This article is devoted to the study of pseudo-differential operators generated by Dunkl operators, focusing primarily on their boundedness properties. We establish that, under a set of suitable assumptions on the symbols and the underlying function spaces, these operators are bounded on specific Banach spaces. In addition, we define the composition of pseudo-differential operators generated by Dunkl operators and rigorously prove that this composition also inherits boundedness properties under appropriate conditions. The analysis is carried out using techniques based on the Dunkl transform, which provides a powerful tool for handling operators associated with reflection groups and allows for the derivation of precise estimates. Beyond the theoretical development, we illustrate an application of the obtained results, demonstrating how these boundedness properties can be employed to address complex problems in mathematical physics and harmonic analysis. Overall, the work contributes to a deeper understanding of Dunkl analysis and the structure of pseudo-differential operators in this context. The results presented not only consolidate existing knowledge but also open new perspectives for further investigations in the field, providing a solid foundation for future research on Dunkl operators and their applications in both theoretical and applied analysis.
In this paper, the initial value problem for a third-order partial differential equation with time delay within a Hilbert space was analyzed. We establish a key theorem regarding the stability of this problem. Additionally, we demonstrate how this stability theorem can be applied to the third-order partial differential equation with time delay.
We study the asymptotic distribution for eigenvalues of fourth-order fractional Sturm-Liouville with Dirichlet boundary condition. In this work, we use the inverse Laplace transform method and the Asymptotic formula of the Mittag-Leffler function to get an analytical solution of the fractional Sturm-Liouville problems. When the fractional-order approaches 1, our results agree with the classical ones of fourth-order differential equations.
Christoph Wilms, Ann-Kathrin Ekat, Katja Hertha-Dunkel
et al.
Trustworthy volumetric flow measurements are essential in many applications such as power plant controls or district heating systems. Flow metering under disturbed flow conditions, such as downstream of bends, is a challenge and leads to errors of up to 20 %. In this paper, an algorithm based on a shallow neural network (SNN) is developed, leading to a significant error reduction for strongly disturbed flow profiles. To cover a wide range of disturbances, the training dataset was chosen to consist of three base types of elbow configurations. For 83 % of the test data, the SNN produces a smaller error than the state-of-the-art approach. The average error is reduced from 2.25 % to 0.42 %. For the SNN, an error of less than 1 % can be achieved for downstream distances greater than 10 pipe diameters. The SNN demonstrated robustness to various reductions of the training dataset, as well as to noisy input data. Additionally, simulation data of a realistic pipe system with a significantly different geometry compared with the training data was used for testing. In this strong extrapolation, the mean error of the SNN was always smaller than the state-of-the-art approach and an error of less than 1 % could be achieved for more than 10 pipe diameters downstream of the last disturbance.
The object of the present paper is to investigate generalizations of the hypergeometric function and Srivastava fractional integral calculus by using a general version of gamma function(namely (r,k)-gamma function). • Some fundamental results for these new concepts are provided. • We introduced differential subordination and superordination results associated with the defined new fractional integral operator. • Also, we establish sandwich results for p-valent analytic functions involving this operator. • Finally, an application to fluid mechanics is discussed.
N. S. Lagopoulos, G. D. Weymouth, B. Ganapathisubramani
In this work, we describe the impact of aspect ratio ($AR$) on the performance of optimally phased, identical flapping flippers in a tandem configuration. Three-dimensional simulations are performed for seven sets of single and tandem finite foils at a moderate Reynolds number, with thrust producing, heave-to-pitch coupled kinematics. Increasing slenderness (or $AR$) is found to improve thrust coefficients and thrust augmentation but the benefits level off towards higher values of $AR$. However, the propulsive efficiency shows no significant change with increasing $AR$, while the hind foil outperforms the single by a small margin. Further analysis of the spanwise development and propagation of vortical structures allows us to gain some insights into the mechanisms of these wake interactions and provide valuable information for the design of novel biomimetic propulsion systems.
Numerous physics theories are rooted in partial differential equations (PDEs). However, the increasingly intricate physics equations, especially those that lack analytic solutions or closed forms, have impeded the further development of physics. Computationally solving PDEs by classic numerical approaches suffers from the trade-off between accuracy and efficiency and is not applicable to the empirical data generated by unknown latent PDEs. To overcome this challenge, we present KoopmanLab, an efficient module of the Koopman neural operator (KNO) family, for learning PDEs without analytic solutions or closed forms. Our module consists of multiple variants of the KNO, a kind of mesh-independent neural-network-based PDE solvers developed following the dynamic system theory. The compact variants of KNO can accurately solve PDEs with small model sizes, while the large variants of KNO are more competitive in predicting highly complicated dynamic systems govern by unknown, high-dimensional, and non-linear PDEs. All variants are validated by mesh-independent and long-term prediction experiments implemented on representative PDEs (e.g., the Navier–Stokes equation and the Bateman–Burgers equation in fluid mechanics) and ERA5 (i.e., one of the largest high-resolution global-scale climate datasets in earth physics). These demonstrations suggest the potential of KoopmanLab to be a fundamental tool in diverse physics studies related to equations or dynamic systems.
Generalized hypergeometric functions and their natural generalizations in one and several variables appear in many mathematical problems and their applications. Solving partial differential equations encountered in many applied problems of mathematics physics is expressed in terms of such generalized hypergeometric functions. In particular, the Srivastava-Daoust double hypergeometric function (S-D function) has proved its practical utility in representing solutions to a wide range of problems in pure and applied mathematics. In this paper, we introduce two general double-series identities involving bounded sequences of arbitrary complex numbers employing the finite summation theorems of Gessel-Stanton and Andrews for terminating 3F2 hypergeometric series with arguments 3/4 and 4/3, respectively. Using these double-series identities, we establish two reduction formulas for the (S-D function) with arguments z, 3z/4 and z, −4z/3 expressed in terms of two generalized hypergeometric function of arguments proportional to z3 and −z3 respectively. All the results mentioned in the paper are verified numerically using Mathematica Program.
In the mathematical literature, a scalar integral equation with a degenerate kernel is well described (see below (1)), where all the written functions are scalar quantities). The authors are not aware of publications where systems of integral equations of (1) type with kernels in the form of a product of matrices would be considered in detail. It is usually said that the technique for solving such systems is easily transferred from the scalar case to the vector one (for example, in the monograph A.L. Kalashnikov "Methods for the approximate solution of integral equations of the second kind" (Nizhny Novgorod: Nizhny Novgorod State University, 2017), a brief description of systems of equations with degenerate kernels is given, where the role of degenerate kernels is played by products of scalar rather than matrix functions). However, as the simplest examples show, the generalization of the ideas of the scalar case to the case of integral systems with kernels in the form of a sum of products of matrix functions is rather unclear, although in this case the idea of reducing an integral equation to an algebraic system is also used. At the same time, the process of obtaining the conditions for the solvability of an integral system in the form of orthogonality conditions, based on the conditions for the solvability of the corresponding algebraic system, as it seems to us, has not been previously described. Bearing in mind the wide applications of the theory of integral equations in applied problems, the authors considered it necessary to give a detailed scheme for solving integral systems with degenerate kernels in the multidimensional case and to implement this scheme in the Maple program. Note that only scalar integral equations are solved in Maple using the intsolve procedure. The authors did not find a similar procedure for solving systems of integral equations, so they developed their own procedure.
M.T. Jenaliyev, A.S. Kassymbekova, M.G. Yergaliyev
et al.
This paper considers an initial boundary value problem for a one-dimensional Boussinesq-type equation in a domain, that is, a trapezoid. Using the methods of the theory of monotone operators, we establish theorems on their unique weak solvability in Sobolev classes.
The coronavirus disease 2019 (COVID-19) has been responsible for over three million reported cases worldwide. The construction of an appropriate mathematical (epidemiological) model for this disease is a challenging task. In this paper, we first consider susceptible - infectious - recovered (SIR) model with constant parameters and obtain an approximate solution for the SIR model with varying coefficient as it is one of the simplest models and many models are derived from this framework. The numerical experiments confirm that the proposed formulation demonstrates similar characteristic behaviour with the well-known approximations.
Maurizio Porfiri, Mert Karakaya, Raghu Ram Sattanapalle
et al.
Mathematical models promise new insights into the mechanisms underlying the emergence of collective behaviour in fish. Here, we establish a mathematical model to examine collective behaviour of zebrafish, a popular animal species in preclinical research. The model accounts for social and hydrodynamic interactions between individuals, along with the burst-and-coast swimming style of zebrafish. Each fish is described as a system of coupled stochastic differential equations, which govern the time evolution of their speed and turn rate. Model parameters are calibrated using experimental observations of zebrafish pairs swimming in a shallow water tank. The model successfully captures the main features of the collective response of the animals, by predicting their preference to swim in-line, with one fish leading and the other trailing. During in-line swimming, the animals share the same orientation and keep a distance from each other, owing to hydrodynamic repulsion. Hydrodynamic interaction is also responsible for an increase in the speed of the pair swimming in-line. By linearizing the equations of motion, we demonstrate local stability of in-line swimming to small perturbations for a wide range of model parameters. Mathematically backed results presented herein support the application of dynamical systems theory to unveil the inner workings of fish collective behaviour.
N.T. Orumbayeva, T.D. Tokmagambetova, Zh.N. Nurgalieva
In this paper, by means of a change of variables, a nonlinear semi-periodic boundary value problem for the Goursat equation is reduced to a linear gravity problem for hyperbolic equations. Reintroducing a new function, the obtained problem is reduced to a family of boundary value problems for ordinary differential equations and functional relations. When solving a family of boundary value problems for ordinary differential equations, the parameterization method is used. The application of this approach made it possible to establish the coefficients of the unique solvability of the semi-periodic problem for the Goursat equation and to propose constructive algorithms for finding an approximate solution.
A time dependent source identification problem for parabolic equation with involution and Neumann condition is studied. The well-posedness theorem on the differential equation of the source identification parabolic problem is established. The stable difference scheme for the approximate solution of this problem and its stability estimates are presented. Numerical results are given.
М.Т. Jenaliyev, M.I. Ramazanov, A.Kh. Attaev
et al.
In this paper we consider the stabilization problem of the solution of a boundary value problem for the heat equation with a loaded two-dimensional Laplace operator. The loaded terms represent the values of the required function and traces of the first-order partial derivatives of the required function at fixed points. An algorithm for constructing boundary control functions is proposed.
In this work we study trigonometric series with general monotone coefficients. Also, we consider Lqϕ ( Lq ) space. In particular, when ϕ ( t ) ≡ 1 the space Lqϕ ( Lq ) coincides with Lq . Well known the theorem of Hardy and Littlewood about trigonometric series with monotone coefficients. Also known various generalizations of this theorem. In 1982 this theorem was generalized by M.F. Timan for the spaces Lqϕ ( Lq ). And in 2007 S.Tikhonov proved Hardy-Littlewood theorem for trigonometric series with general monotone coefficients. In this work we have generalized Hardy-Littlewood theorem for Fourier series of functions f ∈ Lqϕ ( Lq ) with general monotone coefficients. Also, obtained upper - bound estimate of best approximation of functions f ∈ Lq through its Fourier’s coefficients which are general monotone.