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This research paper deals with an extension of the non-central Wishart introduced in 1944 by Anderson and Girshick, that is the non-central Riesz distribution when the scale parameter is derived from a discrete vector. It is related to the matrix of normal samples with monotonous missing data. We characterize this distribution by means of its Laplace transform and we give an algorithm for generating it. Then we investigate, based on the method of the moment, the estimation of the parameters of the proposed model. The performance of the proposed estimators is evaluated by a numerical study. [ABSTRACT FROM AUTHOR]

A high-order time discretization scheme to approximate the time-fractional wave equation with the Caputo fractional derivative of order α ∈ (1 , 2) is studied. We establish a high-order formula for approximating the Caputo fractional derivative of order α ∈ (1 , 2) . Based on this approximation, we propose a novel numerical method to solve the time-fractional wave equation. Remarkably, this method corrects only one starting step and demonstrates second-order convergence in both homogeneous and inhomogeneous cases, regardless of whether the data is smooth or nonsmooth. We also analyze the stability region associated with the proposed numerical method. Some numerical examples are given to elucidate the convergence analysis. [ABSTRACT FROM AUTHOR]

Various constitutive models have been proposed to quantify a wide range of non-Newtonian fluids, but there is lack of a systematic classification and evaluation of these competing models, such as the quantitative comparison between the classical integer-order constitutive models and the newly proposed fractional derivative equations for non-Newtonian fluids. This study reviews constitutive equation models for non-Newtonian fluids, including time-independent fluids, viscoelastic fluids, and time-dependent fluids. A comparison between fractional derivative non-Newtonian fluid constitutive equations and traditional constitutive equations is also provided. Results show that the space fractional derivative model is equivalent to some classical constitutive models under reasonable assumptions. Further discussions are made from the perspective of the industrial and biomedical applications of non-Newtonian fluids. Advantages and limitations of the constitutive models are also explored to help users to select proper models for real-world applications. [ABSTRACT FROM AUTHOR]

In this article, convolution-type fractional derivatives generated by Dickman subordinator and inverse Dickman subordinator are discussed. The Dickman subordinator and its inverse are generalizations of stable and inverse stable subordinators, respectively. The series representations of densities of the Dickman subordinator and inverse Dickman subordinator are also obtained, which could be helpful for computational purposes. Moreover, the space and time-fractional Poisson-Dickman processes, space-fractional Skellam Dickman process and non-homogenous Poisson-Dickman process are introduced and their main properties are studied. [ABSTRACT FROM AUTHOR]

In this study, using convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein’s theorem for completely monotonic functions, and other analytic techniques, the authors verify decreasing property of a ratio between three derivatives of a function involving trigamma function and find the necessary and sufficient conditions for a function defined by three derivatives of a function involving trigamma function to be completely monotonic. These results confirm previous guesses posed by Qi and generalize the corresponding known conclusions.

In this article, we formulate and solve the representation problem for diffusion equations: giving a discretization of the Laplace transform of a diffusion equation under a space discretization over a space scale determined by an increment h > 0 , can we construct a continuous in h family of Cauer ladder networks whose constitutive equations match for all h > 0 the discretization. It is proved that for a finite differences discretization over a uniform geometric space scale, the representation problem over fractal Cauer networks is possible if and only if the coefficients of the diffusion are exponential functions in the space variable. Such diffusion equations admit a (Laplace) transfer function with a fractional behavior whose exponent is explicit. This allows us to justify previous works made by Sabatier and co-workers in [15-16] and Oustaloup and co-workers [14]. [ABSTRACT FROM AUTHOR]

Sufficient conditions are given for a function F ⁢ ( p ) {F(p)} to be the Laplace transform of a function f ⁢ ( t ) {f(t)} or a distribution f. No assumption on f is given a priori. It is not even assumed that f = 0 {f=0} for t < 0 {t<0} . [ABSTRACT FROM AUTHOR]

In this paper, we solve the question completely raised by Feng Qi and Dongkyu Lim in paper "Monotonicity properties for a ratio of finite many gamma functions" published in Advances in Difference Equations and get some properties about ratios of finitely many gamma functions such as complete monotonicity, logarithmically complete monotonicity, the Bernstein function property, null point and extreme value. [ABSTRACT FROM AUTHOR]

We develop a new generalized form of the fractional kinetic equation involving the generalized Mathieu series. By using the Sumudu transform, a solution of these generalized fractional kinetic equation is obtained in terms of the Mittag-Leffler function. The numerical results and graphical interpretation are also presented. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to revisit the proof of the Gearhardt–Prüss–Huang–Greiner theorem for a semigroup S(t), following the general idea of the proofs that we have seen in the literature and to get an explicit estimate on the operator norm of S(t) in terms of bounds on the resolvent of the generator. In Helffer and Sjöstrand (From resolvent bounds to semigroup bounds. , math. FA, 2010) by the first two authors, this was done and some applications in semiclassical analysis were given. Some of these results have been subsequently published in three books written by the two first authors Helffer (Spectral theory and its applications. Cambridge University Press, Cambridge, 2013) and Sjöstrand (Lecture notes : Spectral properties of non-self-adjoint operators. Journées équations aux dérivées partielles (2009), article no. 1), (Non self-adjoint differential operators, spectral asymptotics and random perturbations. Pseudo-differential Operators and Applications, Birkhäuser (2018)). A second work Helffer and Sjöstrand (Integral Equ Oper Theory 93(3), 2021) presents new improvements partially motivated by a paper of Wei (Sci China Math 64:507–518, 2021). In this third paper, we continue the discussion on whether the aforementioned results are optimal, and whether one can improve these results through iteration. Numerical computations will illustrate some of the abstract results. [ABSTRACT FROM AUTHOR]

In this article, we introduce and study a one sided tempered stable first order autoregressive model called TAR(1). Under the assumption of stationarity of the model, the marginal probability density function of the error term is found. It is shown that the distribution of the error term is infinitely divisible. Parameter estimation of the introduced TAR(1) process is done by adopting the conditional least square and method of moments based approach and the performance of the proposed methods are evaluated on simulated data. Also, we study an autoregressive model of order one with tempered stable innovations. Using appropriate test statistic it is shown that the model fits very well on real and simulated data. Our models generalize the inverse Gaussian and one-sided stable autoregressive models existing in the literature. [ABSTRACT FROM AUTHOR]

A full-discrete scheme is presented for a Langevin equation involving Caputo fractional derivative and additive white noise. Based on a spectral truncation of white noise, the fractional Langevin equation is converted to an approximate equation with random parameters and a finite difference scheme is constructed. Consistency of the approximate equation as well as error estimate of the finite difference scheme are obtained. It is proved that when spectral truncation level and the step size are inversely proportional, the convergence order of the finite difference scheme is 1.5 in mean-square sense and independent of the value of fractional derivative order. Numerical examples verify the theoretical analysis. Moreover, to fulfill long-time simulation, a scheme based on piecewise spectral approximation of white noise is further developed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse Laplace transform of the Volterra μ-function (the Bromwich integral) and evaluate it with respect to the Hankel, Schläfli and Bourguet complex contours. In this sense, we establish the generalized Ramanujan's integral representations of the Volterra μ-function for general variations of the parameters. We also discuss the asymptotic analysis of this function with large parameters using the steepest descent method. Further, we show that the solution of Volterra integral equation with differentiated-order fractional integral operator is the Volterra μ-function. [ABSTRACT FROM AUTHOR]

Recently, integral transforms are a powerful tool used in many areas of mathematics, physics, engineering, and other fields and disciplines. This article is devoted to the study of one important integral transform, which is called the modified degenerate Laplace transform (MDLT). The fundamental formulas and properties of the MDLT are obtained. Furthermore, as an application of the acquired MDLT, we solved a simple differential equation and fractional-order kinetic equations. The outcomes covered here are general in nature and easily reducible to new and known outcomes.

Like qq-calculus, Hahn calculus (or q,ωq,\omega -calculus) is constructed by defining a difference derivative operator and an integral operator. The q,ωq,\omega -analogs of the integral representations of the Laplace transform and related special functions, such as gamma and beta, are proposed in this article. Then, some basic properties similar to classical and qq-analogs are investigated. Finally, a few examples are given to solve q,ωq,\omega -initial value problems via the newly introduced q,ωq,\omega -Laplace transform.

In this paper, we study the heat transfer modeling during freezing of a biological tissue and present an analytical approach for solving the heat transfer problem in cryosurgery. We consider a time-fractional bio-heat equation in the cylindrical coordinate and employ the Wiener–Hopf technique to find the temperature of tissue in two different domains by the factorization of associated Wiener–Hopf kernel. We discuss the fundamental roles of the Bessel and Wright functions in determining the analytical solution of fractional cryosurgery problem. [ABSTRACT FROM AUTHOR]

The derivation of the sample covariance is difficult compared to that of the distribution of the sample correlation coefficient. This paper deals with the distributions of covariance structures appearing in real scalar/vector/matrix variables. Covariance structure is a bilinear structure. Consider a bilinear form u = X ′ A Y where X and Y are p × 1 and q × 1 real vectors and A is a constant p × q matrix. The basic aim in this paper is to derive the distribution of such a structure when the components are scalar/vector/matrix Gaussian variables. The procedure used is to examine the Laplace transform or the moment generating function (mgf) coming from such a bilinear form in real scalar/vector/matrix variables. Covariance structures in several situations are shown to produce a mgf of the type (1 − λ 2 t 2) − α , λ > 0 , α > 0 , − 1 λ < R (t) < 1 λ where t is the mgf parameter, R (·) means the real part of (·) , and λ and α are real scalar parameters. Explicit evaluation of the density of u is considered when α is a positive integer as well as for a general α. It is shown that the exact densities can be written as linear functions of double gamma densities and double exponential or Laplace densities when α is a positive integer. For the general value of α, it is shown that the exact density can be written in terms of double Mittag-Leffler or a double confluent hypergeometric function. [ABSTRACT FROM AUTHOR]

This paper is concerned with the approximate controllability of a class of semi-linear neutral integro-differential equation with nonlocal condition. The basic tools of this study are the theory of resolvent operators of linear neutral integro-differential equation and fractional powers. The fundamental solution of linear neutral integro-differential equations is also constructed to deal with the non-uniform boundedness of the extra linear term. Sufficient conditions of approximate controllability are then obtained through the resolvent condition. Finally, an example is provided to illustrate the applications of the obtained results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the heat transfer modeling during freezing of a biological tissue and present an analytical approach for solving the heat transfer problem in cryosurgery. We consider a time-fractional bio-heat equation in the cylindrical coordinate and employ the Wiener–Hopf technique to find the temperature of tissue in two different domains by the factorization of associated Wiener–Hopf kernel. We discuss the fundamental roles of the Bessel and Wright functions in determining the analytical solution of fractional cryosurgery problem.

In this article, with the help of an integral representation of the Dirichlet lambda function, by means of a monotonicity rule for the ratio of two integrals with a parameter, and by virtue of complete monotonicity and another property of an elementary function involving the exponential function, the authors find increasing property and logarithmic convexity of two functions containing the gamma function and the Dirichlet lambda function.

We investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterization of such operators is performed in the Laplace domain, it is necessary to resort to accurate numerical methods to derive the corresponding behaviours in the time domain. In this regard, we develop a computational procedure to solve variable-order fractional differential equations of this novel class. Furthermore, we provide some numerical experiments to show the effectiveness of the proposed technique. [ABSTRACT FROM AUTHOR]

In this paper, we propose a methodology for computing the analytic solutions of linear retarded delay-differential equations and neutral delay-differential equations that include Dirac delta function inputs. In numerous applications, the delta function serves as a convenient and effective surrogate for modeling high voltages, sudden shocks, large forces, impulse vaccinations, etc., applied over a short period of time. The solutions are obtained using the Laplace transform method, in conjunction with the Cauchy residue theorem. The accuracy of these solutions are assessed by comparing them with the ones provided by the method of steps. Numerical examples illustrating the methodology are presented and discussed. These examples show that the Laplace transform solution is very reliable for linear retarded delay-differential equations, because the analytic solution, for a single delta function input, is continuous. However, for linear neutral delay-differential equations with a delta function input the analytic solution is discontinuous. Consequently, the well-known Gibbs phenomenon is observed in the vicinity of the discontinuities. However, for neutral delay differential equations, we show that in some cases, the magnitude of the jumps at the discontinuities decrease, as time increases. Therefore, the Gibbs phenomenon of the Laplace solution dissipates. [ABSTRACT FROM AUTHOR]

Inspired by Jaming's characterization of the Fourier transform on specific groups via the convolution property, we provide a novel approach which characterizes the Fourier transform on any locally compact abelian group. In particular, our characterization encompasses Jaming's results. Furthermore, we demonstrate that the cosine transform as well as the Laplace transform can also be characterized via a suitable convolution property. [ABSTRACT FROM AUTHOR]

The main aim of this article is to establish a closed-form expression for the Neumann series built with Macdonald functions in which order contains the summation index and the coefficients include the summation index exponentially and factorially. Motivated by this problem which occurs in several problems of physics or electrochemistry such as diffusion phenomena, we derive two integral representations of a general Neumann series of this type containing Macdonald functions as building blocks. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain some interesting infinite series representations for the Apostol-type parametrically generalized polynomials with the aid of the Laplace transform and generating functions. In particular, by using the method of generating functions, we derive not only recurrence relations, but also several other formulas, identities, and relations as well as combinatorial sums for these parametrically generalized numbers and polynomials and for other known special numbers and polynomials. These identities, relations and combinatorial sums are related to the two-parameter types of the Apostol–Bernoulli polynomials of higher order, the two-parameter types of Apostol–Euler polynomials of higher order, the two-parameter types of Apostol–Genocchi polynomials of higher order, the Apostol–Bernoulli polynomials of higher order, the Apostol–Euler polynomials of higher order, the Apostol–Genocchi polynomials of higher order, the cosine- and sine-Bernoulli polynomials, the cosine- and sine-Euler polynomials, the λ -array-type polynomials, the λ -Stirling numbers, the polynomials C n x , y , and the polynomials S n x , y . Finally, we present several new recurrence relations for these special polynomials and numbers. [ABSTRACT FROM AUTHOR]

An electrostatic precipitator is an essential piece of equipment to eliminate atmospheric dust pollution. Its power supply is of great significance to the working efficiency and reliability of the dust collector. This paper uses field-programmable gate array (FPGA) technology to design an electrostatic precipitator power controller. In this paper, a fuzzy control scheme with variable scale factors is adopted to make the highest average voltage between the two poles of the precipitator better approach the critical breakdown voltage. In this paper, the transfer function is derived based on the finite element differential equation and the method of three-dimensional drawing. In this way, the influence of parameters such as the coupling inductance of the pulse power supply, the coupling coefficient, and the parasitic resistance of the dust removal chamber on the load oscillation can be obtained. It is found that the control system can control the electrostatic precipitator power well. It can significantly improve dust removal efficiency.

It is more difficult to give Laplace transform directly in a defined form or derive it by Fourier transform in mathematics teaching. The article gives a solution for solving high exponential series sum by using Laplace transform. With the help of Laplace transform, calculus operations can be transformed into complex plane algebra operations. The application of the algorithm to the option hedging strategy verifies the applicability of the algorithm proposed in this article.

In this article, we present a novel extended exponential kernel Laplace-type integral transform. The Laplace, natural, and Sumudu transforms are all included in the suggested transform. The existence theorem, Parseval-type identity, inversion formula, and other fundamental aspects of the new integral transform are examined in this article. Integral identities define the connections between the new transforms and the established transforms. In order to solve specific initial-boundary value problems, the new transforms are used.

In this paper we study the global solvability of Riemann-Liouville and Caputo time-fractional integro-differential equations in a space of functions with square average power growth. We prove the uniqueness of corresponding inverse problems for one point observation. [ABSTRACT FROM AUTHOR]

This article is about finding the possible solutions of fractional kinetic equation associated with the generalized multiindex Bessel function using the method of Laplace transform. Numerical results and graphical solutions of the main theorems are also presented. [ABSTRACT FROM AUTHOR]

The composition operators preserving total non-negativity and totalpositivity for various classes of kernels are classified, following threethemes. Letting a function act by post composition on kernels with arbitrarydomains, it is shown that such a composition operator maps the set of totallynon-negative kernels to itself if and only if the function is constant orlinear, or just linear if it preserves total positivity. Symmetric kernels arealso discussed, with a similar outcome. These classification results are abyproduct of two matrix-completion results and the second theme: an extensionof A.M. Whitney's density theorem from finite domains to subsets of the realline. This extension is derived via a discrete convolution with modulatedGaussian kernels. The third theme consists of analyzing, with tools fromharmonic analysis, the preservers of several families of totally non-negativeand totally positive kernels with additional structure: continuous Hankelkernels on an interval, P\'olya frequency functions, and P\'olya frequencysequences. The rigid structure of post-composition transforms of totallypositive kernels acting on infinite sets is obtained by combining severalspecialized situations settled in our present and earlier works.

COVID-19, a novel coronavirus disease, is still causing concern all over the world. Recently, researchers have been concentrating their efforts on understanding the complex dynamics of this widespread illness. Mathematics plays a big role in understanding the mechanism of the spread of this disease by modeling it and trying to find approximate solutions. In this study, we implement a new technique for an approximation of the analytic series solution called the multistep Laplace optimized decomposition method for solving fractional nonlinear systems of ordinary differential equations. The proposed method is a combination of the multistep method, the Laplace transform, and the optimized decomposition method. To show the ability and effectiveness of this method, we chose the COVID-19 model to apply the proposed technique to it. To develop the model, the Caputo-type fractional-order derivative is employed. The suggested algorithm efficacy is assessed using the fourth-order Runge-Kutta method, and when compared to it, the results show that the proposed approach has a high level of accuracy. Several representative graphs are displayed and analyzed in two dimensions to show the growth and decay in the model concerning the fractional parameter α values. The central processing unit computational time cost in finding graphical results is utilized and tabulated. From a numerical viewpoint, the archived simulations and results justify that the proposed iterative algorithm is a straightforward and appropriate tool with computational efficiency for several coronavirus disease differential model solutions.

The Laplace transform method is applied in this article to study the semi-Hyers-Ulam-Rassias stability of a Volterra integro-differential equation of order n, with convolution-type kernel. This kind of stability extends the original Hyers-Ulam stability whose study originated in 1940. A general integral equation is formulated first, and then some particular cases (polynomial function and exponential function) for the function from the kernel are considered.

By using the Laplace transform method, we revisit the multivariate Mittag-Leffler function as an effective tool to construct a solution for some classes of fractional differential equations with constant coefficients. To support our results, we discuss several particular cases related to classical fractional differential operators. The techniques are not only restricted to fractional derivative operators but also can be applied to general constant coefficient differential equations, including high-order ordinary differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the exact solutions of linear homogeneous and nonhomogeneous fractional differential equations with double delays. Firstly, a new concept of double-delayed Mittag-Leffler type matrix function is introduced, which is the promotion of the double-delayed matrix exponential. Secondly, we apply the double-delayed Mittag-Leffler type matrix function and Laplace transform approach to obtain the exact solutions of fractional differential equations with double delays. Furthermore, the solution is used to investigate the Hyers-Ulam stability of the system. Lastly, we illustrate our techniques by an example. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of Caputo time fractional pseudo-hyperbolic equations of higher order with purely nonlocal conditions of integral type. We use an a priori estimate method; the so-called energy inequalities method, based on some functional analysis tools, is developed for a Caputo time fractional of 2 ⁢ m -th and (2 ⁢ m + 1) -th order and the density of the range of the operator generated by the considered problem. Using the Laplace transform and homotopy perturbation, we find a semi-analytical solution. Finally, we give some examples for illustration. [ABSTRACT FROM AUTHOR]

In this paper, we investigate and evaluate the analytical expressions for some definite integrals of Srinivasa Ramanujan in terms of Meijer's G-function by using the Laplace transforms of sin ⁡ (β ⁢ x 2) , cos ⁡ (β ⁢ x 2) , x ⁢ sin ⁡ (β ⁢ x 2) and x ⁢ cos ⁡ (β ⁢ x 2) . In addition, we investigate a number of infinite summation formulas involving Meijer's G-function and closed-form evaluation of some related infinite series. [ABSTRACT FROM AUTHOR]

This article is devoted to propose and investigate the fractal battery discharging model, which is one of the well-known models with a memory effect. It is presented as to how non-locality affects the behavior of solutions and how the current state of the system is affected by its past. Firstly, we present a local fractal solution. Then we solve the non-local fractal differential equation and examine the memory effect that includes the Mittag-Leffler function with one parameter. For that aim, the local fractal and non-local fractal Laplace transforms are used to achieve fractional solutions. In addition, the simulation analysis is performed by comparing the underlying fractal derivatives to the classical ones in order to understand the significance of the results. The effects of the fractal parameter and the fractional parameter are discussed in the conclusion section. [ABSTRACT FROM AUTHOR]