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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ontology and Poisson equation Analysis of digital humanistic knowledge production based on natural language processing Design of language assisted learning model and online learning system under the background of artificial intelligence Study on the influence of adolescent smoking on physical training vital capacity in eastern coastal areas Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering The Construction of Judgment Model for Cultivating Intelligent Accounting Professionals in Universities Based on Logarithmic Periodic Power Law Model Action of Aut( G) on the set of maximal subgroups of p-groups Digital Regeneration and Database Construction of Hunan Embroidery Needlework under the Perspective of Artificial Intelligence Research on the practice of innovation and entrepreneurship education in universities under the environment of big data Application of digital technology in environmental art design Research on loyalty prediction of e-commerce customer based on data mining Research on the evaluation system of tourism management classroom teaching based on multimedia big data analysis The application of ARCS motivation model in college oral English teaching Research on Innovation of Translation Teaching and Translation Strategies for College Students in Multimedia Background A multi-channel convolutional neural network based on attention mechanism fusion for facial expression recognition Exploring the development of cultural and creative tourism under the perspective of ecological theory of tourism destination management Modeling Analysis of User Consumption Data of Internet Culture Industry in the Context of Emerging Technologies Reference: An algorithm for recognizing the main melody of orchestral music based on artificial intelligence of music melody contour Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Evaluation and prediction of regional human capital based on optimised BP neural network Study on inefficient land use determination method for cities and towns from a city examination perspective A study of local smoothness-informed convolutional neural network models for image inpainting The construction of a practical platform for party building and ideological and political education based on information fusion A study on the influence of the spread of Yangming Studies in Japan on the psychology of the Japanese people based on big data analysis Research on personalized recommendation algorithm for micromechanical sensors based on cloud model Research on the value of modern vernacular literature in contemporary rural construction based on ADDIE model Robust speech recognition based on deep learning for sports game review IoT-based interaction design of smart home products for elderly families Model Construction of Influencing Factors of Physical Education Core Quality Cultivation Based on Multiple Linear Regression A multidimensional interpretation analysis of Marxism globalization theory based on distributed Internet Research on the operation and management mechanism of cross-border e-commerce shared practice training base Strategy for financing mode optimization in international trade supply chain based on deep learning model Strategies for improving Chinese language proficiency based on artificial intelligence technology Early warning and detection of geological disasters based on intelligent genetic algorithm The success rate model of college students’ new entrepreneurship based on nonlinear differential equations Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Research on the application of marketing strategy of national fitness exercise and dance events in the construction of sports culture based on big data technology Exploring the Application of Modern Fine Arts Artistic Modeling and Color in Modern Art Design Based on Information Fusion Technology Decisions of competing supply chain with altruistic retailer under risk aversion Strategies for constructing mathematical models of nonlinear systems based on multiple linear regression models Research on China’s tourism marketing model in the data perspective: based on ecological perspective Research on human resource management performance evaluation based on BP algorithm Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRA-BiLSTM-CRF Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Innovative research of vertical video creation under the background of mobile communication Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model Genetic algorithm-based congestion control optimisation for mobile data network Law of interest rate changes in financial markets based on the differential equation model of liquidity Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Badminton players’ trajectory under numerical calculation method Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Innovation of Economic Management Risk Control in Retailer Supply Chain Based on Differential Equation Model Statistical Model of College Ideological and Political Learning Based on Fractional Differential Equations Nonlinear Differential Equation in Anti-aging Test of Polymer Nanomaterials Mechanics of Building Structural Materials Based on Lagrangian Mathematical Model Analysis The Mental Health Education Management of Higher Vocational Students Based on Fractional Differential Equations Application of regression function model based on panel data in financial risk management of bank resource allocation University Ideological and Political Learning Model Based on Statistical Memory Curve Mathematical Equation Research on Resonance Properties of Semantic Wave Fractal Fractals Based on Quantitative Analysis of English Corpus Research on urban landscape big data information processing system based on ordinary differential equations Modeling of fractional differential equation in cloud computing image fusion algorithm Conventional Algorithms in Sports Training Based on Fractional Differential Equations Dynamic Nonlinear System Based on Complex System Theory in the Development of Vocational Education The Algorithm Accuracy of Mathematical Model to Improve the Transmission Speed of E-commerce Platform The Security of Database Network Model Based on Fractional Differential Equations Electric Vehicle Mechanical Transmission System Based on Fractional Differential Equations The Practice System of Physics and Electronics Courses in Higher Vocational Colleges Based on Fractional Differential Equations The Economic Model of Rural Supply and Demand Under the Data Analysis Function Based on Ordered Probit 3D Modeling System of Indoor Environment Art Landscape Design under Statistical Equation Electronic Information Security Model of Nonlinear Differential Equations The Optimization Model of College Students' Physical Exercise Motivation and Self-control Ability Based on the Mathematical Model of Probability Theory Graphic Design Optimization Method Based on Deep Reinforcement Learning Model Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas Criminal law imputation path for biometric information Research on composite dynamic disaster prevention and control system of mine earthquake and shock in thick and hard rock mines Research on urban energy Internet planning based on fuzzy comprehensive method Research on innovative strategies of college students’ English teaching under the background of artificial intelligence Electromagnetic interference prediction technology of new energy motor drive system Research on the intelligent path of college 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 Cyclic Convolutional Neural Network Model Based on Artificial Intelligence 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 Application of the ADDIE Model in the Teaching Practice of Basketball Skills Energy Efficiency Evaluation of HVDC Converter Station Based on Deep Neural Network Model Numerical Simulation and Characteristic Analysis of Music Based on Nonlinear Equations Títul Automatic Extraction and Discrimination of Vocal Theme Based on Quadratic Wave Equation Marginal Scale State Analysis of Cloud Computing Economy Based on a Class of First Order Differential Equations Computer-Aided Design of Hand-Drawn art Food Packaging Design Based on Deep Neural Network Model Innovative Development Model of Ideological and Political Education Under “Self-media” Based on Big Data Thoughts and Design of Ideological and Political Construction Based on the ADDIE Model 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 Virtual Museum Scene Design Based on VRAR Realistic Interaction under PMC Artificial Intelligence Model Influencing Factors of Ideological and Political Construction Work based on VSBPC Model 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 Research on innovative cultivation mode of English and American literature translation talents based on internet environment Coupling Coordination of Long Triangle Digitization and High Quality Development of Manufacturing Industry based on Differential Equation Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China The application of graph theory teaching method in ideological and political education of discrete mathematics Analysis of quantitative management of online intelligent monitoring of tailing ponds based on the perspective of safety prevention and control Analysis of rural tourism culture advertising content based on LSTM-CNN model Optimization of college physical education curriculum based on G-DINA Evaluation of capability maturity of digital transformation of manufacturing industry in Yangtze River Delta based on SDAF model Optimization of local ethnic music teaching transmission path based on logistic regression model Online and Offline Audio-visual-oral English Teaching Model Construction Based on ARCS Exploring the Construction of Curriculum System of Theater and Film Studies Based on ADDIE Model Innovative research on propagation modes of short video works based on digital technology Analysis of the effect of compound training for explosive power on improving the psychological resilience of football athletes Exploration on the path of ideological and political education and cultural education in colleges and universities from the perspective of red culture 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 Analysis of tennis techniques and tactics based on multiple linear regression model Design and evaluation of intelligent teaching system on basic movements in PE Automatic Detection of Transformer Health Based on Bayesian Network Model Application and value of painting technique in art education based on improved ARMA model 3D Virtual Art Design Method Based on ARCS Model Construction of intelligent terminal equipment for Electric Internet of Things based on edge computing model Analysis of bridge foundation pile detection based on convolutional neural network model Application of deep learning model in computer data mining intrusion detection Defect prediction of big data computer network based on deep learning model Analysis of News Communication Law Based on PSR Model Formative Assessment of College English Teaching Based on the CLA Model 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: Wei, Wei and Lu, Liu. “The influence of X fuzzy mathematics method in basketball tactics scoring” Applied Mathematics and Nonlinear Sciences, vol.7, no.1, 2022, pp.677-684 Retraction Retraction: Retraction of: Mundewadi, R. A. and S, Kumbinarasaiah. “Numerical Solution of Abel's Integral Equations using Hermite Wavelet” Applied Mathematics and Nonlinear Sciences, vol.4, no.2, 2019, pp.395-406