Conservation laws, exact solutions and nonlinear dispersion: A lie symmetry approach

Adnan Shamaoon; Zartab Ali; Qaisar Maqbool

doi:10.59400/jam.v1i1.95

Conservation laws, exact solutions and nonlinear dispersion: A lie symmetry approach

Adnan Shamaoon Department of Mathematics, Physics and Electrical Engineering, Northumbria University Newcastle, Newcastle upon Tyne NE1 8ST, UK
Zartab Ali Department of Mathematics, Lahore University of Management Sciences, Lahore 54792, Pakistan
Qaisar Maqbool Department of Physics, University of Okara, Okara 56300, Pakistan

Ariticle ID: 95

164 Views, 83 PDF Downloads

DOI: https://doi.org/10.59400/jam.v1i1.95

Keywords: lie symmetries, infinitesimals operator, conservation laws, Euler-Lagrangian operator, nonlinear dispersion, exact solutions, multipliers approach

Abstract

In this study, we investigated a set of equations that exhibit compact solutions and nonlinear dispersion. We used the classical lie symmetry approach to derive ordinary differential equations (ODEs) that are well suited for qualitative study. By examining the dynamic behavior of these ODEs, we gained insights into the intricate nature of the underlying system. We also used a powerful multiplier approach to establish nontrivial conservation laws and exact solutions for these equations. These conservation laws provide essential information regarding the underlying symmetries and invariants of the system, and shed light on its fundamental properties.

References

[1] Wang K. Solitary wave dynamics of the local fractional Bogoyavlensky-Konopelchenko model. Fractals 2023; 31(5): 2350054. doi: 10.1142/S0218348X23500548

[2] Rosenau P, Hyman JM. Compactons: Solitons with finite wavelength. Physical Review Letters 1993; 70(5): 564–567. doi: 10.1103/PhysRevLett.70.564

[3] Bruzón MS, Gandarias ML. Traveling wave solutions of the K(m, n) equation with generalized evolution. Mathematical Methods in the Applied Sciences 2018; 41(15): 5851–5857. doi: 10.1002/mma.1339

[4] Bruzón MS, Gandarias ML, González GA, Hansen R. The K(m, n) equation with generalized evolution term studied by symmetry reductions and qualitative analysis. Applied Mathematics and Computation 2012; 218(20): 10094–10105. doi: 10.1016/j.amc.2012.03.084

[5] Anco SC, Bluman G. Direct construction method for conservation laws of partial differential equations part II: General treatment. European Journal of Applied Mathematics 2002; 13(5): 567–585. doi: 10.1017/S0956792501004661

[6] Naz R. Conservation laws for some compacton equations using the multiplier approach. Applied Mathematics Letters 2012; 25(3): 257–261. doi: 10.1016/j.aml.2011.08.019

[7] Rosenau P, Oron A. On compactons induced by a non-convex convection. Communications in Nonlinear Science and Numerical Simulation 2014; 19(5): 1329–1337. doi: 10.1016/j.cnsns.2013.09.028

[8] Gandarias ML. Conservation laws for some equations that admit compacton solutions induced by a non-convex convection. Journal of Mathematical Analysis and Applications 2015; 430(2): 695–702. doi: 10.1016/j.jmaa.2015.04.071

[9] Anco SC, Bluman G. Direct construction of conservation laws from field equations. Physical Review Letters 1997; 78(15): 2869–2873. doi: 10.1103/PhysRevLett.78.2869

[10] Ibragimov NH. A new conservation theorem. Journal of Mathematical Analysis and Applications 2007; 333(1): 311–328. doi: 10.1016/j.jmaa.2006.10.078

[11] Ibragimov NH. Quasi-self-adjoint differential equations. Archives of ALGA 2007; 4: 55–60.

[12] Gandarias ML. Weak self-adjoint differential equations. Journal of Physics A: Mathematical and Theoretical 2011; 44(26): 262001. doi: 10.1088/1751-8113/44/26/262001

[13] Ibragimov NH. Nonlinear self-adjointness and conservation laws. Journal of Physics A: Mathematical and Theoretical 2011; 44: 432002. doi: 10.1088/1751-8113/44/43/432002

[14] Bruzón MS, Gandarias ML, Ibragimov NH. Self-adjoint sub-classes of generalized thin film equations. Journal of Mathematical Analysis and Applications 2009; 357(1): 307–313. doi: 10.1016/j.jmaa.2009.04.028

[15] Gandarias ML, Bruzón MS. Some conservation laws for a forced KDV equation. Nonlinear Analysis: Real World Applications 2012; 13(6): 2692–2700. doi: 10.1016/j.nonrwa.2012.03.013

[16] Freire IL. Self-adjoint sub-classes of third and fourth-order evolution equations. Applied Mathematics and Computation 2011; 217(22): 9467–9473. doi: 10.1016/j.amc.2011.04.041

[17] Freire IL, Sampaio JCS. On the nonlinear self-adjointness and local conservation laws for a class of evolution equations unifying many models. Communications in Nonlinear Science and Numerical Simulation 2014; 19(2): 350–360. doi: 10.1016/j.cnsns.2013.06.010

[18] Ibragimov N, Torrisi M, Tracinà R. Quasi self-adjoint nonlinear wave equations. Journal of Physics A: Mathematical and Theoretical 2010; 43(44): 442001. doi: 10.1088/1751-8113/43/44/442001

[19] Ibragimov NH, Torrisi M, Tracina R. Self-adjointness and conservation laws of a generalized burgers equation. Journal of Physics A: Mathematical and Theoretical 2011; 44: 145201. doi: 10.1088/1751-8113/44/14/145201

[20] Gandarias ML. Weak self-adjointness and conservation laws for a porous medium equation. Communications in Nonlinear Science and Numerical Simulation 2012; 17(6): 2342–2349. doi: 10.1016/j.cnsns.2011.10.020

[21] Gandarias ML, Redondo M, Bruzón MS. Some weak self-adjoint Hamilton—Jacobi—Bellman equations arising in financial mathematics. Nonlinear Analysis: Real World Applications 2012; 13(1): 340–347. doi: 10.1016/j.nonrwa.2011.07.041

[22] Gandarias ML. Nonlinear self-adjointness through differential substitutions. Communications in Nonlinear Science and Numerical Simulation 2014; 19(10): 3523–3528. doi: 10.1016/j.cnsns.2014.02.013

[23] Torrisi M, Tracinà R. Quasi self-adjointness of a class of third order nonlinear dispersive equations. Nonlinear Analysis: Real World Applications 2013; 14(3): 1496–1502. doi: 10.1016/j.nonrwa.2012.10.013

[24] Deiakeh RA, Arqub OA, Smadi MA, Momani S. Lie symmetry analysis, explicit solutions, and conservation laws of the time-fractional Fisher equation in two-dimensional space. Journal of Ocean Engineering and Science 2022; 7(4): 345–352. doi: 10.1016/j.joes.2021.09.005

[25] Arqub OA, Hayat T, Alhodaly M. Analysis of lie symmetry, explicit series solutions, and conservation laws for the nonlinear time-fractional phi-four equation in two-dimensional space. International Journal of Applied and Computational Mathematics 2022; 8(3): 145. doi: 10.1007/s40819-022-01334-0

[26] Aal MA, Arqub OA. Lie analysis and laws of conservation for the two-dimensional model of Newell—Whitehead—Segel regarding the Riemann operator fractional scheme in a time-independent variable. Arab Journal of Basic and Applied Sciences 2023; 30(1): 55–67. doi: 10.1080/25765299.2023.2172840

[27] Euler N, Euler M. On nonlocal symmetries, nonlocal conservation laws and nonlocal transformations of evolution equations: Two linearisable hierarchies. Journal of Nonlinear Mathematical Physics 2009; 16(4): 489–504. doi: 10.1142/S1402925109000509

[28] Anco SA. Conservation laws of scaling-invariant field equations. Journal of Physics A: Mathematical and General 2003; 36: 8623. doi: 10.1088/0305-4470/36/32/305

Published

2023-06-26

How to Cite

Shamaoon, A., Ali, Z., & Maqbool, Q. (2023). Conservation laws, exact solutions and nonlinear dispersion: A lie symmetry approach. Journal of AppliedMath, 1(1), 95. https://doi.org/10.59400/jam.v1i1.95

Download Citation

Issue

Vol. 1 No. 1 (2023)

Section

Article

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Editor-in-Chief

Assoc. Prof. Ahmad Zeeshan

International Islamic University, Pakistan

eISSN

2972-4805

Publication Frequency

Bi-monthly

About the Publisher

Academic Publishing insists on taking academic exchange and publication as the main line, carrying out comprehensive management based on science and technology, and fully exploring excellent international publishing resources. Within 5 years, it will form a strategic framework and scale with science (S), technology (T), medicine (M), education (E), and humanities and arts (H) as the main publishing fields. Academic Publishing is headquartered in Singapore and based in Malaysia, with the United States and China providing the main scientific and academic resources. At the same time, it has established long-term good cooperative relations with other publishing companies, scientific research communities, and academic organizations in more than a dozen countries and regions. Academic Publishing uses English and Chinese as its main publishing languages, mainly publishing books, journals, and conference papers in print and online. The vast majority of publications follow the international open access policy, providing stable and long-term quality and professional publications. With the joint efforts of the expert team and our professional editorial team, our publications will gradually be indexed by international databases in stages to provide convenient and professional retrieval for various scholars. At the same time, manuscripts we accept will be subject to the peer review principle, and cutting-edge and innovative research articles will be preferentially accepted for peer reference and discussion. All kinds of our publications are welcome for peer to contribute, access, and download.

more

Volume Arrangement

2024

2023

Featured Articles

H∞ hybrid control and MRD in a steel frame building subjected to excessivevibrations caused by the dynamic action of wind and earthquake

The dynamic loads from earthquakes and winds can destroy lives, cause collapse in civil structures, and interrupt basic services provided to the population. In this scenario, structural designs must be developed to decrease the damage induced by these actions. The objective of this work is to design a hybrid controller based on the H∞ optimization via state feedback and the magneto-rheological damper (MRD) to mitigate the excessive vibrations of a three-story steel frame building, represented through the shear building model, subjected to the simultaneous dynamic action of wind and earthquake. All research is based on computational simulation, experimental research and results will not be addressed. In the numerical analysis, digital computer and MATLAB® software are used, and implemented codes generate the expected results based on the mathematical modeling. With the application of the H∞ control technique via state feedback, the displacements were reduced by 77%. With MRD this reduction was 79%. With the hybrid controller, this reduction was 100%. Thus, the verifications in relation to maximum displacements were met for NBR 15421:2006, NBR 8800:2008 and NBR 6118:2014. From the results, it is concluded that the hybrid controller proved to be more efficient and achieved the proposed objective. The exogenous inputs had zero influence on the behavior of the system output.

Topological analysis of multiple tables

The paper proposes a topological approach in order to explore several data tables simultaneously. These data tables of quantitative and/or qualitative variables measured on different homogeneous themes, collected from the same individuals. This approach, called topological analysis of multiple tables (TAMT), is based on the notion of neighborhood graphs in the context of a joint analysis of several data tables. It allows the simultaneous study of possible links between several thematic tables. The structure of the correlations or associations of the variables in each thematic table is analyzed according to quantitative, qualitative or mixed variables considered. Like the multiple factorial analysis (MFA), the TAMT allows several tables of variables to be analyzed simultaneously, and to obtain results, in particular graphical representations, which make it possible to study the relationship between individuals, variables and tables of data. These can also be tables of temporal data, collected at different times on the same individuals. The proposed TAMT approach is illustrated using real data associated with several and different homogeneous themes. Its results are compared to those from the MFA method.

Development of a stacked hybrid Decision Tree model leveraging the NSL-KDD dataset

Intrusion detection in information technology as well as operational technology networks is highly required in modern day systems due to the increased spate of cyber-attacks in both number and complexity. Anomaly-based intrusion detection systems which have the capacity to detect novel or zero-day attacks are highly employed in this regard. One important component of anomaly-based intrusion detection systems which ensures their behaviour is artificial intelligence in general and machine learning in particular. The burden in modern day cybersecurity research is to investigate and develop models that can outperform existing ones. This paper is aimed at developing a hybrid decision tree model using the stacking ensemble approach. Performances were measured on the basis of recall, precision, accuracy, F1-score, receiver operating characteristics and confusion matrices. The hybrid model presented a precision of 97%, accuracy of 81%, F1-score of 80% and AUC score of 0.96, respectively.

The synergistic effect of the multiple parameters of vibro-impact nonlinear energy sink

This article studies the dynamics and efficiency of a vibro-impact damper (single-sided vibro-impact nonlinear energy sink—SSVI NES) depending on the exciting force parameters. The damper is coupled with a linear oscillator—the primary structure. It is shown that the damper is quite effective in a wide range of the exciting force amplitude and in the range of its frequency, which are higher than the resonant frequency; damper efficiency in these regions is fairly stable. The dynamics of the vibro-impact system “primary structure—SSVI NES” is rich and complex, which, however, does not impair the damper efficiency. In complex oscillatory regimes, the damper makes bilateral impacts: it hits both an obstacle and directly against the primary structure, which actually turns a single-sided NES into a double‐sided one. The optimization procedure and the choice of optimal damper parameters play a very important role in damper design. Optimizing multiple damper parameters instead of three shows a synergistic effect and provides better results.

Optimization imposition upon drone gimbal control electronics

The goal of the manuscript is to design a relatively good control structure for the noise suppression of a drone’s camera gimbal action. The gimbal’s movement can be simplified as a rest-to-rest reorientation system that can achieve the boundary result of a dynamic system. Six different control architectures are proposed and evaluated based on their ability to control the trajectory of the dynamic-system position and speed, their running time, and their quadratic cost. The robustness of the control architecture to uncertainties in inertia and sensor noise is also analyzed. Monte Carlo figures are used to assess the performance of the six control systems. The conditions for applying different architectures are identified through this analysis. The analysis and experimental tests reveal the most suitable control of the drone’s camera gimbal rotation.