Search

Many properties of special polynomials, such as recurrence relations, sum formulas, integral transforms and symmetric identities, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we introduce hybrid forms of q-Mittag-Leffler functions. The q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are constructed using a q-symbolic operator. The generating functions, series definitions, q-derivative formulas and q-recurrence formulas for q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are obtained. The N q -transforms and N q -transforms of q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are obtained. These hybrid q-special functions are also studied by plotting their graphs for specific values of the indices and parameters. [ABSTRACT FROM AUTHOR]

The aim of this work is to examine some q-analogs and differential properties of the gamma integral operator and its convolution products. The q-gamma integral operator is introduced in two versions in order to derive pertinent conclusions regarding the q-exponential functions. Also, new findings on the q-trigonometric, q-sine, and q-cosine functions are extracted. In addition, novel results for first and second-order q-differential operators are established and extended to Heaviside unit step functions. Lastly, three crucial convolution products and extensive convolution theorems for the q-analogs are also provided. [ABSTRACT FROM AUTHOR]

By means of relaxation methods, residual stresses can be obtained by introducing a progressive cut or a hole in a specimen and by measuring and elaborating the strains or displacements that are consequently produced. If the cut can be considered a controlled crack-like defect, by leveraging Bueckner's superposition principle, the relaxed strains can be modeled through a weighted integral of the residual stress relieved by the cut. To evaluate residual stresses, an integral equation must be solved. From a practical point of view, the solution is usually based on a discretization technique that transforms the integral equation into a linear system of algebraic equations, whose solutions can be easily obtained, at least from a computational point of view. However, the linear system is often significantly ill-conditioned. In this paper, it is shown that its ill-conditioning is actually a consequence of a much deeper property of the underlying integral equation, which is reflected also in the discretized setting. In fact, the original problem is ill-posed. The ill-posedness is anything but a mathematical sophistry; indeed, it profoundly affects the properties of the discretized system too. In particular, it induces the so-called bias–variance tradeoff, a property that affects many experimental procedures, in which the analyst is forced to introduce some bias in order to obtain a solution that is not overwhelmed by measurement noise. In turn, unless it is backed up by sound and reasonable physical assumptions on some properties of the solution, the introduced bias is potentially infinite and impairs every uncertainty quantification technique. To support these topics, an illustrative numerical example using the crack compliance (also known as slitting) method is presented. The availability of the Linear Elastic Fracture Mechanics Weight Function for the problem allows for a completely analytical formulation of the original integral equation by which bias due to the numerical approximation of the physical model is prevented. [ABSTRACT FROM AUTHOR]

Tidal bores, defined by sudden upstream surges of tidal water in estuaries, exert significant hydrodynamic forces on riverbeds, leading to complex sedimentary responses. This study examines the dynamic response and liquefaction potential of riverbeds subjected to tidal bores in macro-tidal estuaries. An analytical model, developed using the generalized Biot theory and integral transform methods, evaluates the dynamic behavior of riverbed sediments. Key factors such as permeability, saturation, and sediment properties are analyzed for their influence on momentary liquefaction. The results indicate that fine sand reduces liquefaction risk by facilitating pore water discharge, while silt soil increases sediment instability. Additionally, the study reveals that pressure gradients induced by tidal bores can trigger momentary liquefaction, with the maximum liquefaction depth predicted based on horizontal pressure gradients being five times that predicted based on vertical pressure gradients. This research highlights the critical role of sediment characteristics in riverbed stability, providing a comprehensive understanding of the interactions between tidal bores and riverbed dynamics. The findings contribute to the development of predictive models and guidelines for managing the risks of tidal bore-induced liquefaction in coastal and estuarine environments. [ABSTRACT FROM AUTHOR]

A novel and efficient modeling approach has been developed for simulating the responses of triaxial induction logging (TIL) in layered biaxial anisotropic (BA) formations. The core of this innovative technique lies in analytically calculating the primary fields within a homogeneous medium and approximating the scattered fields within layered formations. The former involves employing a two-level subtraction technique. Initially, the first-level subtraction entails altering the direction of the Fourier transform to mitigate the integral singularity of the spectral fields, particularly in high-angle and horizontal wells. Conversely, the second-level subtraction aims to further optimize integral convergence by creating an equivalent unbounded transverse isotropic (TI) formation and eliminating the corresponding spectral fields. With the two-level subtractions, the convergence of the spectral field has been enhanced by more than six orders of magnitude. Additionally, a strict recursive algorithm and approximation method are developed to compute the scattered fields in layered biaxial anisotropic media. The rigorous algorithm is based on a modified amplitude propagator matrix (MAPM) approach and serves as the benchmark for the approximation method. In contrast, the approximation method exploits the similarity between the spectral scattered field of the TI medium and the BA medium, establishing corresponding equivalent layered TI models for each magnetic component. Since the scattered field in TI models only involves a one-dimensional semi-infinite integral, the computational complexity is significantly reduced. Numerical simulation examples demonstrate that the new simulation method is at least two orders of magnitude faster than the current modeling approach while maintaining computational precision error within 0.5%. This significantly improved simulation efficiency provides a solid foundation for expediting the logging data processing. [ABSTRACT FROM AUTHOR]

In deep excavation engineering, the implementation of cut-off walls stands as a crucial measure to ensure structural support stability. However, the existing theories for dewatering design often overlook the variations in cut-off wall penetration depths, potentially compromising the efficacy of groundwater control strategies. Addressing this gap, this study conducts a comprehensive investigation into the dynamics of groundwater levels within confined aquifers during the dewatering process of strip excavation pits with cut-off walls. Central to this inquiry is the conceptualization of the entire excavation pit as a singular large-diameter well, with the open part beneath the cut-off wall in the confined aquifer serving as a constant-flux boundary. Employing an advanced analytical modeling approach, the study formulates a robust framework to describe the intricate interplay of unsteady groundwater flow phenomena. Leveraging the techniques of the Laplace and finite cosine transform methods, a semi-analytical solution is derived to elucidate groundwater drawdown patterns over time. The validation of the proposed solution against finite element method results underscores its fidelity and applicability. The parametric analysis reveals a dynamic evolution in drawdown characteristics within the confined aquifer, transitioning from initially cone-shaped distributions to more linear profiles that eventually stabilize with prolonged dewatering. This evolution is governed by the aquifer's inherent anisotropy and the barrier effect exerted by the depth of cut-off wall penetration. The parametric research also underscores the critical role of lateral boundary distance in influencing groundwater drawdown patterns. The presented solutions can be used to identify optimal penetration depth ratios tailored to specific parameters, thereby offering insights for optimizing dewatering strategies for deep excavation groundwater control. Moreover, a case study was included in this study using the proposed analytical solution, and a comparison was made with field data to validate the practical applicability of the approach. [ABSTRACT FROM AUTHOR]

In this study, we derive multiple incomplete matrix Mittag-Leffler (ML) functions. We systematically investigate several properties of these incomplete matrix ML functions, which include some general properties and distinct representations of integral transforms. We further study the properties of the Riemann–Liouville fractional integrals and derivatives related to the incomplete matrix ML functions. Additionally, some interesting special cases of this work are highlighted. Finally, we establish the solution to the kinetic equations as an application. [ABSTRACT FROM AUTHOR]

The Euler characteristic transform (ECT) is an integral transform used widely in topological data analysis. Previous efforts by Curry et al. and Ghrist et al. have independently shown that the ECT is injective on all compact definable sets. In this work, we first study the injectivity of the ECT on definable sets that are not necessarily compact and prove a complete classification of constructible functions that the Euler characteristic transform is not injective on. We then introduce the quadric Euler characteristic transform (QECT) as a natural generalization of the ECT by detecting definable shapes with quadric hypersurfaces rather than hyperplanes. We also discuss some criteria for the injectivity of QECT. [ABSTRACT FROM AUTHOR]

This research aims to develop generalized fractional integral inequalities by utilizing multiple Erdélyi–Kober (E–K) fractional integral operators. Using a set of j, with (j ∈ N) positively continuous and decaying functions in the finite interval a ≤ t ≤ x , the Fox-H function is involved in establishing new and novel fractional integral inequalities. Since the Fox-H function is the most general special function, the obtained inequalities are therefore sufficiently widespread and significant in comparison to the current literature to yield novel and unique results. [ABSTRACT FROM AUTHOR]

This article introduces (p , q) -analogs of the gamma integral operator and discusses their expansion to power functions, (p , q) -exponential functions, and (p , q) -trigonometric functions. Additionally, it validates other findings concerning (p , q) -analogs of the gamma integrals to unit step functions as well as first- and second-order (p , q) -differential operators. In addition, it presents a pair of (p , q) -convolution products for the specified (p , q) -analogs and establishes two (p , q) -convolution theorems. [ABSTRACT FROM AUTHOR]

Polyurethane (PU) mixture, which is a new pavement mixture, exhibits different dynamic properties compared to a hot-mixed asphalt mixture (HMA). This paper analyzed whether the Kramers–Kronig (K–K) relation and thermorheologically simple properties applied to the PU mixture. Based on the results, the PU mixture exhibited thermorheologically simple properties within the test conditions. The time–temperature superposition principle (TTSP) was applicable for the PU mixture to construct a dynamic modulus master curve using the standard logistic sigmoidal (SLS) model, the generalized logistic sigmoidal (GLS) model, and the Havriliak–Negami (HN) model. The Hilbert integral transformed SLS and GLS models for the phase angle can accurately fit the measured phase angle data with newly fitted shift factors and predict the phase angle within the viscoelastic range. The core–core and black space diagrams both displayed single continuous smooth curves, which can be utilized to characterize the viscoelastic property of the PU mixture. The K–K relation is applicable for the PU mixture to obtain the phase angle master curve model, storage modulus, and loss modulus from the complex modulus test results with the test temperatures and loading frequencies. The phase angle of the PU mixture at extremely high or low test temperatures cannot be derived from the dynamic modulus data. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the multi-dimensional integral transform with the Fox H-function in the kernel in weighted spaces with integrable functions in the domain R + n with positive coordinates. Due to the generality of the Fox H-function, many special integral transforms have the form studied in this paper, including operators with such kernels as generalized hypergeometric functions, classical hypergeometric functions, Bessel and modified Bessel functions and so on. Moreover, most important fractional integral operators, such as the Riemann–Liouville type, are covered by the class under consideration. The mapping properties in Lebesgue-weighted spaces, such as the boundedness, the range and the representations of the considered transformation, are established. In special cases, it is applied to the specific integral transforms mentioned above. We use a modern technique based on the extensive use of the Mellin transform and its properties. Moreover, we generalize our own previous results from the one-dimensional case to the multi-dimensional one. The multi-dimensional case is more complex and needs more delicate techniques. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define the linear operator based on the generalized Mittag-Leffler function and the Lambert series. By using this operator, we introduce a new subclass of β-uniformly starlike functions Τ J (α i) . Further, we obtain coefficient estimates, convex linear combinations, and radii of close-to-convexity, starlikeness, and convexity for functions f ∈ Τ J (α i) . In addition, we investigate the inclusion conditions of the Hadamard product and the integral transform. Finally, we determine the second Hankel inequality for functions belonging to this subclass. [ABSTRACT FROM AUTHOR]

This study introduces a novel process identification method aimed at overcoming the challenge of accurately estimating process models when faced with deterministic disturbances, a common limitation in conventional identification methods. The proposed method tackles the difficult modeling problems due to deterministic disturbances by representing the disturbances as a linear combination of Laguerre polynomials and applies an integral transform with frequency weighting to estimate the process model in a numerically robust and stable manner. By utilizing a least squares approach for parameter estimation, it sidesteps the complexities inherent in iterative optimization processes, thereby ensuring heightened accuracy and robustness from a numerical analysis perspective. Comprehensive simulation results across various process types demonstrate the superior capability of the proposed method in accurately estimating the model parameters, even in the presence of significant deterministic disturbances. Moreover, it shows promising results in providing a reasonably accurate disturbance model despite structural disparities between the actual disturbance and the model. By improving the precision of process models under deterministic disturbances, the proposed method paves the way for developing refined and reliable control strategies, aligning with the evolving demands of modern industries and laying solid groundwork for future research aimed at broadening application across diverse industrial practices. [ABSTRACT FROM AUTHOR]

With the advancement of technology, mining trucks are gradually becoming larger, imposing higher performance requirements on the front suspension. There is a need to transform the original integral non-independent front axle of mining dump trucks with a payload exceeding 300 tons into an independent front suspension with a double-wishbone suspension. The changing of the front suspension is bound to have an impact on the overall vehicle's handling stability and ride comfort. Therefore, the following research is conducted to investigate and analyze these effects. Firstly, the paper proposes a method for optimizing the parameters of the double-wishbone front suspension. The double-wishbone front suspension is modeled, and a comparison with a kinematic model is conducted to validate the accuracy of the model. Secondly, unreasonable hardpoint parameters are optimized. Thirdly, a dynamic model of the entire vehicle is established based on the optimized parameters, and an analysis of handling stability and ride comfort for the entire vehicle is performed. Finally, simulation results are compared and analyzed against experimental data. The results indicate that the optimized positioning parameters not only effectively enhance the suspension performance of the mining dump truck but also meet the requirements for handling stability and smoothness. The overall smoothness of the vehicle is significantly improved after the modification. This study not only holds significant engineering value in reducing vibrations in dump trucks and enhancing driver comfort, but also provides theoretical support for subsequent research and development in the industry. [ABSTRACT FROM AUTHOR]

The studied configuration is a two-dimensional, very thin parabolic reflector made of graphene and illuminated by an H-polarized electromagnetic plane wave. We present basic scattering and focusing properties of such a graphene reflector depending on the graphene parameters at microwave frequencies, using the resistive boundary condition for very thin sheets. The scattering is formulated as an electromagnetic boundary-value problem; it is transformed to a singular integral equation that is further treated with the method of analytical regularization (MAR) based on the known solution of the Riemann–Hilbert Problem (RHP). The numerical results are computed by using a Fredholm second-kind matrix equation that guarantees convergence and provides easily controlled accuracy. Compared to THz range, in microwaves, the scattering pattern of reflector and the field level at geometrical focus can be controlled in a wide range by adjusting the chemical potential of graphene. Even though here no dielectric substrate supporting the graphene is considered, the practical realization can also be possible as a thin layer graphene material in GHz range. As we demonstrate, the variation of the chemical potential from 0 to 1 eV can improve the focusing ability within the factor of three. The high accuracy of the used method and the full wave formulation of the problem support our findings. [ABSTRACT FROM AUTHOR]

Integral transform methods are a common tool employed to study the Hyers–Ulam stability of differential equations, including Laplace, Kamal, Tarig, Aboodh, Mahgoub, Sawi, Fourier, Shehu, and Elzaki integral transforms. This work provides improved techniques for integral transforms in relation to establishing the Hyers–Ulam stability of differential equations with constant coefficients, utilizing the Kamal transform, where we focus on first- and second-order linear equations. In particular, in this work, we employ the Kamal transform to determine the Hyers–Ulam stability and Hyers–Ulam stability constants for first-order complex constant coefficient differential equations and, for second-order real constant coefficient differential equations, improving previous results obtained by using the Kamal transform. In a section of examples, we compare and contrast our results favorably with those established in the literature using means other than the Kamal transform. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of high-order, accurate difference schemes' numerical solutions of local and non-local problems for ordinary differential equations of the fourth order. Local and non-local problems for ordinary differential equations with constant coefficients can be solved by classical integral transform methods. However, these classical methods can be used simply in the case when the differential equation has constant coefficients. We study fourth-order differential equations with dependent coefficients and their corresponding boundary value problems. Novel compact numerical solutions of high-order, accurate finite difference schemes generated by Taylor's decomposition on five points have been studied in these problems. Numerical experiments support the theoretical statements for the solution of these difference schemes. [ABSTRACT FROM AUTHOR]

Internal gravity waves (IGWs) in the middle atmosphere are the main source of mesoscale fluctuations of wind and temperature. The parameterization of IGWs and study of their climatology is necessary for the development of global atmospheric circulation models. In this review, we focus on the application of Radio Occultation (RO) observations for the retrieval of IGW parameters. (1) The simplest approach employs the retrieved temperature profiles. It is based on the fact that IGWs are highly anisotropic structures and can be accurately retrieved by RO. The basic assumption is that all the temperature fluctuations are caused by IGWs. The smoothed background temperature profile defines the the Brunt–Väisälä frequency, which, together with the temperature fluctuations, defines the IGW specific potential energy. Many studies have derived the distribution and climatology of potential energy, which is one of the most important characteristics of IGWs. (2) More detailed analysis of the temperature profiles is based on the derivation of the temperature fluctuation spectra. For saturated IGWs, the spectra must obey the power law with an exponent of − 3 . Such spectra are obtained by using Wave Optical (WO) processing. (3) More advanced analysis employs space–frequency analysis. It is based on phase-sensitive techniques like cross S- or wavelet transforms in order to identify propagating IGWs. (4) Another direction is the IGW parameter estimate from separate temperature profiles applying the stability condition in terms of the Richardson number. In this framework, a necessary condition is formulated that defines whether or not the temperature fluctuations can be related to IGW events. The temperature profile retrieval involves integral transforms and filtering that constitute the observation filter. (5) A simpler filter is implemented by the analysis of the RO amplitude fluctuation spectra, based on the diffraction theory in the framework of the phase screen and weak fluctuation approximations. The two spectral parameters, the external scale and the structural characteristic, define the specific potential energy. This approach allows the derivation of the spacial and seasonal distributions of IGW activity. We conclude that the success of IGW study by RO is stimulated by a large number of RO observations and advanced techniques based on Fourier and space–time analysis, physical equations describing IGWs, and diffraction theory. [ABSTRACT FROM AUTHOR]

In this paper, the interaction of a mass moving uniformly on an infinite beam on a three-layer viscoelastic foundation is analyzed with the objective of determining the lowest velocity at the stability limit, called, in this context, the critical velocity. This issue is important for rail transport and, in particular, for the high-speed train, because the moving mass is the basic model of a vehicle, and the infinite beam on a three-layer viscoelastic foundation is the usual mechanical representation of the railway track. In addition to this, the advantages and disadvantages of the two implemented methods, namely, the semi-analytical approach and the Green's function method, are summarized in terms of computational time, the precision of the obtained results, limitations, and the feasibility of implementation. All results are presented in a dimensionless form to cover a wide range of possible scenarios. Some results may be considered academic, however, results related to a particular railway track are also included. Particular attention is paid to the influence of the damping of materials in the foundation upon the critical velocity of the moving mass. Regarding the semi-analytical approach, it is demonstrated that the critical velocities can be obtained in an exact manner by tracing the branches of the so-called instability lines in the velocity–moving-mass plane. This analysis can be maintained within the real domain. As for the time series, they can be determined by a numerical inverse Laplace transform. Moreover, thanks to the analytical form of the final result in the Fourier domain, each value corresponding to a specific time instant can be obtained directly, that is, without the previous time history. Regarding the Green's function method, this is used to verify a few points delimiting the stable and unstable regions of the moving mass with the help of the D-decomposition approach. Additionally, a numerical algorithm based on the Green's function and convolution integral written for dimensionless quantities is used to calculate the time series of the moving mass. In addition to identifying the critical velocity of the moving mass, its connection with the critical velocity of the moving force is emphasized, and the possibility of validating the results on long finite beams using modal expansion is presented and described. [ABSTRACT FROM AUTHOR]

This manuscript focuses on new generalizations of q-Mittag-Leffler functions, called generalized hyper q-Mittag-Leffler functions, and discusses their regions of convergence and various fractional q operators. Moreover, the solutions to the q-fractional kinetic equations in terms of the investigated generalized hyper q-Mittag-Leffler functions are obtained by applying the q-Sumudu integral transform. Furthermore, we present solutions obtained as numerical graphs using the MATLAB 2018 program. [ABSTRACT FROM AUTHOR]

Under the revised market risk framework of the Basel Committee on Banking Supervision, the model validation regime for internal models now requires that models capture the tail risk in profit-and-loss (P&L) distributions at the trading desk level. We develop multi-desk backtests, which simultaneously test all trading desk models and which exploit all the information available in the presence of an unknown correlation structure between desks. We propose a multi-desk extension of the spectral test of Gordy and McNeil, which allows the evaluation of a model at more than one confidence level and contains a multi-desk value-at-risk (VaR) backtest as a special case. The spectral tests make use of realised probability integral transform values based on estimated P&L distributions for each desk and are more informative and more powerful than simpler tests based on VaR violation indicators. The new backtests are easy to implement with a reasonable running time; in a series of simulation studies, we show that they have good size and power properties. [ABSTRACT FROM AUTHOR]

The editorial discusses the use of integral transforms, specifically the Laplace transform, in converting functions from the time domain to the complex domain. It highlights the effectiveness of this method in analyzing initial value problems and the dynamic characteristics of linear systems. The editorial also mentions the challenges in achieving convergence and binding the kernel in integral transforms. Additionally, it explores the connection between integral transforms and convolutional neural networks in artificial intelligence (AI) and proposes the development of new algorithms to accelerate processes and reduce heating issues in AI computations. The editorial concludes by acknowledging the contributions of various researchers in the field of integral transforms. [Extracted from the article]

An analytical model to find the temperature field that has been developed for friction systems consists of a strip and semi-space. The strip is made of a two-component functionally graded material (FGM) with an exponentially changing coefficient of thermal conductivity. In contrast, the material of the semi-space is homogeneous. An appropriate boundary-value problem of heat conduction with constant specific friction power was formulated and solved using the Laplace integral transform method. The model takes into consideration the imperfect thermal friction contact between the strip and the semi-space, and also the convective cooling on the exposed surface of the strip. The appropriate asymptotic solutions to this problem for low and high values of Fourier number were obtained. It is shown how the determined exact solution can be generalized using Duhamel's formula for the case of a linearly reduction in time-specific friction power (a braking process with constant deceleration). Numerical analysis for selected materials of the friction pair was carried out in terms of examining the mutual impact on the temperature of the two Biot numbers, characterizing the intensity of the thermal contact conductivity and convective heat exchange on the exposed surface of the strip. The obtained results can be used to predict the temperature of friction systems containing elements made of FGM. In particular, such systems include modern disc braking systems. [ABSTRACT FROM AUTHOR]

In this paper, we extend the scope of the Tate and Ormerod Lemmas to the Dunkl setting, revealing a profound interconnection that intricately links the Dunkl transform and the Mellin transform. This illumination underscores the pivotal significance of the Mellin integral transform in the realm of fractional calculus associated with differential-difference operators. Our primary focus centers on the Dunkl–Laplace operator, which serves as a prototype of a differential-difference second-order operator within an unbounded domain. Following influential research by Pagnini and Runfola, we embark on an innovative exploration employing Bochner subordination approaches tailored for the fractional Dunkl Laplacian (FDL). Notably, the Mellin transform emerges as a robust and enlightening tool, particularly in its application to the FDL. [ABSTRACT FROM AUTHOR]

Ballasted railway tracks can be modeled using reduced/simplified models composed of several layers of discrete components. This paper deals with the two-layer model, which is very popular due to its computational efficiency. In order to provide some recommendations for track design, it is necessary to identify which set of parameters leads to some irregular/unexpected behavior. In this paper, irregularities are investigated at three levels, namely, (i) the critical velocity of a moving constant force, (ii) the instability of one moving mass, and (iii) the instability of two moving masses. All results are presented in a dimensionless form to cover a wide range of real parameters. Irregular cases are identified by sets of parameters leading to them, which is the main finding of this paper; then, general conclusions are drawn. Regarding the method, all results are obtained analytically or semi-analytically, where "semi" refers to solving the roots of a given polynomial using predefined numerical procedures in symbolic software. No numerical integration is involved in any of the results presented. This means that the results are highly accurate and refer to exact values, so any kind of parametric or sensitivity analyses is readily possible. [ABSTRACT FROM AUTHOR]

The main aim of this study is to implement the general integral transform technique to determine Ulam-type stability and Ulam–Hyers–Mittag–Leffer stability. We are given suitable examples to validate and support the theoretical results. As an application, the general integral transform is used to find Ulam stability of differential equations arising in Thevenin equivalent electrical circuit system. The results are graphically represented, which provides a clear and thorough explanation of the suggested method. [ABSTRACT FROM AUTHOR]

Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that Γ z e − s z cannot be integrated over positive real numbers. Secondly, Dirac delta function is a linear functional under which every function f is mapped to f (0) . This article combines both functions to solve the problems that have remained unsolved for many years. For instance, it has been demonstrated that the power law feature is ubiquitous in theory but challenging to observe in practice. Since the fractional derivatives of the delta function are proportional to the power law, we express the gamma function as a complex series of fractional derivatives of the delta function. Therefore, a unified approach is used to obtain a large class of ordinary, fractional derivatives and integral transforms. All kinds of q-derivatives of these transforms are also computed. The most general form of the fractional kinetic integrodifferential equation available in the literature is solved using this particular representation. We extend the models that were valid only for a class of locally integrable functions to a class of singular (generalized) functions. Furthermore, we solve a singular fractional integral equation whose coefficients have infinite number of singularities, being the poles of gamma function. It is interesting to note that new solutions were obtained using generalized functions with complex coefficients. [ABSTRACT FROM AUTHOR]

We present a systematic introduction to a class of functions that provide fundamental solutions for autonomous linear integer-order and fractional-order delay differential equations. These functions, referred to as delay functions, are defined through power series or fractional power series, with delays incorporated into their series representations. Using this approach, we have defined delay exponential functions, delay trigonometric functions and delay fractional Mittag-Leffler functions, among others. We obtained Laplace transforms of the delay functions and demonstrated how they can be employed in finding solutions to delay differential equations. Our results, which extend and unify previous work, offer a consistent framework for defining and using delay functions. [ABSTRACT FROM AUTHOR]

Fractional integrodifferential diffusion equations play a significant role in describing anomalous diffusion phenomena. In this paper, we study the existence and uniqueness of mild solutions to these equations. Firstly, we construct an appropriate resolvent family, through which the related equicontinuity, strong continuity, and compactness properties are studied using the convolution theorem of Laplace transform, the probability density function, the Cauchy integral formula, and the Fubini theorem. Then, we construct a reasonable mild solution for the considered equations. Finally, we obtain some sufficient conditions for the existence and uniqueness of mild solutions to the considered equations by some fixed point theorems. [ABSTRACT FROM AUTHOR]

The transportation of seawater on a grand scale via an ultra-large cold-water pipe situated within the context of ocean thermal energy conversion (OTEC) floating installations inherently presents challenges associated with instability and potential malfunction in the face of demanding operational circumstances. This study endeavors to augment the stability and security of cold-water pipe (CWP) operations by scrutinizing their vibrational attributes across diverse boundary configurations. Initially, we invoke Euler–Bernoulli beam theory to forge the analytical framework and proffer a semi-analytical resolution by utilizing the generalized integral transform technique (GITT). Subsequently, we authenticate the convergence and precision of our proposed approach through comparative analysis with extant theories. Our findings underscore the conspicuous influence of boundary conditions on the convergence of transverse displacement. The influence of internal flow on the transverse displacement and the natural frequency manifests substantial variability under different boundary conditions. Significantly, an escalation in the internal flow velocity triggers a concomitant reduction in the natural frequency, ultimately culminating in instability once the critical velocity threshold is reached. Additionally, the reliance of the transverse displacement and the natural frequency on the clump weight at the bottom is markedly pronounced. Our discoveries propose that pipe stability can be ameliorated by adjusting the clump weight at the bottom. Furthermore, the novel insights obtained through our proposed approach can significantly aid in the early-stage design and analysis of CWP. [ABSTRACT FROM AUTHOR]

Rapid technological advances have accelerated offshore and onshore tight oil extraction to meet growing energy demand. Reliable tools to carry out production prediction are essential for development of unconventional reservoirs. The existed tri-linear analytical solutions are verified to be versatile enough to capture fundamental flow mechanisms and make accurate production predictions. However, these solutions are obtained in Laplace space with the Laplace transform and numerical inversion, which may lead to uncertainty in the solution. In this paper, a general analytical solution is derived in real-time space through integral transform and average pressure substitution. Namely, the partial differential equations describing subsurface fluid flow are firstly triple-integrated and then the obtained volume average pressure are replaced with the rate-dependent expressions. Furthermore, the ordinary differential equations related to oil rate are solved analytically in real-time space. To validate our model, this derived solution is verified against two numerical models constructed with two typical physical configurations. The great match indicates the accuracy and applicability of the analytical solution. According to the developed workflow, two field cases including offshore and onshore tight oilfield data are selected for history matching and production prediction. This new approach not only makes the obtained solution more simplified, but also helps field engineers diagnose flow patterns more quickly to better optimize production schemes. [ABSTRACT FROM AUTHOR]

Tales of Times Now Past (Konjaku Monogatarishū 今昔物語集) is a Japanese setsuwa (anecdotes) collection dating from the early twelfth century. Originally contained in 31 volumes, it includes more than one thousand systematically arranged tales from India, China, and Japan. Despite the fact that Daoism was rejected by the Japanese imperial court as an organized religion, Daoist philosophies and practical systems found their ways into Japan, having a significant and profound influence over Japanese esoteric cosmology and folk beliefs. This article takes the Lord of Mount Tai (CN: Taishan Fujun; JP: Taizan Fukun 泰山府君) as the focus and examines two pertinent stories in Tales of Times Now Past. By placing the texts in a broader historical, religious, and comparative Japan–China perspective, I examine the reshaping of Daoist elements and traditional Chinese philosophical principles in these two stories and thus demonstrate how the Daoist fragments transmitted to Japan were transformed into an integral part of the orthodox structure by the time of the late Heian period in the pursuit of a more organized form of government. [ABSTRACT FROM AUTHOR]

This paper provides both analytical and numerical solutions of (PDEs) involving time-fractional derivatives. We implemented three powerful techniques, including the modified variational iteration technique, the modified Adomian decomposition technique, and the modified homotopy analysis technique, to obtain an approximate solution for the bounded space variable ν. The Laplace transformation is used in the time-fractional derivative operator to enhance the proposed numerical methods' performance and accuracy and find an approximate solution to time-fractional Fornberg–Whitham equations. To confirm the accuracy of the proposed methods, we evaluate homogeneous time-fractional Fornberg–Whitham equations in terms of non-integer order and variable coefficients. The obtained results of the modified methods are shown through tables and graphs. [ABSTRACT FROM AUTHOR]

The wave resistance caused by a point load steadily moving on an infinitely extended viscoelastic plate floating on an inviscid fluid is analytically studied, which can be used to describe the response due to the motion of amphibious air-cushion vehicles on the continuous ice sheet on the ocean. The action of concentrated and distributed point loads are both considered. Under the assumptions that the fluid is incompressible and homogeneous and the motion of the fluid is irrotational, the Laplace equation is taken as the governing equation. For the floating plate, the Kelvin–Voigt viscoelastic model is employed. At the plate–fluid interface, linearized boundary conditions are used when the wave amplitude generated is less than its wavelength. The Fourier integral transform is performed to achieve the formal solution. The residue theorem is applied to derive the response of flexural–gravity wave resistance. It is indicated that for a point load with a uniform rectilinear motion, the wave resistance shows a sharp decrease with the increase in the moving speed when the load velocity is greater than the minimum phase velocity. There is no steady wave resistance when the load velocity is smaller than the minimum phase velocity. The effects of different parameters are obtained. Wave resistance decreases with the increasing plate thickness, viscoelastic parameter, and Poisson's ratio, especially for a small value of viscoelastic parameter. [ABSTRACT FROM AUTHOR]

In this paper, we consider the modified Mellin transform of the product of the square of the Riemann zeta function and the exponentially decreasing function, and we discuss its probabilistic and approximation properties. It turns out that this Mellin transform approximates the identical zero in the strip { s ∈ C : 1 / 2 < σ < 1 } . [ABSTRACT FROM AUTHOR]

In this article, we study the fractional SIR epidemic model with the Atangana–Baleanu–Caputo fractional operator. We explore the properties and applicability of the ZZ transformation on the Atangana–Baleanu–Caputo fractional operator as the ZZ transform of the Atangana–Baleanu–Caputo fractional derivative. This study is an application of two power methods. We obtain a special solution with the homotopy perturbation method (HPM) combined with the ZZ transformation scheme; then we present the problem and study the existence of the solution, and also we apply this new method to solving the fractional SIR epidemic with the ABC operator. The solutions show up as infinite series. The behavior of the numerical solutions of this model, represented by series of the evolution in the time fractional epidemic, is compared with the Adomian decomposition method and the Laplace–Adomian decomposition method. The results showed an increase in the number of immunized persons compared to the results obtained via those two methods. [ABSTRACT FROM AUTHOR]

Theis' theory (1935), later improved by Hantush & Jacob (1955) and Moench (1971), is a technique designed to study the water level in aquifers. The key formula in this theory is a certain integral transform H [ g ] (r , t) of the pumping function g that depends on the time t and the relative position r to the pumping point as well as on other physical parameters. Several analytic approximations of H [ g ] (r , t) have been investigated in the literature that are valid and accurate in certain regions of r, t and the mentioned physical parameters. In this paper, the analysis of possible analytic approximations of H [ g ] (r , t) is completed by investigating asymptotic expansions of H [ g ] (r , t) in a region of the parameters that is of interest in practical situations, but that has not yet been investigated. Explicit and/or recursive algorithms for the computation of the coefficients of the expansions and estimates for the remainders are provided. Some numerical examples based on an actual physical experiment conducted by Layne-Western Company in 1953 illustrate the accuracy of the approximations. [ABSTRACT FROM AUTHOR]

A mathematical model for determining the temperature field of a bimetallic plate with plane-parallel boundaries during short-term induction heating by a non-stationary electromagnetic field is proposed. Initial boundary value problems for determining the parameters of a non-stationary electromagnetic field and temperature are formulated. The temperature and the component of the magnetic field intensity vector that is tangential to the plate base were selected as defining functions. We used an approximation of the defining functions in each layer of the plate with quadratic polynomials by the thickness coordinate and Laplace transform of the integral over time. General solutions to the formulated problems under uniform non-stationary electromagnetic action were obtained. Based on them, the temperature during short-term induction heating by a non-stationary electromagnetic field was numerically analyzed depending on its amplitude-frequency parameters and duration. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel integral transform known as the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT). We first define the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT) of integrable (and square integrable) functions on R. Later on, we show that 1D-QQPFT satisfies all the respective properties such as inversion formula, linearity, Moyal's formula, convolution theorem, correlation theorem and uncertainty principle. Moreover, we use the proposed transform to obtain an inversion formula for two-dimensional quaternion quadratic-phase Fourier transform. Finally, we highlight our paper with some possible applications. [ABSTRACT FROM AUTHOR]

In this manuscript, the nonlinear Burgers equations are studied via a fractal fractional (FF) Caputo operator. The results of coupled fixed point theorems in cone metric space are used to discuss the uniqueness of solution to the proposed coupled equations. The solution of the proposed equation is computed via Natural transform associated with the Adomian decomposition method (NADM). The acquired results are graphically presented for some values of fractional order and fractal dimensions. The accuracy and consistency of the applied method is verified through error analysis. [ABSTRACT FROM AUTHOR]

Blockchain 3.0, an advanced iteration of blockchain technology, has emerged with diverse applications encompassing various sectors such as identity authentication, logistics, medical care, and Industry 4.0/5.0. Notably, the integration of blockchain with industrial automation and control systems (IACS) holds immense potential in this evolving landscape. As industrial automation and control systems gain popularity alongside the widespread adoption of 5G networks, Internet of Things (IoT) devices are transforming into integral nodes within the blockchain network. This facilitates decentralized communication and verification, paving the way for a fully decentralized network. This paper focuses on showcasing the implementation and execution results of data preservation from industrial automation and control systems to IOTA, a prominent distributed ledger technology. The findings demonstrate the practical application of IOTA in securely preserving data within the context of industrial automation and control systems. The presented numerical results validate the effectiveness and feasibility of leveraging IOTA for seamless data preservation, ensuring data integrity, confidentiality, and transparency. By adopting IOTA's innovative approach based on Directed Acyclic Graph (DAG), the paper contributes to the advancement of blockchain technology in the domain of Industry 4.0/5.0. [ABSTRACT FROM AUTHOR]

In this paper, we consider a generalized Mittag-Leffler (ML)-type function and establish several integral formulas involving Jacobi and related transforms. We also establish some of the composition of generalized fractional derivative formulas associated with the generalized Mittag-Leffler (ML)-type function. Additionally, certain special cases of generalized fractional derivative formulas involving the Mittag-Leffler (ML)-type function are corollarily presented. [ABSTRACT FROM AUTHOR]

In this paper, a combined Laplace transform (LT) and boundary element method (BEM) is used to find numerical solutions to problems of anisotropic functionally graded media that are governed by the transient diffusion–convection–reaction equation. First, the variable coefficient governing equation is reduced to a constant coefficient equation. Then, the Laplace-transformed constant coefficients equation is transformed into a boundary-only integral equation. Using a BEM, the numerical solutions in the frame of the Laplace transform may then be obtained from this integral equation. Then, the solutions are inversely transformed numerically back to the original time variable using the Stehfest formula. The numerical solutions are verified by showing their accuracy and steady state. For symmetric problems, the symmetry of solutions is also justified. Moreover, the effects of the anisotropy and inhomogeneity of the material on the solutions are also shown, to suggest that it is important to take the anisotropy and inhomogeneity into account when performing experimental studies. [ABSTRACT FROM AUTHOR]

Different studies highlight the importance of developing entrepreneurial competence (EC) from the earliest educational levels, as well as during the teacher training stages, as they are considered to be facilitating and transforming agents in the integral development of students. The purpose of this research is to analyse the self-perceived level of knowledge and implementation of EC among active teachers from non-university centres managed by the Sagrada Familia Foundation, Spain. After designing a questionnaire based, among others sources, on EntreComp and EntreCompEdu frameworks, an expert judgement was carried out. A total of 326 participants answered the questionnaire. The result show that more than 70% of respondents had not heard about EntreComp and EntreCompEdu; more than 50% had no knowledge of entrepreneurial education (EE); and the respondents as a whole stand out in terms of perseverance, motivation, autonomy, decision-making and teamwork; only 14–18% of respondents plan subjects that pay attention to ECs. Moreover, almost 50% of respondents state that they use participatory learning methodologies, while only 13% believe that when evaluating pupils' performance, it is necessary to support sharing achievements and progress with internal and external agents; only a further 15% involve students in the assessment process. We must advocate and work for the development of 'teacherpreneurs', and start developing EC in all stages of teacher training. [ABSTRACT FROM AUTHOR]

The development of risk models for managing portfolios of financial institutions and insurance companies requires, both from the regulatory and management points of view, a strong validation of the quality of the results provided by internal risk models. In Solvency II, for instance, regulators ask for independent validation reports from companies who apply for the approval of their internal models. We analyze here various ways to enable management and regulators to gain confidence in the quality of models. It all starts by ensuring a good calibration of the risk models and the dependencies between the various risk drivers. Then, by applying stress tests to the model and various empirical analyses, in particular the probability integral transform, we can build a full and credible framework to validate risk models. [ABSTRACT FROM AUTHOR]

This paper focuses on the finite-time formation-control problem of a multi-AUV formation under unknown perturbations with prescribed performance. First, the nonlinear AUV model is transformed into a second-order integral model using feedback linearization. Suitable prescribed performance functions are selected to constrain the control errors of AUVs within a preset range and convert AUV tracking errors into unconstrained tracking errors using an error-conversion function to facilitate controller design. Finite-time sliding-mode disturbance observers are designed for unknown disturbances in the ocean so that they can accurately estimate the unknown disturbances in finite time. Based on the unconstrained tracking error and the unknown disturbance observer, the fast terminal sliding-mode formation controller is designed so that the multi-AUV formation can converge in finite time. Finally, the simulation experimental results show that the finite-time formation-control method with prescribed performance proposed in this paper can better cancel the unknown disturbance in the ocean in finite time and improve the robustness of the multi-AUV formation control. [ABSTRACT FROM AUTHOR]

Symmetry can play an important role in the study of boundary value problems, which are a type of problem in mathematics that involves finding the solutions to differential equations subject to given boundary conditions. Integral transforms play a crucial role in solving ordinary differential equations (ODEs), partial differential equations (PDEs), and integral equations. This article focuses on extending a single-valued Sawi transform to a double-valued ST, which we call the double Sawi (DS) transform. We derive some fundamental features and theorems for the proposed transform. Finally, we study the applications of the proposed transform by solving some boundary value problems such as the Fourier heat equation and the D'Alembert wave equation. [ABSTRACT FROM AUTHOR]

The focus of the research presented in this paper is on a new generalized family of degenerate three-variable Hermite–Appell polynomials defined here using a fractional derivative. The research was motivated by the investigations on the degenerate three-variable Hermite-based Appell polynomials introduced by R. Alyosuf. We show in the paper that, for certain values, the well-known degenerate Hermite–Appell polynomials, three-variable Hermite–Appell polynomials and Appell polynomials are seen as particular cases for this new family. As new results of the investigation, the operational rule for this new generalized family is introduced and the explicit summation formula is established. Furthermore, using the determinant formulation of the Appell polynomials, the determinant form for the new generalized family is obtained and the recurrence relations are also determined considering the generating expression of the polynomials contained in the new generalized family. Certain applications of the generalized three-variable Hermite–Appell polynomials are also presented showing the connection with the equivalent results for the degenerate Hermite–Bernoulli and Hermite–Euler polynomials with three variables. [ABSTRACT FROM AUTHOR]