scholarly journals Optical Solutions of the Date–Jimbo–Kashiwara–Miwa Equation via the Extended Direct Algebraic Method

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Ghazala Akram ◽  
Naila Sajid ◽  
Muhammad Abbas ◽  
Y. S. Hamed ◽  
Khadijah M. Abualnaja
Keyword(s):  
Water Waves ◽  
Three Dimensional ◽  
Algebraic Method ◽  
Mathematical Physics ◽  
Integration Scheme ◽  
Hyperbolic Functions ◽  
Dynamical Characteristics ◽  
Complex Valued ◽  
Djkm Equation ◽  
Model Water

In this study, the solutions of 2 + 1 -dimensional nonlinear Date–Jimbo–Kashiwara–Miwa (DJKM) equation are characterized, which can be used in mathematical physics to model water waves with low surface tension and long wavelengths. The integration scheme, namely, the extended direct algebraic method, is used to extract complex trigonometric, rational and hyperbolic functions. The complex-valued solutions represent traveling waves in different structures, such as bell-, V-, and W-shaped multiwaves. The results obtained in this article are novel and more general than those contained in the literature (Wang et al., 2014, Yuan et al., 2017, Pu and Hu 2019, Singh and Gupta 2018). Furthermore, the mechanical features and dynamical characteristics of the obtained solutions are demonstrated by three-dimensional graphics.

Diverse wave propagation in shallow water waves with the Kadomtsev–Petviashvili–Benjamin–Bona–Mahony and Benney–Luke integrable models

Open Physics ◽  
2021 ◽  
Vol 19 (1) ◽  
pp. 808-818
Author(s):  
Usman Younas ◽  
Aly R. Seadawy ◽  
Muhammad Younis ◽  
Syed T. R. Rizvi ◽  
Saad Althobaiti
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Three Dimensional ◽  
Wave Model ◽  
Algebraic Method ◽  
Integrable Models ◽  
Computational Technique ◽  
Shallow Water Waves ◽  
Physical Problems ◽  
Contour Plots

Abstract The shallow water wave model is one of the completely integrable models illustrating many physical problems. In this article, we investigate new exact wave structures to Kadomtsev–Petviashvili–Benjamin–Bona–Mahony and the Benney–Luke equations which explain the behavior of waves in shallow water. The exact structures are expressed in the shapes of hyperbolic, singular periodic, rational as well as solitary, singular, shock, shock-singular solutions. An efficient computational strategy namely modified direct algebraic method is employed to construct the different shapes of wave structures. Moreover, by fixing parameters, the graphical representations of some solutions are plotted in terms of three-dimensional, two-dimensional and contour plots, which explain the physical movement of the attained results. The accomplished results show that the applied computational technique is valid, proficient, concise and can be applied in more complicated phenomena.

scholarly journals Variational modelling of extreme waves through oblique interaction of solitary waves: application to Mach reflection

2017 ◽  
Vol 24 (1) ◽  
pp. 43-60 ◽  
Author(s):  
Floriane Gidel ◽  
Onno Bokhove ◽  
Anna Kalogirou
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Mach Reflection ◽  
Temporal Integration ◽  
Three Dimensional ◽  
Integration Scheme ◽  
Extreme Waves ◽  
Flow Equations ◽  
Dispersive Model ◽  
Incident Waves

Abstract. In this work, we model extreme waves that occur due to Mach reflection through the intersection of two obliquely incident solitary waves. For a given range of incident angles and amplitudes, the Mach stem wave grows linearly in length and amplitude, reaching up to 4 times the amplitude of the incident waves. A variational approach is used to derive the bidirectional Benney–Luke equations, an asymptotic equivalent of the three-dimensional potential-flow equations modelling water waves. This nonlinear and weakly dispersive model has the advantage of allowing wave propagation in two horizontal directions, which is not the case with the unidirectional Kadomtsev–Petviashvili (KP) equation used in most previous studies. A variational Galerkin finite-element method is applied to solve the system numerically in Firedrake with a second-order Störmer–Verlet temporal integration scheme, in order to obtain stable simulations that conserve the overall mass and energy of the system. Using this approach, we are able to get close to the 4-fold amplitude amplification predicted by Miles.

scholarly journals Variational modelling of extreme waves through oblique interaction of solitary waves

2016 ◽  
Author(s):  
Floriane Gidel ◽  
Onno Bokhove
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Mach Reflection ◽  
Temporal Integration ◽  
Three Dimensional ◽  
Integration Scheme ◽  
Extreme Waves ◽  
Flow Equations ◽  
Dispersive Model ◽  
Incident Waves

Abstract. In this work, we model extreme waves that occur due to Mach reflection through the intersection of two obliquely incident solitary waves. For a given range of incident angles and amplitudes, the Mach stem wave grows linearly in length and amplitude, reaching up to four times the amplitude of the incident waves. A variational approach is used to derive the bidirectional Benney-Luke equations, an asymptotic equivalent of the three-dimensional potential-flow equations modelling water waves. This nonlinear and dispersive model has the advantage of allowing wave propagation in two horizontal directions, which is not the case with the unidirectional Kadomtsev-Petviashvili (KP) equation used in most previous studies. A variational Galerkin finite element method is applied to solve the system numerically in Firedrake with a second-order Stormer-Verlet temporal integration scheme in order to obtain stable simulations that conserve the overall mass and energy of the system. Using this approach, we are able to get close to the fourfold amplitude amplification predicted by Miles.

scholarly journals Machine-readable Yin–Yang imbalance: traditional Chinese medicine syndrome computer modeling based on three-dimensional noninvasive cardiac electrophysiology imaging

2019 ◽  
Vol 47 (4) ◽  
pp. 1580-1591 ◽  
Author(s):  
Wei Cen ◽  
Ralph Hoppe ◽  
Aiwu Sun ◽  
Hongyan Ding ◽  
Ning Gu
Keyword(s):  
Chinese Medicine ◽  
Traditional Chinese Medicine ◽  
Electrical Activity ◽  
Computer Modeling ◽  
Cardiac Electrophysiology ◽  
Three Dimensional ◽  
Mathematical Physics ◽  
Syndrome Differentiation ◽  
Physics Model ◽  
Machine Readable

Objectives The principal diagnostic methods of traditional Chinese medicine (TCM) are inspection, auscultation and olfaction, inquiry, and pulse-taking. Treatment by syndrome differentiation is likely to be subjective. This study was designed to provide a basic theory for TCM diagnosis and establish an objective means of evaluating the correctness of syndrome differentiation. Methods We herein provide the basic theory of TCM syndrome computer modeling based on a noninvasive cardiac electrophysiology imaging technique. Noninvasive cardiac electrophysiology imaging records the heart’s electrical activity from hundreds of electrodes on the patient’s torso surface and therefore provides much more information than 12-lead electrocardiography. Through mathematical reconstruction algorithm calculations, the reconstructed heart model is a machine-readable description of the underlying mathematical physics model that reveals the detailed three-dimensional (3D) electrophysiological activity of the heart. Results From part of the simulation results, the imaged 3D cardiac electrical source provides dynamic information regarding the heart’s electrical activity at any given location within the 3D myocardium. Conclusions This noninvasive cardiac electrophysiology imaging method is suitable for translating TCM syndromes into a computable format of the underlying mathematical physics model to offer TCM diagnosis evidence-based standards for ensuring correct evaluation and rigorous, scientific data for demonstrating its efficacy.

scholarly journals Three-Dimensional Elastodynamic Analysis Employing Partially Discontinuous Boundary Elements

Algorithms ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 129
Author(s):  
Yuan Li ◽  
Ni Zhang ◽  
Yuejiao Gong ◽  
Wentao Mao ◽  
Shiguang Zhang
Keyword(s):  
Side Effect ◽  
Domain Boundary ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Integration Scheme ◽  
Computational Accuracy ◽  
Time Step ◽  
Corner Points ◽  
Discontinuous Elements ◽  
The Time Domain

Compared with continuous elements, discontinuous elements advance in processing the discontinuity of physical variables at corner points and discretized models with complex boundaries. However, the computational accuracy of discontinuous elements is sensitive to the positions of element nodes. To reduce the side effect of the node position on the results, this paper proposes employing partially discontinuous elements to compute the time-domain boundary integral equation of 3D elastodynamics. Using the partially discontinuous element, the nodes located at the corner points will be shrunk into the element, whereas the nodes at the non-corner points remain unchanged. As such, a discrete model that is continuous on surfaces and discontinuous between adjacent surfaces can be generated. First, we present a numerical integration scheme of the partially discontinuous element. For the singular integral, an improved element subdivision method is proposed to reduce the side effect of the time step on the integral accuracy. Then, the effectiveness of the proposed method is verified by two numerical examples. Meanwhile, we study the influence of the positions of the nodes on the stability and accuracy of the computation results by cases. Finally, the recommended value range of the inward shrink ratio of the element nodes is provided.

scholarly journals Computation of Three-Dimensional, Rotational Flow Through Turbomachinery Blade Rows for Improved Aerodynamic Design Studies

1986 ◽  
Author(s):  
S. V. Subramanian ◽  
R. Bozzola ◽  
Louis A. Povinelli
Keyword(s):  
Three Dimensional ◽  
Maximum Flow ◽  
Cost Effective ◽  
Computer Code ◽  
Integration Scheme ◽  
Aerodynamic Design ◽  
Design Studies ◽  
Flow Equations ◽  
Transonic Compressor ◽  
Numerical Predictions

The performance of a three dimensional computer code developed for predicting the flowfield in stationary and rotating turbomachinery blade rows is described in this study. The four stage Runge-Kutta numerical integration scheme is used for solving the governing flow equations and yields solution to the full, three dimensional, unsteady Euler equations in cylindrical coordinates. This method is fully explicit and uses the finite volume, time marching procedure. In order to demonstrate the accuracy and efficiency of the code, steady solutions were obtained for several cascade geometries under widely varying flow conditions. Computed flowfield results are presented for a fully subsonic turbine stator and a low aspect ratio, transonic compressor rotor blade under maximum flow and peak efficiency design conditions. Comparisons with Laser Anemometer measurements and other numerical predictions are also provided to illustrate that the present method predicts important flow features with good accuracy and can be used for cost effective aerodynamic design studies.

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