Computing solutions of nonlinear partial differential equations (NLPDEs) is of importance to solve and understand the physical meaning of a given problem. To obtain the exact solutions of NLPDEs, researchers have come up with several methods, such as Kudryshov method , double reduction method , simplest equation method [3–5], homotopy analysis transform method , direct method , etc.
The Lie’s theory [8, 9], introduced by Sophus Lie (1842–1899), is used to construct a symmetry reduction, and it can also play a major role in finding the exact solutions of partial differential equations (PDEs). The application of the Lie’s method can also be extended to fields such as relativity, classical mechanics, numerical analysis and so on [10–12].
Conservation laws [13–18] play a significant role in finding the solutions of PDEs. Noether’s theorem helps us with an innovation formula for deriving conservation laws by appealing to the symmetries of the action.
The Kadomtsev-Petviashvili (KP) equation emerged pursuant to the efforts made when Kadomtsev and Petviashvili were learning the growth of lengthy ion-acoustic surfs with minor amplitude broadcasting in plasmas falling short of the result of lengthy diagonal worries . The KP equation defines the development of non-linear, lengthy surfs of minor amplitude with sluggish requirement on the transversal coordinate. It is the simplification of the Kortewegde Vries (KdV) equation.
In this paper, we investigate the extended Kadomtsev-Petviashvili (eKP) equation , which reads as:
The Lie symmetry technique is used to achieve symmetry reduction of the eKP equation and solve the resulting ordinary differential equation we obtain by applying straightforward integration in conjunction with simplest equation technique to derive results for the eKP equation. Furthermore, we derive the conservation laws of the eKP equation using multiplier methods, and then give concluding remarks.
The direction ground can be expressed as follows:
which, when used, reduce the eKP equation to the following form:
We further perform symmetry reductions by utilising the Lie symmetries of Eq. (4). Eq. (4) has three Lie symmetries, namely:
By considering the symmetry (Υ = Υ1 +
which reduces the eKP equation to the following ODE:
Eq. (5) can be written as
Integrating Eq. (6) yields the following expression:
Furthermore, the integral of Eq. (7) is expressed as:
The integral of Eq. (9) yields:
which in its simplest form can be expressed as:
Assuming that the cubic equation
has real roots
We now solve Eq. (5) by employing the simplest equation method . By doing so, we are facilitated to ascertain the exact solutions for Eq. (1). We will use the Bernoulli and Riccati equations as our simplest equations.
Recalling the simplest equation method briefly, let
be the solutions of Eq. (1). In this case,
By considering the Riccati equation
where Θ2 =
The Bernoulli equation is expressed as:
The matching technique produces
We now proceed to replace Φ(
Solving the system of equations, we get
As a result, a solution for Eq. (1) is obtained, as the following:
By solving the resultant system of equations, we acquire:
As a consequence, results for Eq. (1) are obtained in the following form:
The resident conservation law for eKP Eq. (1) can be given as:
being a space-time deviation.
which holds good for all official results of Eq. (1). We achieve a multiplier Λ, which is expressed in terms of the equation:
Corresponding to Eq. (27), we then obtain the resulting conservation laws:
In this research paper, we studied the eKP equation by using the Lie symmetry analysis, which reduced the eKP equation to an ODE; furthermore, we used straightforward integration and the simplest equation method to obtain exact solutions for the eKP equation. Straightforward integration gave us the Jacobi cosine function solution, whereas using the simplest equation method, we obtained the trigonometric hyperbolic solutions. Finally, using the multiplier approach and with the aid of Maple, conservation laws were derived for the eKP equation.
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students’ network ideological and political education based on big data mining technology Research on the application of PLC technology in electrical automation engineering Application of matrix multiplication in signal sensor image perception Analysing the variation of metadiscourse verb patterns in English academic papers from within and between disciplines Impact of COVID-19 policy on stock prices of listed property companies Research on the experimental principle of deep integration of LETS software and criminal procedure under the background of artificial intelligence VR-based computer maintenance practical training platform development design and application research Design and implementation of an integrated management system for discipline construction and postgraduate education in universities Research on innovative human capital for China’s economic development based on STI model In-depth analysis of the artistic expression of paper-cut elements in the design of boat space Knowledge graph construction and Internet of Things optimisation for power grid data knowledge extraction Modeling the pathway of breast cancer in the Middle East Construction and intelligent analysis of power grid physical data knowledge graph based on Internet of Things for power system Research on industrial Internet of Things and power grid technology application based on knowledge graph and data asset relationship model Research on the effects of financial literacy on rural household credit constraint Research on educational resource recommendation system based on MRLG Rec Visual analysis of the trend of mental health changes among Chinese college students in the context of the normalisation of the new crown epidemic A study on the phenomenon of anaphoric correction in college students’ English conversation Limit cycles of a generalised Mathieu differential system Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Application of data mining in basketball statistics The nonlinear effects of ageing on national savings rate – An Empirical Study based on threshold model Design of fitness walker for the elderly based on ergonomic SAPAD model Research on the innovation of collaborative education model of scientific research and teaching in colleges and universities based on the theory of process reengineering AtanK-A New SVM Kernel for Classification The innovative path of diversified vocal music education reform in colleges and universities under the advanced educational concepts of China and the West Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Rhino application in ceramic packaging design teaching Study of a linear-physical-programming-based approach for web service selection under uncertain service quality Analysis and countermeasures of cultivating independent learning ability in colleges teaching English based on OBE theory A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Projection of Early Warning Identification of Hazardous Sources of Gas Explosion Accidents in Coal Mines Based on NTM Deep Learning Network Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China RETRACTION NOTE Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Research on non-linear visual matching model under inherent constraints of images Good congruences on weakly U-abundant semigroups Can policy coordination facilitate unimpeded trade? An empirical study on factors influencing smooth trade along the Belt and Road Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Research on design of customer portrait system for E-commerce Research on rule extraction method based on concept lattice of intuitionistic fuzzy language Fed-UserPro: A user profile construction method based on federated learning Design and evaluation of intelligent teaching system on basic movements in PE Research on identifying psychological health problems of college students by logistic regression model based on data mining Research on evaluation system of cross-border E-commerce platform based on the combined model A mathematical model of PCNN for image fusion with non-sampled contourlet transform Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Retraction