shapes of closed and open cylindrical shells based on the Flügge theory, the analytical power spectral densities of responses for the two shells were derived via the pseudo excitation method (PEM), where the calculus operation in the spatial domain and the integral in frequency domain were involved. To take full advantage of PEM for matrix operation and improve computational efficiency, the discrete analytical method (DAM), implementing analytically the calculus operation in the spatial domain and discretizing the frequency domain, was proposed. Furthermore, the exact stationary stochastic responses of three shell examples under point, surface, base acceleration and spacecraft specification spectrum excitations were achieved. By comparing the root mean squares of responses, especially twisting moment and membrane shear force with those by finite element analysis software, the high accuracy and efficiency of DAM were demonstrated. The remarkable effect of frequency bandwidth of excitation on stochastic responses was also revealed. The exact stochastic responses calculated by DAM can be taken as benchmark solutions to verify the numerical methods for the high-frequency wideband random vibration problem.

Explicit analytical solutions of incommensurate fractional differential equation systems

Fractional differential equations have been studied due to their applications in modelling, and solved using various mathematical methods. Systems of fractional differential equations are also used, for example in the study of electric circuits, but they are more difficult to analyse mathematically. We present explicit solutions for several families of such systems, both homogeneous and inhomogeneous cases, both commensurate and incommensurate. The results can be written, in several interesting special cases, in terms of a recently defined bivariate Mittag-Leffler function and the associated operators of fractional calculus.

Composition Formulae for thek-Fractional Calculus Operators Associated withk-Wright Function

In this article, thek-fractional-order integral and derivative operators including thek-hypergeometric function in the kernel are used for thek-Wright function; the results are presented for thek-Wright function. Also, some of special cases related to fractional calculus operators andk-Wright function are considered.

Special Issue of Symmetry: “Integral Transformations, Operational Calculus and Their Applications”

This Special Issue consists of a total of 14 accepted submissions (including several invited feature articles) to the Special Issue of the MDPI’s journal, Symmetry on the general subject-area of “Integral Transformations, Operational Calculus and Their Applications” from different parts of the world [...]

The Poincare Lemma, Antiexact Forms, and Fermionic Quantum Harmonic Oscillator

The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract derivative and the homotopy operator plays the role of an abstract integral. This operator calculus can be used to formulate abstract differential equations. An example of the eigenvalue problem that resembles the fermionic quantum harmonic oscillator is presented. The second part presents the dual complex to the Dolbeault bicomplex generated by the homotopy operator on complex manifolds.

A Modified Generalization of Fractional Calculus Operators in A Complex Domain

This investigation deals with a new generalization for fractional calculus operators in a complex domain based on the well-known hypergeometric function. Conditions are forced for these generalized operators such as the upper bounds. Other properties for the above operators are also presented. Besides, the employment of these operators is proposed in the geometric function theory.

The Marichev-Saigo-Maeda Fractional Calculus Operators Pertaining to the Generalized K-Struve Function

Abstract In the present paper, we establish some compositions formulas for Marichev-Saigo-Maeda (MSM) fractional calculus operators with k-Struve function S ν , c k S_{\nu ,c}^k as of the kernel. The results are presented in terms of generalized k-Wright function p Ψ q k _p\Psi _q^k .