Search Results (6)

The fifth-order WENO-Z scheme proposed by Borges et al., using a linear combination of low-order smoothness indicators, is designed to provide a low numerical dissipation to solve hyperbolic conservation laws, while the power q in the framework of WENO-Z plays a key role in its performance. In this paper, a novel global smoothness indicator with fifth-order accuracy, which is based on several lower-order smoothness indicators on two-point sub-stencils, is presented, and a new lower-dissipation WENO-Z scheme (WENO-NZ) is developed. The spectral properties of the WENO-NZ scheme are studied through the ADR method and show that this new scheme can exhibit better spectral results than WENO-Z no matter what the power value is. Accuracy tests confirm that the accuracy of WENO-Z with q = 1 would degrade to the fourth order at first-order critical points, while WENO-NZ can recover the optimal fifth-order convergence. Furthermore, numerical experiments with one- and two-dimensional benchmark problems demonstrate that the proposed WENO-NZ scheme can efficiently decrease the numerical dissipation and has a higher resolution compared to the WENO-Z scheme. Full article

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 $\alpha \in (0,2)$ 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 $t\ge 0$ 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 $\alpha \in (0,2)$ and is amenable to the numerical evaluation using uniformly accurate quadrature-based algorithms. Full article

This paper seeks to find optical soliton solutions for Lakshmanan–Porsezian–Daniel (LPD) model with the parabolic law of nonlinearity. The spatiotemporal dispersion is included to the model, as it can contribute to handling the problem of internet bottleneck. This study was performed analytically using the traveling wave hypothesis to reduce the model to an integrable form. Then, the resulting equation was handled with two approaches, namely, the auxiliary equation method and the Bernoulli subordinary differential equation (sub-ODE) method. With an intentional focus on hyperbolic function solutions, abundant optical soliton waves including W-shaped, bright, dark, kink-dark, singular, kink, and antikink solitons were derived with the existing conditions. Furthermore, the behaviors of some optical solitons are illustrated. The spatiotemporal dispersion was found to significantly affect the pulse propagation dynamics. Finally, the modulation instability (MI) of the LPD model is explained in detail along with the extraction of the expression of MI gain. Full article

We present an exponentially convergent numerical method to approximate the solution of the Cauchy problem for the inhomogeneous fractional differential equation with an unbounded operator coefficient and Caputo fractional derivative in time. The numerical method is based on the newly obtained solution formula that consolidates the mild solution representations of sub-parabolic, parabolic and sub-hyperbolic equations with sectorial operator coefficient A and non-zero initial data. The involved integral operators are approximated using the sinc-quadrature formulas that are tailored to the spectral parameters of A, fractional order $\alpha$ and the smoothness of the first initial condition, as well as to the properties of the equation’s right-hand side $f\left(t\right)$. The resulting method possesses exponential convergence for positive sectorial A, any finite t, including $t=0$ and the whole range $\alpha \in (0,2)$. It is suitable for a practically important case, when no knowledge of $f\left(t\right)$ is available outside the considered interval $t\in [0,T]$. The algorithm of the method is capable of multi-level parallelism. We provide numerical examples that confirm the theoretical error estimates. Full article

Background: The present in vitro study aimed to investigate the fatigue performance of different dental fixtures in two different emergence profiles. Biological failures are frequently reported because the problem canonly be solved by replacing a failing implant with a new one. Clinicians addressed minor mechanical failures, such as bending, loosening or the fracture of screws, abutment, or the entire prosthesis, by simply replacing or fixing them. Methods: Transmucosal and submerged bone-level dental implants underwent fatigue strength tests (statical and dynamical performance) by a standardized test: UNI EN ISO 14801:2016. Two types of emergence profiles (Premium sub-crestal straight implant with a cylindrical-shaped coronal emergence or Prama one-piece cylindrical-shape implant with transmucosal convergent neck and hyperbolic geometry) were tested, and dynamic fatigue were run to failure. Data was analyzed by a suitable statistical tool. Results: The Wöhler curve of 0.38 cm Premium group c2, appeared to be significantly different from that of the 0.38 cm Prama group c3 (nonparametric one-way ANOVA χ² = 6; degree of freedom = 1; probability = 0.0043) but not from that of the 0.33 cm Premium group c1 (nonparametric one-way ANOVA χ² = 0.62; degree of freedom = 1; probability = 0.4328). Fatigue performance of configuration 2 was one and a half times better than that of configuration 3. Group c3 had a better ultimate failure load (421.6 ± 12.5 N) than the other two settings i.e., c1 (324.5 ± 5.5 N) and c2 (396.3 ± 5.6) reaching almost a nonsignificant level. Conclusions: It was observed that a transmucosal implant design could provide the highest resistance to static fracture. On the other hand, an equicrestal implant design could increase dynamic endurance. Full article

In this paper, non-Fourier heat conduction in a cylinder with non-homogeneous boundary conditions is analytically studied. A superposition approach combining with the solution structure theorems is used to get a solution for equation of hyperbolic heat conduction. In this solution, a complex origin problem is divided into, different, easier subproblems which can actually be integrated to take the solution of the first problem. The first problem is split into three sub-problems by setting the term of heat generation, the initial conditions, and the boundary condition with specified value in each sub-problem. This method provides a precise and convenient solution to the equation of non-Fourier heat conduction. The results show that at low times (t = 0.1) up to about r = 0.4, the contribution of T₁ and T₃ dominate compared to T₂ contributing little to the overall temperature. But at r > 0.4, all three temperature components will have the same role and less impact on the overall temperature (T). Full article