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In this paper, we propose an efficient numerical method for the solution of one and two dimensional Riesz space fractional advection–dispersion equation. To this end, we use the Crank–Nicolson scheme to discretize this equation in temporal direction and prove that the semi-discrete scheme is unconditionally stable. Then, we apply the spectral element method in spatial directions and obtain the fully discrete scheme. We present an error estimate for the fully discrete scheme. The presented numerical results demonstrate the accuracy and efficiency of the proposed method in comparison with other schemes in literature.

First-arrival traveltime tomography has been widely used for upper crustal velocity modeling, but it usually suffers from the problem of complex surface topography. To overcome this problem, we have developed a new topography-dependent eikonal tomography scheme that combines a developed accurate and efficient traveltime modeling method and introduces a flexible and robust adjoint inversion scheme in the presence of irregular topography. A surface-flattening scheme is used to handle the irregular surface, where the real model is discretized by curvilinear grids and the irregular free surface is mathematically flattened through the transformation from Cartesian to curvilinear coordinates. Based on this parameterization, the forward traveltime modeling is conducted by a monotone fast-sweeping method that discretizes the factored topography-dependent eikonal equation with a point-source condition. This algorithm can circumvent the source-singularity problem and decrease the numerical error in the vicinity of a point source in the curvilinear system. Then, the gradient-based inversion is used to minimize the misfit function, which is achieved by a matrix-free adjoint-state method without cumbersome ray tracing and explicit estimation of the Fréchet derivative matrix in the curvilinear coordinate system. The new tomographic scheme is evaluated through numerical examples with different seismic structures with complex topography, and then applied to a wide-angle profile acquired in the northeastern Tibetan Plateau. The results validate the effectiveness and efficiency of our tomography scheme in constructing shallow crustal velocity models with irregular topography.

Here, we use the modified Kudryashov’s algorithm and addendum to Kudryashov’s technique to obtain new optical solitons of the Ginsburg–Landau fractional complex of non-linearity (CGLE) laws in Kerr, which demonstrate various phenomena in physics such as non-linear waves, the transformation of the secondary phase, superconductivity, surfaces, liquid crystals, and field-theory strings. Dark, bright, singular solitons and combo bright-singular solitons are retrieved. The modified Kudryashov’s algorithm provides dark, singular solitons, and combo bright-singular solitons, although the addendum to Kudryashov’s technique ensures bright and singular solitons. These solutions are practiced for three optional definitions of derivative viz. conformable derivative, beta derivative, and M-truncated. Also shown are graphic representations of the solutions obtained.

Direct Z-scheme g-C3N4/V2O5 photocatalysts were prepared through a sonication-assisted calcination method. The obtained samples were characterised by X-ray diffraction (XRD),Ultraviolet-visible diffuse reflectance spectroscopy (UV-Vis-DRS), Scanning electron microscope (SEM), andPhotoluminescence spectroscopy (PL). Oxidations of tetracycline hydrochloride (TC) were employed to evaluate the photocatalytic activities of the obtained g-C3N4/V2O5materials. Different weight ratios (5, 10, 15, and 20%) of g-C3N4/V2O5 loaded composites were prepared, in which a 15% (CV-15) loaded composite was found to show optimal catalytic performance for the reaction. The degradation conversation of TC has achieved approximately 79.67% in CV-15 after a 2-hour reaction. g-C3N4/V2O5 photocatalystwas more active than the individual g-C3N4 and V2O5 materials, which could be attributed to the efficient separation of photogenerated electron-hole pairs shown in the photocatalytic mechanism of TC degradation.

In this article, a hybrid Haar wavelet collocation method (HWCM) is proposed for the ill-posed inverse problem with unknown source control parameters. Applying numerical techniques to such problems is a challenging task due to the presence of nonlinear terms, unknown control parameter sources along the solution inside the given region. To find the numerical solution, derivatives are discretized adopting implicit finite-difference scheme and Haar wavelets. The computational stability and theoretical rate of convergence of the proposed HWCM are discussed in detail. Two numerical experiments are incorporated to show the well-condition behavior of the matrix obtained from HWCM and hence not required to supplement some regularization procedures. Moreover, the numerical solutions of the considered experiments illustrate the reliability, suitability, and correctness of HWCM.

We demonstrate that in the mass-independent renormalization scheme, the renormalization group equations associated with the unphysical parameters that characterize the renormalization scheme and the mass scale leads to summation that results in a cancellation between the implicit and explicit dependence on these parameters. The resulting perturbative expansion is consequently independent of these arbitrary parameters. We illustrate this by considering R, the cross section for e+e– → hadrons.

We show that the one- and two-loop $\beta$-functions of the closed, bosonic string can be written in a manifestly O($D$,$D$)-covariant form. Based on this result, we prove that 1) Poisson-Lie symmetric $\sigma$-models are two-loop renormalisable and 2) their $\beta$-functions are invariant under Poisson-Lie T-duality. Moreover, we identify a distinguished scheme in which Poisson-Lie symmetry is manifest. It simplifies the calculation of two-loop $\beta$-functions significantly and thereby provides a powerful new tool to advance into the quantum regime of integrable $\sigma$-models and generalised T-dualities. As an illustrating example, we present the two-loop $\beta$-functions of the integrable $\lambda$- and $\eta$-deformation., Comment: 44 pages, 2 tables

The recent growing interest in urban air mobility (UAM) worldwide has led to the demand for physical analyses of the aerodynamic performance and aeroacoustic characteristics of electric vertical ta...

In this paper, we study the global existence of periodic solutions to an isothermal relativistic Euler system in BV space. First, we analyze some properties of the shock and rarefaction wave curves in the Riemann invariant plane. Based on these properties, we construct the approximate solutions of the isothermal relativistic Euler system with periodic initial data by using a Glimm scheme, and prove that there exists an entropy solution V(x, t) which belongs to L∞ ∩ BVloc (ℝ × ℝ+).

Various models were proposed to calculate the van der Waals (vdW) forces, but they have generally been complicated and time-consuming. In this paper, a spherical volume elements scheme to calculating the vdW force distributions is proposed, on the basis of the infinitesimal dividing modeling scheme. We apply this scheme to calculate the vdW force distributions between sphere and three types of surfaces with different roughness and also to irregular particles (cylinders, circular cone) orientated differently with respect to the surface. The simulation results reveal that the surface with low roughness has the narrowest force distributions. The average force for the circular cone bottom case is 1, 2, 37 times higher than the case of horizontal, vertical cylinder and circular cone top, respectively. The calculated force distributions are good agreement with exist experimental data. By defining the effective zones of both particles and surface, the computational time of the scheme is an order of magnitude faster than the case of considering the whole surface. The proposed model established the effective zone of the particles in contact from a completely new perspective and significantly simplifies the process of force evaluation.

We examine the perturbative calculation of the inclusive semi-leptonic decay rate Γ for the b quark, using mass-independent renormalization. To finite order of perturbation theory the series for Γ will depend on the unphysical renormalization scale parameter μ and on the particular choice of mass-independent renormalization scheme; these dependencies will only be removed after summing the series to all orders. In this paper we show that all explicit μ-dependence of Γ, through powers of ln(μ), can be summed by using the renormalization group equation. We then find that this explicit μ-dependence can be combined together with the implicit μ-dependence of Γ (through powers of both the running coupling a(μ) and the running b quark mass m(μ)) to yield a μ-independent perturbative expansion for γ in terms of a(μ) and m(μ) both evaluated at a renormalization scheme independent mass scale [Formula: see text] which is fixed in terms of either the “[Formula: see text] mass” [Formula: see text] of the b quark or its pole mass mpole. At finite order the resulting perturbative expansion retains a degree of arbitrariness associated with the particular choice of mass-independent renormalization scheme. We use the coefficients ci and gi of the perturbative expansions of the renormalization group functions β(a) and γ(a), associated with a(μ) and m(μ) respectively, to characterize the remaining renormalization scheme arbitrariness of Γ. We further show that all terms in the expansion of Γ can be written in terms of the ci and gi coefficients and a set of renormalization scheme independent parameters τi. A second set of renormalization scheme independent parameters σi is shown to play a very similar role in the perturbative expansion of mpole in terms of m(μ) and a(μ). We illustrate our approach by a perturbative computation of Γ using the [Formula: see text] renormalization scheme. Two other particular mass independent renormalization schemes are briefly considered.

In this paper, the determination of eccentric anomaly (E) for Kepler’s satellite orbit using Perturbation-Based Seeded Secant (PBSS) iteration algorithm is presented. The solution is meant for Kepler’s orbit with the value of eccentricity (e) in the range 0 ≤ e ≤ 1. Such orbits are either circular or elliptical. The demonstration of the applicability of the PBSS iteration is presented using sample numerical examples with different values of mean anomaly (M) and eccentricity (e). The summary of the results of E for M = 30° and e in the range 0.001 ≤ e ≤1 showed that the convergence cycle (n) increases as e increases. Particularly, n increased from 2 at e = 0.01 to n = 8 at e =1. The implication is that it takes more iterations to arrive at the value of E with the desired accuracy or error performance (which in this case is set to 10^(-12)). Another implication is that a good choice of the initial value of E is essential especially as the value of e increases. As such, effort should be made to develop a means of estimating the initial value of E which will reduce the convergence cycle for higher values of e.

In this paper, we solve analytically the Schrodinger equation for s-wave and arbitrary angular momenta with the Hua potential is investigated respectively. The wave function as well as energy equation are obtained in an exact analytical manner via the Nikiforov Uvarov method using two approximations scheme. Some special cases of this potentials are also studied.

In this paper, we investigate the approximate bound states solutions of Klein–Gordon equation with position-dependent mass for scalar and vector central Hulthen-type potentials by applying an approximation scheme to deal with the centrifugal potential for any l-states. The supersymmetric quantum mechanics (SUSYQM) and shape invariance approaches are used in the calculations. We obtain straightforwardly the relativistic energies and their corresponding normalized wavefunctions, expressed in terms of Jacobi polynomials. Special cases of interest of this problem are also deduced and it is found that our results are in good agreement with those obtained in the literature.

In a companion paper, we present the theoretical development of a new robust, relaxation-free iteration scheme for multiphysics k-eigenvalue problems. These types of problems are essential to the s...

This paper presents a new robust scheme for coupled physics nuclear reactor calculations. We focus specifically on high-fidelity whole-core transport calculations with coarse mesh finite difference...

We derive and describe a very accurate variational scheme for the ground state of the system of a few ultra-cold bosons confined in one-dimensional traps of arbitrary shapes. It is based on assumption that all inter-particle correlations have two-body nature. By construction, the proposed ansatz is exact in the noninteracting limit, exactly encodes boundary conditions forced by contact interactions, and gives full control on accuracy in the limit of infinite repulsions. We show its efficiency in a whole range of intermediate interactions for different external potentials. Our results manifest that for generic non-parabolic potentials mutual correlations forced by interactions cannot be captured by distance-dependent functions., The authors' version accepted for publication

Given a fat point scheme $\mathbb {W}=m_{1}P_{1}+\cdots +m_{s}P_{s}$ in the projective n-space $\mathbb {P}^{n}$ over a field K of characteristic zero, the modules of Kahler differential k-forms of its homogeneous coordinate ring contain useful information about algebraic and geometric properties of $\mathbb {W}$ when $k\in \{1,\dots , n+1\}$ . In this paper, we determine the value of its Hilbert polynomial explicitly for the case k = n + 1, confirming an earlier conjecture. More precisely this value is given by the multiplicity of the fat point scheme $\mathbb {Y} = (m_{1}-1)P_{1} + {\cdots } + (m_{s}-1)P_{s}$ . For n = 2, this allows us to determine the Hilbert polynomials of the modules of Kahler differential k-forms for k = 1,2,3, and to produce a sharp bound for the regularity index for k = 2.

With the help of a successful scheme for overcoming difficulties arising for l ≠ 0 in the centrifugal part of the Rosen–Morse potential with bound states, a solution of the modified Klein–Fock–Gordon equation is found. The energy eigenvalues and the corresponding radial wave functions are found for an arbitrary value of the orbital quantum number ( l ≠ 0 ).

This brief proposes a new discrete-time realization of the super-twisting algorithm based observer. While keeping the property of chattering suppression, the estimation precision of the proposed realization scheme is consistent with the continuous-time super-twisting algorithm (STA), that is, in the absence of noises, the estimation errors satisfy the standard asymptotical accuracy level, i.e., $|e_{1}| \sim \mu _{1}h^{2}$ and $|e_{2}|\sim \mu _{2}h$ with sampling time $h$ and two positive constants $\mu _{1}$ and $\mu _{2}$ . Comparing to the conventional explicit Euler method, the proposed scheme is more exact in the term of smaller constants $\mu _{1}$ and $\mu _{2}$ . The superiority of the proposed scheme is demonstrated through an example of typical buck converter circuit system.

Without four-wave mixing terms in birefringent fibers, the extended trial function scheme was used to obtain optical soliton solutions for the coupled system corresponding to the Gerdjikov-Ivanov equation. The procedure reveals singular soliton solutions, bright soliton solutions, and highly important solutions in terms of Jacobi’s elliptic function. And in the limiting case of the modulus of ellipticity, singular and singular-periodic soliton solutions, along with their respective existence criteria.

A modification of the homotopy perturbation method is suggested with three effective expansions to solve a nonlinear oscillator with damping terms to expand the solution, the frequency and the amplitude. The Duffing equation with linear damping is used as an example to illustrate the simple solution process and effective results. The analysis exhibits that the amplitude behaves as an exponential decay with the damping parameter. This scheme yields a more effective result for the nonlinear oscillators and overcomes the shortcoming in some problems.

The present study presents the application of a structural optimization scheme for a Double Split Hollow Sphere (DSHS) based acoustic cloaking structure for irregular surfaces. For acoustic wave cl...

The generalized minimal subtraction scheme for ultraviolet renormalization (Kuznetsov and Tkachov, 1988) is fine-tuned with applications in mind. The resulting $\text{MS}^4$ scheme obviates extraneous regularizations and renders momentum integrands integrable by subtracting troublesome asymptotic terms in a physically correct fashion due to the use of special minimal subtraction operators defined congruously with the physically natural Polchinski cutoffs. A direct derivation of the Callan-Symanzik equations avoids divergent renormalization factors or counterterms, and automatically yields explicit exact finite integral expressions for renormalization group functions., Comment: 15 pages

In the original Predicted Particle Properties (P3) bulk microphysics scheme, all ice-phase hydrometeors are represented by one or more “free” ice categories, where the physical properties evolve smoothly through changes to the four prognostic variables (per category), and with a two-moment representation of the particle size distribution. As such, the spectral dispersion cannot evolve independently, which thus results in limitations in representation of ice—in particular, hail—due to necessary constraints in the scheme to prevent excessive gravitational size sorting. To overcome this, P3 has been modified to include a three-moment representation of the size distribution of each ice category through the addition of a fifth prognostic variable, the sixth moment of the size distribution. The details of the three-moment ice parameterization in P3 are provided. The behavior of the modified scheme, with the single-ice-category configuration, is illustrated through simulations in a simple 1D kinematic model framework as well as with near large-eddy-resolving (250-m grid spacing) 3D simulations of a hail-producing supercell. It is shown that the three-moment ice configuration controls size sorting in a physically based way and leads to an improved capacity to simulate large, heavily rimed ice (hail), including mean and maximum sizes and reflectivity, and thus an overall improvement in the representation of ice-phase particles in the P3 scheme.

This paper develops a high order adaptive scheme for solving nonlinear Schrodinger equations. The solutions to such equations often exhibit solitary wave and local structures, which makes adaptivity essential in improving the simulation efficiency. Our scheme uses the ultra-weak discontinuous Galerkin (DG) formulation and belongs to the framework of adaptive multiresolution schemes. Various numerical experiments are presented to demonstrate the excellent capability of capturing the soliton waves and the blow-up phenomenon.

Quantum logic gate operates on a number of qubits, where a controlled Z gate operates on two qubit data. Similarly, the square root of controlled Z (SRCZ) operates also on two qubit data. The SRCZ gate is very much advantageous in quantum computing. Different quantum gates are implemented by using the physical properties of light such as phase, polarization, intensity, and frequency. Here, the authors propose a photonic scheme for implementing quantum SRCZ logic gate using phase as well as intensity encoding. The non‐linear property of Pockels material with proper external biasing voltage is used to develop the scheme.

This paper deals with solution methods for singularly perturbed delay differential equations having delay on the convection and reaction terms. The considered problem exhibits an exponential boundary layer on the left or right side of the domain. The terms with the delay are treated using Taylor?s series approximation and the resulting singularly perturbed boundary value problem is solved using a specially designed exponentially finite difference method. The stability of the scheme is analysed and investigated using a comparison principle and solution bound. The formulated scheme converges uniformly with linear order of convergence. The theoretical findings are validated using three numerical test examples.

In this paper, a six-point finite-difference numerical scheme for calculating the Prandtl equation of a laminar boundary layer is proposed to determine the point of separation of flows with large Reynolds numbers when flowing around smooth bodies. The input data for this scheme are the results of modeling by the method of discrete vortices within the model of an ideal fluid. The velocity profile around the critical point is determined from the analytical solution. The resulting system of linear algebraic equations is solved by the run method. Because the coefficients of the system are nonlinear, the iteration method is used to find the solution. The thickness of the boundary layer is determined during the solution process. The point of separation and circulation of descending vortices is determined from the calculation of the boundary layer. Then at the point of separation the rise of several free vortices is modeled, the dynamics of which is modeled within the method of discrete vortices. The scheme was tested on the problem of the flow around the cylinder and showed good results in comparison with the experimental data and calculations of other authors.