Formula For Solution Research Articles

The [formula omitted]-asymptotic behavior of strong solutions to the incompressible magneto-hydrodynamic equations in half-spaces

We give the L1-decay estimates on the fractional spatial derivatives of the strong solution to the incompressible magneto-hydrodynamic (MHD) equations in a half-space. Moreover, by using the Stokes solution formula and the decomposition of the nonlinear terms in MHD equations, we derive asymptotic expansion properties of the strong solution in L1(R+n).

The Riemann problem in a half-plane for generalized analytic functions with a supersingular point on the contour of the boundary condition

In this paper, we study an inhomogeneous Riemann boundary value problem with a finite index and a boundary condition on the real axis for a generalized equation Cauchy–Riemann with supersingular coefficients. To solve the problem, it was necessary to derive a structural formula for the general solution of this equation and to investigate the solvability of the Riemann boundary value problem of the theory of analytic functions with an infinite index due to the power-order vorticity point.

Stability Study of a Compounded Sublingual Buprenorphine Solution for Neonatal Opioid Withdrawal Syndrome

OBJECTIVE Sublingual (SL) buprenorphine is a cornerstone of care in the treatment of adult opioid use disorder. Recent studies have demonstrated its advantages in the management of neonatal opioid withdrawal syndrome (NOWS). Commercially available SL tablets and transdermal patches are not amenable to neonatal use, and published compounding formulas of SL solutions contained undesirable excipients, including ethanol, sugars, and preservatives. The objective of this research is to explore the stability of a novel SL buprenorphine formulation free of alcohol, sugars, and preservatives. METHODS A 0.075 mg/mL buprenorphine solution was prepared by diluting the commercial injectable solution with normal saline and packaged into polyethylene terephthalate amber prescription bottles and polypropylene amber oral syringes and stored in refrigeration. Quality assessments were conducted by visual, pH, and high-performance liquid chromatography (HPLC) analysis immediately after preparation, and at 7 and 14 days of storage. RESULTS There were neither visual nor pH changes detected through 14 days. HPLC analysis indicated that all samples retained >99% initial buprenorphine concentration. Drug concentration increased slightly in the oral syringe after day 7, probably due to moisture loss. No degradation peaks were observed in chromatograms. CONCLUSIONS This novel buprenorphine is free of alcohol, sugar, and preservatives, and it may offer a significant safety advantage for NOWS patients. Additional clinical studies are recommended to verify the bioavailability and efficacy of this formulation.

Angular non-critical phase-matching second harmonic generation with the Ba3(ZnB5O10)PO4 crystal

The angular non-critical phase-matching (A-NCPM) second-harmonic-generation (SHG) properties of Ba3(ZnB5O10)PO4 (BZBP) crystal were investigated. Based on the small signal solution formula of SHG, the effective nonlinear coefficient deff along z-axis was corrected to 1.43 p.m./V. Combining the theoretical SHG PM conditions, BZBP crystal can be achieved type-I A-NCPM with the fundamental wavelength of 750 nm on y-axis, and 1012 nm on z-axis. Using an OPO laser as pump source, the BZBP crystal could realize the 506 nm and 375 nm SHG lasers with the conversion efficiencies of 35 % and 45.5 % respectively. The acquired results demonstrated that BZBP is a promising NLO crystal for A-NCPM frequency conversion applications in the output spectral regions of Ti:sapphire laser and near-infrared lasers.

Inertial power balance system with nonlinear time-derivatives and periodic natural frequencies

In this paper, we study the asymptotic behavior of the macroscopic power grid system derived from energy conservation. We assume that its coupling is all-to-all and that the energy supply and consumption are periodic. This system contains nonlinear time-derivative terms representing inertia and energy dissipation. We provide a sufficient condition for the existence of a special solution. This solution is a basin of attraction and its phase difference is periodic. To prove the existence of this solution, we employ a transformation to extract periodic variables. We consider nine functionals to show the asymptotic stability of the solution. A nine-dimensional vector whose components are the nine functionals satisfies a new nonlinear system. This property with the a priori estimate method yields the desired stability. We also obtain a sharp asymptotic formula for general solutions by combining this result and the existence of the solution with periodicity.

Theoretical prediction of ground settlements due to shield tunneling in multi-layered soils considering process parameters

This paper conducts a theoretical analysis of ground settlements due to shield tunneling in multi-layered soils which are usually encountered in urban areas. The proposed theoretical solution which is based on the general form of the Mindlin's solution and Loganathan-Poulos formula can comprehensively consider the in-process tunneling parameters including: unbalanced face pressure, shield-soil friction, unbalanced tail grouting pressure, unbalanced secondary grouting pressure, overloading during tunneling and the ground volume loss. The method is verified by comparing with the field data from the Qinghuayuan Tunnel Project in terms of the ground surface settlements along the longitudinal and transverse direction. Due to the local settlement or heave caused by the certain tunneling parameters, the ground surface settlements calculated using current solution along the longitudinal direction presents an irregular S-shaped curve instead of the traditional S-shaped curve. Results also find that the effect of the unbalanced secondary grouting pressure and the overloading during tunneling cannot be ignored.

Estimation of Fracture Size and Probability Density Function by Setting Scanlines in Rectangular Sampling Window

Abstract Rock masses are very important materials in geotechnical engineering. In engineering rock mass, fracture is the relatively weak part of mechanical strength in rock mass and is the most important factor controlling the deformation, damage, and permeability of rock mass. Therefore, investigating fractures is very important for characterizing rock mass. This paper proposed a new approach by using uniformly equidistant orthogonal scanlines. Within the study context, the solution formula of fracture size is derived by establishing the space intersection model of arbitrary fracture and scanline, rectangular window, and a rectangular box with a rectangular window. Then, fractures were randomly generated in a certain size cube and compared with the traditional Kulatilake trace length integral evaluation method. The study results have shown that the proposed method is more reasonable and accurate. Then, this method was applied to an adit of Songta Hydropower Station. Finally, a new fracture diameter probability density estimation method was proposed, the fracture diameter of the normal distribution was verified, and the parameters of the probability density function obtained by the scanlines method were in agreement with the initial set parameters. In summary, the proposed scanlines method can well estimate the mean value of the fracture diameter and the probability density function of the fracture size.

The Riemann–Hilbert approach for the Chen–Lee–Liu equation with higher-order poles

This paper considers the Riemann–Hilbert problem (RHP) with higher-order poles for the Chen–Lee–Liu equation. By assuming the scattering coefficients have Nth order zero points, we obtain expressions of these scattering coefficients, the theta condition, and parallelization conditions between Jost solutions and their derivatives. Based on these parallelization conditions, it yields generalizations of residue conditions, which are essential for solving the RHP with higher-order poles. The explicit formulae for solutions of CLL equation are reconstructed from asymptotic behaviors of the RHP. As an example, we present the second-order pole soliton and illustrate that parameter b0 plays a principal role to determine the structures of solutions of the CLL equation.

On the Solution of the Dirichlet Problem for Second-Order Elliptic Systems in the Unit Disk

The role played by explicit formulas for solving boundary value problems for elliptic equations and systems is well known. In this paper, explicit formulas for a general solution of the Dirichlet problem for second-order elliptic systems in the unit disk are given. In addition, an iterative method for solving this problem for systems with respect to two unknown functions is described, and an integral representation of the Poisson type is obtained by applying this method.

Double and triple-pole solutions for the third-order flow equation of the Kaup-Newell system with zero/nonzero boundary conditions

In this work, the double and triple-pole solutions for the third-order flow equation of Kaup-Newell system (TOFKN) with zero boundary conditions (ZBCs) and non-zero boundary conditions (NZBCs) are investigated by means of the Riemann-Hilbert (RH) approach stemming from the inverse scattering transformation. Starting from spectral problem of the TOFKN, the analyticity, symmetries, asymptotic behavior of the Jost function and scattering matrix, the matrix RH problem with ZBCs and NZBCs are constructed. Then the obtained RH problem with ZBCs and NZBCs can be solved in the case of scattering coefficients with double or triple zeros, and the reconstruction formula of potential, trace formula as well as theta condition are also derived correspondingly. Specifically, the general formulas of N-double and N-triple poles solutions with ZBCs and NZBCs are derived systematically by means of determinants. The vivid plots and dynamics analyses for double and triple-pole soliton solutions with the ZBCs as well as double and triple-pole interaction solutions with the NZBCs are exhibited in details. Compared with the most classical second-order flow Kaup-Newell system, we find the third-order dispersion and quintic nonlinear term of the Kaup-Newell system change the trajectory and velocity of solutions. Furthermore, the asymptotic states of the 1-double poles soliton solution and the 1-triple poles soliton solution are analyzed when t tends to infinity.

On The Solution Set of Euler's Systems Using Elzaki Transform

In this work, general formula of set solutions for system of Euler's equation in dimension m are derived by using Elzaki transformation. Also, supported examples are presented utilizing formula derived.

Novel Transform for Mathematical Systems of Second Order Whether Under or Not Initial Conditions

In this work, the formulas of set solution for linear mathematical systems of the second order are derived whether subject to or not subject to initial conditions. In addition to a set of supporting examples have been resolved.

Abstract Fractional Cauchy Problem: Existence of Propagators and Inhomogeneous Solution Representation

We consider a Cauchy problem for the inhomogeneous differential equation given in terms of an unbounded linear operator A and the Caputo fractional derivative of order α∈(0,2) in time. The previously known representation of the mild solution to such a problem does not have a conventional variation-of-constants like form, with the propagator derived from the associated homogeneous problem. Instead, it relies on the existence of two propagators with different analytical properties. This fact limits theoretical and especially numerical applicability of the existing solution representation. Here, we propose an alternative representation of the mild solution to the given problem that consolidates the solution formulas for sub-parabolic, parabolic and sub-hyperbolic equations with a positive sectorial operator A and non-zero initial data. The new representation is solely based on the propagator of the homogeneous problem and, therefore, can be considered as a more natural fractional extension of the solution to the classical parabolic Cauchy problem. By exploiting a trade-off between the regularity assumptions on the initial data in terms of the fractional powers of A and the regularity assumptions on the right-hand side in time, we show that the proposed solution formula is strongly convergent for t≥0 under considerably weaker assumptions compared to the standard results from the literature. Crucially, the achieved relaxation of space regularity assumptions ensures that the new solution representation is practically feasible for any α∈(0,2) and is amenable to the numerical evaluation using uniformly accurate quadrature-based algorithms.

Generalizations of the drift Laplace equation over the quaternions in a class of Grushin-type spaces

Beals, Gaveau, and Greiner established a formula for the fundamental solution to the Laplace equation with drift term in Grushin-type planes. The first author and Childers expanded these results by invoking a p-Laplace type generalization that encompasses these formulas while the authors explored a different natural generalization of the p-Laplace equation with drift term that also encompasses these formulas. In both, the drift term lies in the complex domain. We extend these results by considering a drift term in the quaternion realm and show our solutions are stable under limits as p tends to infinity.

Local well‐posedness of the higher‐order nonlinear Schrödinger equation on the half‐line: Single‐boundary condition case

AbstractWe establish local well‐posedness in the sense of Hadamard for a certain third‐order nonlinear Schrödinger equation with a multiterm linear part and a general power nonlinearity, known as higher‐order nonlinear Schrödinger equation, formulated on the half‐line . We consider the scenario of associated coefficients such that only one boundary condition is required and hence assume a general nonhomogeneous boundary datum of Dirichlet type at . Our functional framework centers around fractional Sobolev spaces with respect to the spatial variable. We treat both high regularity () and low regularity () solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, this is no longer the case and, instead, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial‐boundary value problems, as it involves proving boundary‐type Strichartz estimates that are not common in the study of Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher‐order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; and (iii) complicated oscillatory kernels in the weak solution formula for the linear initial‐boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial‐boundary value problem for a partial differential equation associated with a multiterm linear differential operator.

A numerical approach to solve 2D fractional RADE of variable-order with Vieta–Lucas polynomials

Here, we present a novel numerical algorithm based on the operational matrix and collocation approach for solving the 2D non-linear fractional reaction–advection diffusion equations (RADE) of variable order that is related to the groundwater pollution problem. In this article, the operational matrices of variable-order derivatives are derived with the aid of Vieta–Lucas polynomials for 2D problem, which is used in the proposed algorithm. To find a solution to the considered problem, the terms are approximated by a series of triple-shifted Vieta–Lucas polynomials to construct the residual of the RADE. Then the collocation technique is applied to transform the problem into algebraic equations, which are solved by Newton’s method. Moreover, the convergency and upper bound of the derived error formula for the approximate solution are discussed. Finally, some examples are presented to show the efficiency of the considered scheme, and the results are shown by using graphical presentation and tabular representations.

Theoretical and experimental investigations on free vibration characteristics of arbitrary spatially closed-coupled plates

A semi-analytical method and related experiments are present in this paper to investigate the free vibration characteristics of the closed-coupled plates with arbitrary spatial coupling angles and curved edges, which are the widely used structural form but researched for the first time. By combining the penalty function method with flexible setting virtual spring stiffness functions and the comprehensive relative motion relation matrices along the edges of the plates system, various types of boundary and spatial coupling connection conditions in practical engineering can be constructed. The semi-analytical solution formulas for the entire vibration model are established by using the first-order shear deformation theory (FSDT) and the two-dimensional Jacobian differential quadrature method (JDQM). To fulfill the requirements of integral operations for displacement approximation functions, the irregular domain mapping technique is employed to transform the irregular solving domain into the regular basic domain. The accuracy and applicability of the present method in predicting eigenfrequencies and mode shapes are proved by hammer modal tests concerning four pyramid/frustum-like built-up plates, comparisons with results from the open literature, as well as the finite element method (FEM) simulations of several specially designed cases. Moreover, many meaningful conclusions are drawn regarding the effects of spatial geometric parameters on the free vibration characteristics of spatially closed-coupled plates with curved edges.

Compact formula for skew-symmetric system of matrix equations

In this paper, we consider skew-Hermitian solution of coupled generalized Sylvester matrix equations encompassing *-hermicity over complex field. The compact formula of the general solution of this system is presented in terms of generalized inverses when some necessary and sufficient conditions are fulfilled. An algorithm and a numerical example are provided to validate our findings. A numerical example is carried out using determinantal representations of the Moore–Penrose inverse.

A Riemann–Hilbert method to algebro-geometric solutions of the Korteweg–de Vries equation

A proof without using Dubrovin’s equations is given for the Its–Matveev formula for algebro-geometric solutions of the Korteweg–de Vries (KdV) equation. It is shown that the Baker–Akhiezer (BA) function of the KdV equation can be described in terms of two types of holomorphic vector Riemann–Hilbert problems with σ1-symmetry condition in the complex k-plane. Further, a sufficient condition on determining whether the algebro-geometric KdV solution has mKdV-relevant property is presented. This provides an explanation for the classical Miura transformation in terms of the deformations of associated algebro-geometric datum. Our main tools include algebraic curve and Riemann surface, Riemann theta function, Riemann–Hilbert problem and associated deformation procedure.

Two-parameters formulas for general solution to planar weakly delayed linear discrete systems with multiple delays, equivalent non-delayed systems, and conditional stability

Weakly delayed planar linear discrete systems with multiple delaysx(k+1)=Dx(k)+∑l=1nHlx(k−ml),k=0,1,… are considered where m1,m2,…,mn are fixed integers, D, H1, …, Hn are nonzero 2×2 real constant matrices and x:{−mn,−mn+1,…}→R2. Formulas for general solutions are found and simplified, equivalent non-delayed planar linear discrete systems are constructed and conditional stability is analyzed. Results are illustrated by examples.