Search Results (135)

For $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, this paper derives refined conditions for the inclusion of special functions in lemniscate and nephroid domains focusing on solutions to the differential equations of the form $z^n{F}^{″}(z)+a\left(z\right)z^{n-1}{F}^{\prime }\left(z\right)+b\left(z\right)F\left(z\right)+d\left(z\right)=0,n\in \{1,2\},z\in \mathbb{D},$ with normalization $F\left(0\right)=1$, where $a\left(z\right)$, $b\left(z\right)$ and $d\left(z\right)$ are analytic in $\mathbb{D}$. Using advanced techniques from geometric function theory, we generalize and improve existing results, particularly for classes of functions defined by differential equations. Specific applications include generalized Bessel functions, regular Coulomb wave functions, and associated Laguerre polynomials, where we derive improved bounds for their inclusion in lemniscate domains. Additionally, we present open problems, supported by numerical experiments, to guide future research in this direction. Full article

This manuscript presents a study of the Stress-Driven integral Model (SDM) for the bending response of Bernoulli–Euler nanobeams. Unlike conventional approaches that reformulate the nonlocal integral problem into an equivalent differential form, a direct numerical strategy is developed to solve the integral equation. The proposed framework enables a systematic comparison of six different convolution kernels (Helmholtz, Gaussian, Lorentzian, triangular, Bessel and hyperbolic cosine), highlighting how their mathematical properties influence the structural response. To address issues related to long-range interactions and the potential ill-posedness of the integral operator, an adaptive regularization procedure based on the Morozov discrepancy principle is introduced. Furthermore, a local clipping and renormalization technique is proposed to properly account for boundary effects while preserving the weighted averaging property of the kernels. Validation against available analytical solutions for the Helmholtz kernel demonstrates high accuracy, with errors below 1%. The results show that the nonlocal parameter significantly affects structural rigidity depending on the kernel shape and that the proposed approach ensures consistent convergence to the local solution as the nonlocal parameter tends to zero. Full article

This work analyzes the non-Newtonian electromagnetohydrodynamic (EMHD) flow in an irregular circular porous microchannel while incorporating the consequences of surface charge-dependent slip boundary conditions. The Jeffrey fluid is employed to examine the non-Newtonian behavior, such as elasticity. The boundary walls of the channel are considered in the form of periodic sinusoidal wave function. The mathematical formulation is developed using the momentum equation, modified Darcy’s law, the continuity equation, and Ohm’s law. The perturbation method is used to derive the solutions up to second-order approximation. The analytical expression for the velocity field and volumetric flow rate are explicitly presented. At the zeroth-order, a nonhomogeneous partial differential equation is solved, and the solutions are presented in terms of Bessel functions. The first-order problem defined by a homogeneous partial differential equation is solved using the method of separation of variables. At the second-order, a homogeneous partial differential equation is obtained, and the solution form is prescribed by the boundary conditions, consisting of a radially varying mean component and a second-harmonic angular contribution. Two- and three-dimensional plots are used to analyze and discuss the impacts of key parameters, namely the Reynolds, Darcy, and Hartmann numbers, channel corrugation amplitude and wave number, surface charge density, and the relaxation and retardation times on the velocity field and flow rate. It is found that elastic memory causes a proportional growth between the flow rate and the relaxation time, emphasizing the consequences of surface charge application in conjunction with corrugations. Conversely, maintaining a short retardation time mitigates changes in wave amplitude and surface charge. While prolonging it lessens the flow rate and diminishes corrugations and surface charge effects. The Darcy number dampens the velocity and the flow rate, while its enhancement reduces the impact of surface charge density and corrugations amplitude. For high Reynolds number, a ring phenomenon emerges which is attenuated by increased Darcy number, preventing the formation of trapped boluses close to the border. Ignoring surface charge amplifies the flow rate while its consideration diminishes the latter with reinforced impacts of surface charge and wall corrugations at higher Reynolds number. Full article

Electromagnetic interference (EMI) rectification of grid-tied inverters is crucial for their practical application, and the variable frequency modulation (VFM) technique is a low-cost and simple way for EMI reduction. However, changes in loss and harmonic behaviors make it hard for parameter determination of VFM. In this paper, the parameters required for switching frequency (SF) function are determined for loss optimization of MOSFETs and inductors, while total harmonic distortion (THD) and temperature rise in MOSFETs and inductor core are constrained to guarantee the feasibility of the calculated parameters. Current transient is derived through multidimensional Fourier decomposition (MFD) and characteristics of Bessel function for loss estimation of MOSFET and inductor. Modified Steinmetz equation (MSE) is applied for core loss estimation and AC resistance is considered for copper loss estimation. With the constraints of THD and temperature, the loss optimization problem is solved by the augmented Lagrangian (AL) method. With the assistance of the proposed method, total loss optimization can be realized in feasible regions while the temperature rise in essential components can be restricted to the preset values. Full article

In this paper, we consider a partial differential equation with mixed derivatives of first order in time and second order in the spatial variable. Such equations are usually referred to as one-dimensional pseudoparabolic equations. We prove the existence and uniqueness of a classical solution to problems for a pseudoparabolic equation with a second-order differential operator involving pure involution, under certain requirements imposed on the initial data. The possibility of applying the Fourier method is based on the Riesz basis property of the eigenfunctions of the considered non-self-adjoint second-order differential operator with pure involution. Bessel-type inequalities are established for new systems of functions. The presence of a Bessel inequality for Fourier coefficients facilitates the proof of the uniform convergence of differentiated Fourier series. The solutions are obtained explicitly in the form of a Fourier series. Such representations can be used for performing numerical computations. Full article

Two-dimensional (2D) correlation functions are central to understanding structural crossovers in soft-core fluids; however, their asymptotic analysis is hindered by the Hankel-transform kernel, whose asymptotic representation introduces a term that breaks the natural conjugate symmetry of the poles. To address this, we present a symmetric contour integration scheme that restores symmetry at the level of the integration path. By employing quarter-circle contours in the first and fourth quadrants, the method captures conjugate pole pairs simultaneously and evaluates the sine term from the Bessel-function asymptotic without variable transformation or real-part extraction, yielding closed-form analytic expressions for the long-range decay of the density–density correlation function. The approach is demonstrated for a 2D Gaussian-charge one-component plasma under the random phase approximation at intermediate coupling, where the pole analysis provides direct access to the oscillation wavelength and decay length. In the high-density regime, the pole equations simplify to a form amenable to a Lambert W-function approximation, revealing a logarithmic scaling of correlation lengths even at moderate coupling. These findings establish symmetric contour integration as a transparent and versatile framework for pole-resolved asymptotics in 2D liquids. Full article

In the theory of mixed-type equations, there are many works in bounded domains with smooth boundaries bounded by a normal curve for first- and second-kind mixed-type equations. In this paper, for a second-kind mixed-type equation in an unbounded domain whose elliptic part is a horizontal half-strip, a Bitsadze–Samarskii-type problem is investigated. The uniqueness of the solution is proved using the extremum principle, and the existence of the solution is proved by the Green’s function method and the integral equations method. When constructing the Green’s function, the properties of Bessel functions of the second kind with imaginary argument and the properties of the Gauss hypergeometric function are widely used. Visualization of the solution to the Bitsadze–Samarskii-type problem is performed, confirming its correctness from both mathematical and physical points of view. Full article

Integral transforms play a fundamental role in science and engineering. Above all, the Fourier transform is the most vital, which has some specifications—Laplace transform, Mellin transform, etc., with their inverse transforms. In this paper, we restrict ourselves to the use of a few versions of the Mellin transform, which are best suited to the treatment of zeta functions as Dirichlet series. In particular, we shall manifest the underlying principle that automorphy (which is a modular relation, an equivalent to the functional equation) is intrinsic to lattice (or Epstein) zeta functions by considering some generalizations of the holomorphic and non-holomorphic Eisenstein series as the Epstein-type Eisenstein series, which have been treated as totally foreign subjects to each other. We restrict to the modular relations with one gamma factor and the resulting integrals reduce to a form of the modified Bessel function. In the H-function hierarchy, what we work with is the second simplest $H_{1,1}^{1,1}\leftrightarrow H_{0,2}^{2,0}$, with H denoting the Fox H-function. Full article

We demonstrate the time evolution of a free particle in a three-dimensional space given that the initial condition is any arbitrary radial function $f\left(r\right)$. The solution is expressed as a series expansion in terms of generalized Bessel functions derived using a 3D recursion formula for Bessel functions in the radial coordinate r. Additionally, we establish that these generalized Bessel functions can be represented through intricate double series, which ultimately enable the construction of the full solution. This work presents a novel solution to the problem, as previous approaches were limited to expressions involving only spherical Bessel functions. Full article

For the function solution to the well-known homogeneous Bessel differential equation, we utilized a normalized form of this function to define a certain operator on a subclass of analytic functions. Using this operator, we introduced various subordination properties. We also examined the sufficient starlikeness conditions and provided some estimates for a specific subclass of univalent functions defined in the unit disc. Full article

The moisture content of wood components varies with changes in the external environment, which significantly affects the mechanical properties, moisture stress, decay, drying shrinkage, and cracking of wood components. Therefore, calculating the moisture content distribution of the cross-section of wood components is an important basis for in-depth research on wood components. First, a hygroscopicity test was performed on 45° sector-shaped Chinese fir thin-plate specimens. The specimens were treated to an absolutely dry state and placed in two different environments. The average moisture content and moisture content gradient on the cross-section of the specimens were measured, and the spatial distribution and temporal variation in the moisture content were studied. A theoretical model for the moisture content distribution of wood was then derived based on food drying theory. Finally, the applicability of the theoretical model was verified through experiments, and the effects of the root order μ_n of the characteristic equation of key parameters, the size of the component, and the position of the component on the moisture content distribution were discussed for the theoretical model. During the hygroscopic process, the average moisture content of wood components increased continuously, but the growth rate gradually slowed. The surface moisture content rapidly reached the level of the external moisture content first, followed by the equilibrium moisture content within a few hours. Hygroscopic hysteresis evidently occurred within the wood, which may take dozens or even hundreds of days. When calculating the average moisture content model of cylindrical components, as well as those of the models of the spatial and temporal variation in the moisture content, it is sufficient to take the first 3 orders of the root μ_n of the characteristic equation of the first Bessel function J. The rate of moisture release of cylindrical components is faster than that of laminates because the ratio of the surface area to the volume of a cylinder is greater than that of a plate, and the former is twice that of the latter. The results revealed that the theoretical model for the moisture content distribution of wood has good accuracy and applicability. Full article

In this study, we consider and analyze an inhomogeneous Whittaker-type differential equation of the form ${\displaystyle \frac{d^{2}y\left(x\right)}{d{x}^2}}+{\displaystyle \frac{1}{x}}{\displaystyle \frac{dy\left(x\right)}{dx}}-{\alpha }^2\left(x^2-{\beta }^2\right)y\left(x\right)=g\left(x\right)$, where $\alpha$ and $\beta$ are given parameters. We investigate the analytical structure of its solution through the application of the Whittaker integral representation. The analysis encompasses both initial value problems (IVPs) and boundary value problems (BVPs), wherein appropriate conditions are imposed within a unified analytical framework. Furthermore, a systematic methodology is developed for constructing explicit solutions within the framework of Whittaker function theory. This approach not only elucidates the functional behaviour of the solutions but also provides a foundation for extending the analysis to more general classes of second-order linear differential equations. Full article

Schrödinger’s operator relations combined with Einstein’s special relativistic energy-momentum equation produce the linear Klein–Gordon partial differential equation. Here, we extend both the operator relations and the energy-momentum relation to determine new families of nonlinear partial differential relations. The Planck–de Broglie duality principle arises from Planck’s energy expression $e=h\nu$, de Broglie’s equation for momentum $p=h/\lambda$, and Einstein’s special relativity energy, where h is the Planck constant, $\nu$ and $\lambda$ are the frequency and wavelength, respectively, of an associated wave having a wave speed $w=\nu \lambda$. The author has extended these relations to a family that is characterised by a second fundamental constant $h^{\prime }$ and underpinned by Lorentz invariant power-law particle energy-momentum expressions. In this note, we apply generalized Schrödinger operator relations and the power-law relations to generate a new family of nonlinear partial differential equations that are characterised by the constant $\kappa =h^{\prime }/h$ such that $\kappa =0$ corresponds to the Klein–Gordon equation. The resulting partial differential equation is unusual in the sense that it admits a stretching symmetry giving rise to both similarity solutions and simple harmonic travelling waves. Three simple solutions of the partial differential equation are examined including a separable solution, a travelling wave solution, and a similarity solution. A special case of the similarity solution admits zeroth-order Bessel functions as solutions while generally, it reduces to solving a nonlinear first-order ordinary differential equation. Full article

This study develops a novel analytical solution for three-dimensional axisymmetric consolidation of unsaturated soils incorporating radial–vertical composite seepage mechanisms and anisotropic permeability characteristics. A groundbreaking dual orthogonal expansion framework is established, utilizing innovative Bessel–trigonometric function coupling to solve the inherently complex spatiotemporal coupled partial differential equations in cylindrical coordinate systems. The mathematical approach synergistically combines modal expansion theory with Laplace transform methodology, achieving simultaneous spatial expansion of gas–liquid two-phase pressure fields through orthogonal function series, thereby transforming the three-dimensional problem into solvable ordinary differential equations. Rigorous validation demonstrates exceptional accuracy with coefficient of determination R² exceeding 0.999 and relative errors below 2% compared to numerical simulations, confirming theoretical correctness and practical applicability. The analytical solutions reveal four critical findings with quantitative engineering implications: (1) dual-directional drainage achieves 28% higher pressure dissipation efficiency than unidirectional drainage, providing design optimization criteria for vertical drainage systems; (2) normalized matric suction variation exhibits characteristic three-stage evolution featuring rapid decline, plateau stabilization, and slow recovery phases, while water phase follows bidirectional inverted S-curve patterns, enabling accurate consolidation behavior prediction under varying saturation conditions; (3) gas-water permeability ratio k_a/k_w spanning 0.1 to 1000 produces two orders of magnitude time compression effect from 10⁻² s to 10⁻⁴ s, offering parametric design methods for construction sequence control; (4) initial pressure gradient parameters λ_a and λ_w demonstrate opposite regulatory mechanisms, where increasing λ_a retards consolidation while λ_w promotes the process, providing differentiated treatment strategies for various geological conditions. The unified framework accommodates both uniform and gradient initial pore pressure distributions, delivering theoretical support for refined embankment engineering design and construction control. Full article

In this article, we investigate the regularization and qualitative properties of parabolic Ginzburg–Landau equations in variable exponent Herz spaces. These spaces capture both local and global behavior, providing a natural framework for our analysis. We establish boundedness results for fractional Bessel–Riesz operators, construct examples highlighting their advantage over classical Riesz potentials, and recover several known theorems as special cases. As an application, we analyze a parabolic Ginzburg–Landau operator with VMO coefficients, showing that our estimates ensure the boundedness and continuity of solutions. Full article