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In this article, the stochastic Hirota–Maccari system forced in the Itô sense by multiplicative noise is considered. We just use the He’s semi-inverse method, sine–cosine method, and Riccati–Bernoulli sub-ODE method to get new stochastic solutions which are hyperbolic, trigonometric, and rational. The major benefit of these three approaches is that they can be used to solve similar models. Furthermore, we plot 3D surfaces of analytical solutions obtained in this article by using MATLAB to illustrate the effect of multiplicative noise on the exact solution of the Hirota–Maccari method.

Purpose It is extremely difficult to establish a variational principle for plasma. Kalaawy obtained a variational principle by using the semi-inverse method in 2016, and Li and He suggested a modification in 2017. This paper aims to search for a generalized variational formulation with a free parameter. Design/methodology/approach The semi-inverse method is used by suitable construction of a trial functional with some free parameters. Findings A modification of Li-He’s variational principle with a free parameter is obtained. Originality/value This paper suggests a new approach to construction of a trial-functional with some free parameters.

Purpose A three-dimensional (3D) unsteady potential flow might admit a variational principle. The purpose of this paper is to adopt a semi-inverse method to search for the variational formulation from the governing equations. Design/methodology/approach A suitable trial functional with a possible unknown function is constructed, and the identification of the unknown function is given in detail. The Lagrange multiplier method is used to establish a generalized variational principle, but in vain. Findings Some new variational principles are obtained, and the semi-inverse method can easily overcome the Lagrange crisis. Practical implications The semi-inverse method sheds a promising light on variational theory, and it can replace the Lagrange multiplier method for the establishment of a generalized variational principle. It can be used for the establishment of a variational principle for fractal and fractional calculus. Originality/value This paper establishes some new variational principles for the 3D unsteady flow and suggests an effective method to eliminate the Lagrange crisis.

PurposeOn a microgravity condition, a motion of soliton might be subject to a microgravity-induced motion. There is no theory so far to study the effect of air density and gravity on the motion property. Here, the author considers the air as discrete molecules and a motion of a soliton is modeled based on He’s fractal derivative in a microgravity space. The variational principle of the alternative model is constructed by semi-inverse method. The variational principle can be used to establish the conservation laws and reveal the structure of the solution. Finally, its approximate analytical solution is found by using two-scale method and homotopy perturbation method (HPM).Design/methodology/approachThe author establishes a new fractal model based on He’s fractal derivative in a microgravity space and its variational principle is obtained via the semi-inverse method. The approximate analytical solution of the fractal model is obtained by using two-scale method and HPM.FindingsHe’s fractal derivative is a powerful tool to establish a mathematical model in microgravity space. The variational principle of the fractal model can be used to establish the conservation laws and reveal the structure of the solution.Originality/valueThe author proposes the first fractal model for the soliton motion in a microgravtity space and obtains its variational principle and approximate solution.

Doppler current profilers on autonomous underwater gliders measure water velocity relative to the moving glider over vertical ranges of O (10) m. Measurements obtained with 1-MHz Nortek acoustic Doppler dual current profilers (AD2CPs) on Spray gliders deployed off Southern California, west of the Galápagos Archipelago, and in the Gulf Stream are used to demonstrate methods of estimating absolute horizontal velocities in the upper 1000 m of the ocean. Relative velocity measurements nearest to a glider are used to infer dive-dependent flight parameters, which are then used to correct estimates of absolute vertically averaged currents to account for the accumulation of biofouling during months-long glider missions. The inverse method for combining Doppler profiler measurements of relative velocity with absolute references to estimate profiles of absolute horizontal velocity is reviewed and expanded to include additional constraints on the velocity solutions. Errors arising from both instrumental bias and decreased abundance of acoustic scatterers at depth are considered. Though demonstrated with measurements from a particular combination of platform and instrument, these techniques should be applicable to other combinations of gliders and Doppler current profilers.

The optimization of the landscape ecological security pattern aims to construct a suitable ecological environment and promote the harmonious development between humans and nature. The optimization model of the ecological security pattern for the main urban area of Chongqing was constructed with the granularity inverse method, minimum cumulative resistance model, and spatial network analysis method. We used ecological nodes to optimize the landscape ecological security pattern by organically combining the landscape quantity and spatial structure, and analyzed the effectiveness of the optimized pattern. The results were as follows: (1) The optimal granularity for selecting the ecological source in the study area was 500 m. There were 220 ecological sources with a total area of 188691.03 hm 2 and a minimum area of 75.15 hm 2 . (2) The ecological buffer zone, protection and utilization zone, key development zone, coordinated control zone, and restricted development zone accounted for 57.78%, 20.87%, 12.36%, 6.48%, and 2.50%, respectively, of the total area. (3) The construction of the landscape ecological security pattern contained 70 ecological corridors with a total length of 415.89 km. The longest and shortest ecological corridors had lengths of 20.33 km and 1153.23 m, respectively. There were 17 ecological nodes of corridor-resistance and 27 ecological nodes of corridor-corridor. (4) 41 ecological node service areas were constructed, with a total area of approximately 236.0723 hm 2 , accounting for 0.04% of the total study area, and the largest and smallest ecological node areas were 6.0744 hm 2 and 0.0057 hm 2 , respectively. (5) The optimized result of the landscape ecological security pattern converted 209.1384 hm 2 of nonecological land into ecological land.

The well-known nonlinear Schrödinger equation (NLSE) plays a crucial role in describing the temporal evolution of disturbances in marginally stable or unstable media. However, when the media is a fractal form, it becomes ineffective. Thus, the fractal modification to the NLSE is presented based on the fractal derivative in this work for the first time. The semi-inverse method is employed to establish the fractal variational principle. The entire process of deriving the fractal variational principle is presented in detail. To our knowledge, the fractal variational principle mentioned in this article is the first exploration and report to date. The fractal variational principle established in this paper is expected to deepen our understanding of the essence of physical phenomena in the fractal space and offer new ideas for the application and exploration of the variational approaches.

This paper proposes an inverse method to design two-dimensional freeform imaging systems. The mathematical model to design a parallel-to-point double-reflector imaging system using inverse methods from nonimaging optics is presented. An imaging condition is imposed on the energy distributions at the source and target of the optical system. The inverse freeform design is compared to the classical Schwarzschild telescope, which is well-known for minimizing third-order aberrations. A raytracer using quasi-interpolation is employed to test the performance of both designs by comparing the spot sizes corresponding to on-axis and off-axis light rays. A numerical example demonstrates that the inverse freeform design outperforms the classical design.

Aortic dissection is the most common catastrophe of the thoracic aorta, with a very high rate of mortality. Type A dissection is often associated with an ascending thoracic aortic aneurysm (ATAA). However, it is widely acknowledged that the risk of type A dissection cannot be reliably predicted simply by measuring the ATAA diameter and there is a pressing need for more reliable risk predictors. It was previously shown that there is a significant correlation between a rupture criterion based on the ultimate stretch of the ATAA and the local extensional stiffness of the aorta. Therefore, reconstructing regional variations of the extensional stiffness across the aorta appears highly important. In this paper, we present a novel noninvasive inverse method to identify the patient-specific local extensional stiffness of aortic walls based on preoperative gated CT scans. Using these scans, a structural mesh is defined across the aorta with a set of nodes attached to the same material points at different time steps throughout the cardiac cycle. For each node, time variations of the position are analyzed using Fourier series, permitting the reconstruction of the local strain distribution (fundamental term). Relating these strains to tensions with the extensional stiffness, and writing the local equilibrium satisfied by the tensions, the local extensional stiffness is finally derived at every position. The methodology is applied onto the ascending and descending aorta of three patients. Interestingly, the regional distribution of identified stiffness properties appears heterogeneous across the ATAA. Averagely, the identified stiffness is also compared with values obtained using other nonlocal methodologies. The results support the possible noninvasive prediction of stretch-based rupture criteria in clinical practice using local stiffness reconstruction.

In this report, a combination of the diffusion multiple technique and the recently developed pragmatic numerical inverse method was employed for a high-throughput determination of interdiffusivity matrices in Co-Cr-FeMn-Ni high-entropy alloys (HEAs). Firstly, one face-centered cubic (fcc) quinary Co-Cr-Fe-Mn-Ni diffusion multiple at 1373 K was carefully prepared by means of the hot-pressing technique. Based on the composition profiles measured by the field emission electron probe micro analysis (FE-EPMA), the composition-dependent interdiffusivity matrices in quinary Co-Cr-Fe-Mn-Ni system at 1373 K were then efficiently determined using the pragmatic numerical inverse method. The determined interdiffusivities show good agreement with the limited results available in the literature. Moreover, the further comparison with the interdiffusivities in the lower-order systems indicates the sluggish diffusion effect in Co-Cr-Fe-Mn-Ni HEAs, which is however not observed in tracer diffusivities. In order for the convenience in further analysis, a generalized transformation relation among interdiffusivities with different dependent components in multicomponent systems was finally derived.