This paper applies a powerful scheme, namely Bernoulli sub-equation function method, to some partial differential equations with high non-linearity. Many new travelling wave solutions, such as mixed dark-bright soliton, exponential and complex domain, are reported. Under a suitable choice of the values of parameters, wave behaviours of the results obtained in the paper – in terms of 2D, 3D and contour surfaces – are observed.

#### Keywords

- Integro-partial differential equation
- Fifth-Order integrable model
- Analytical method
- Rational function solution
- Complex Solution
- Contour surface
- Travelling wave solutions
- Mixed dark-bright soliton

Mathematical models have been used to explain many real-world problems, in the past decade. In this sense, Qi et al. [1] have investigated some important models describing certain waves in physics. Colucci et al. [2] have introduced a new partial differential equation to define the ice crystal size delivery. Another novel model considered to explain the nucleation of spherical agglomerates using the immersion mechanism has been developed by Tash et al. [3]. Baleanu et al. [4] have presented a new study about people's liver using Caputo–Fabrizio fractional model. Pignotti et al. [5] have given another novel differential model related to the project of extraction in mines. Pompa et al. [6] proposed some important models about gastrointestinal absorption for availability of drugs to biological mechanisms. Compression of main electrocardiography signals using a new genetic programming-based mathematical modelling algorithm has been studied by Feli and Abdali-Mohammadi [7]. With the aim of assessing the Bang-Bang model related to hysteresis influences on heat and mass transmit in spongy building material, another important article has been proposed by Berger et al. [8]. Tsur et al. [9] studied the reaction of melanoma patients to the immune checkpoint surrounding (including understandings collected) in an assessment of a new mathematical mechanistic sample. Camaraza-Medina et al. [10] presented a new study on the mathematical inference of computation of heat transmission by thickenings inside tubes. Another powerful model involving chemical reaction systems has been proposed by Amin et al. [11]. Kortcheva et al. [12] explored new ways and differential equations related to peripheral risk administration in harbours. Aiming to get data using rubrics, Sahin and Baki have developed a new model for measuring mathematical success [13]. Meena et al. [14] composed a new mathematical model about the influencing agent in biofilms under toxic situations to discuss the values of parameters. Hamzehlou et al. have explored a unique way to model and predict the active progress of particle morphology mathematically [15]. There are many other such studies [16,17,18,19,20, 29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44].

The remainder of this current paper is constructed in the following parts. In Section 2, we introduce the Bernoulli sub-equation function method (BSEFM) in detail. In Section 3, as a first application, we apply BSEFM to the (1+1)-dimensional integro-differential Ito equation (ITOE) defined as follows [21]:

Gepreel et al. [21] have applied the modified simple equation method to Eq. (1) for getting some important properties. Wazwaz has investigated the physical meaning of Eq. (1) [22]. Further, Eq. (1) has been investigated by using meshless discrete collocation method, numerically in another paper [23].

As a second application, we consider the (2+1)-dimensional fifth-order integrable equation (FOIE) given as follows [24]:

In this sub-section of the paper, the scheme considered herein is introduced [25,26,27].

According to the balance principle, we can obtain a relationship between

Solving this system, the values of _{0},_{1},_{2},_{n}

When we use a complete discrimination system for polynomial, we get the solutions to Eq. (5) through computational programs and classify certain solutions to Eq. (5). For a better understanding of the results obtained in this manner, we can draw two- and three-dimensional surfaces of solutions by taking into consideration appropriate values of parameters.

This section applies BSEFM to the governing models, such as ITOE and FOIE models, to find new travelling wave solutions.

If we take _{x}

If we consider the travelling wave transformation as

When ^{′} =

Balancing,

From Eq. (15), we can get many entirely new travelling wave solutions to Eq. (1).

_{4} ≠ 0,

Putting these into Eq. (16) by considering Eq. (10), we get the following new exponential function solution for Eq. (1):

Here,

Putting these variables into Eq. (16) by taking into account Eq. (10), we get the following new exponential function solution for Eq. (1):

^{2/3}^{1/3}^{−2/3}, ^{2/3}^{1/3}^{−2/3}, and

This sub-section applies BSEFM to the FOIE model for finding some new travelling wave solutions. First of all, considering the travelling wave transformation as

For simplicity, if we take ^{′}, then, we can rewrite Eq. (32) as follows:

With the help of the balance principle, we obtain the following:

This gives many new travelling wave solutions to Eq. (2).

_{4} ≠ 0,

^{2} −^{2} and ^{2} − ^{2}>0; _{2},

Putting these variables into Eq. (35) by taking into account Eq. (10), we get the following exponential function solution for Eq. (2):
^{2} −^{2} and ^{2} −^{2}>0; _{2},

^{1/4}^{−1/2}^{−1/2}, ^{2} + ^{2} and ^{2} − ^{2}>0; here,

^{2} − ^{2}>0.

Here,

Putting these variables into Eq. (35) by considering Eq. (10), we get the following exponential function solution for Eq. (2):
^{2} + ^{2})^{1/4}^{−1/2}^{−1/2}, ^{2} + ^{2} and ^{2} − ^{2}>0 are the strain conditions.

Here, _{2},

^{2} + ^{2})^{1/4}^{−1/2}^{−1/2}, ^{2} + ^{2} and ^{2} − ^{2}>0 are the strain conditions. Here, _{2},

^{2} + ^{2}, ɷ = 1152^{3}^{2dτ}^{3/4}^{2}_{2},
^{2} − ^{2}>0 are the strain conditions. Here, _{2},_{2},

In this paper, we have successfully applied BSEFM to some powerful non-linear models, such as the integro-partial differential equation and fifth-order integrable model. We have reported some strain conditions for the validity of the obtained results. Moreover, the travelling wave solutions, such as Eqs. (16, 18, 20, 29), obtained by using BSEFM are the new exponential function solutions for Eq. (1), compared with the paper previously found in literature [28]. Using powerful computational package programs, we observe that all solutions verify Eqs. (1, 2). Under specific values of parameters, we revise our results into existing solutions. Moreover, we have found many other entirely new analytical and complex travelling wave solutions for governing models. As far as we know, BSEFM has not been applied to Eq. (1) earlier. The projected method in this paper may be used to seek more travelling wave solutions of non-linear evolution equations for some applications, such as easy calculations, writing programs for obtaining variables, and so on.

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 Industrial transfer and regional economy coordination based on multiple regression model Response model for the psychological education of college students based on non-linear finite element equations Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Spatial–temporal graph neural network based on node attention A contrastive study on the production of double vowels in Mandarin Financial accounting measurement model based on numerical analysis of rigid normal differential equation and rigid generalised functional equation Research of cascade averaging control in hydraulic equilibrium regulation of heating pipe network Mathematical analysis of civil litigation and empirical research of corporate governance Health monitoring of Bridges based on multifractal theory College students’ innovation and entrepreneurship ability based on nonlinear model Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Mathematical simulation analysis of optimal testing of shot puter's throwing path Performance evaluation of college laboratories based on fusion of decision tree and BP neural network Application and risk assessment of the energy performance contracting model in energy conservation of public buildings The term structure of economic management rate under the parameter analysis of the estimation model based on common differential equation Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas The Spatial Form of Digital Nonlinear Landscape Architecture Design Based on Computer Big Data Analysis of the relationship between industrial agglomeration and regional economic growth based on the multi-objective optimisation model The improvement of museum information flow based on paste functional mapping method Constraint effect of enterprise productivity based on constrained form variational computing The art design of industrialised manufacturing furniture products based on the simulation of mathematical curves The impact of urban expansion in Beijing and Metropolitan Area urban heat Island from 1999 to 2019 TOPSIS missile target selection method supported by the posterior probability of target recognition Research on Evaluation of Intercultural Competence of Civil Aviation College Students Based on Language Operator Ultrasonic wave promoting ice melt in ice storage tank based on polynomial fitting calculation model The incentive contract of subject librarians in university library under the non-linear task importance Modelling and Simulation of Collaborative Innovation System in Colleges and Universities Based on Interpreted Structural Equation Model Small amplitude periodic solution of Hopf Bifurcation Theorem for fractional differential equations of balance point in group competitive martial arts The Optimal Solution of Feature Decomposition Based on the Mathematical Model of Nonlinear Landscape Garden Features Composite mechanical performance of prefabricated concrete based on hysteresis curve equation Higher education innovation and reform model based on hierarchical probit Application of Fuzzy Mathematics Calculation in Quantitative Evaluation of Students’ Performance of Basketball Jump Shot The teaching of sports science of track and field-based on nonlinear mathematical equations Visual error correction of continuous aerobics action images based on graph difference function Ecological balance model of effective utilization of agricultural water resources based on fractional differential equations Application of Higher Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Precision algorithms in second-order fractional differential equations Application of Forced Modulation Function Mathematical Model in the Characteristic Research of Reflective Intensity Fibre Sensors Fractional differential equations in National Sports Training in Colleges and Universities Radioactive source search problem and optimisation model based on meta-heuristic algorithm Visualized calculation of regional power grid power data based on multiple linear regression equation Application of mathematical probabilistic statistical model of base – FFCA financial data processing Least-squares method and deep learning in the identification and analysis of name-plates of power equipment Research on a method of completeness index based on complex model Distribution network monitoring and management system based on intelligent recognition and judgement Research on the sustainable development and renewal of Macao inner harbour under the background of digitisation Support design of main retracement passage in fully mechanised coal mining face based on numerical simulation Study on the crushing mechanism and parameters of the two-flow crusher Topological optimisation technology of gravity dam section structure based on ANSYS partial differential equation operation Modeling the pathway of breast cancer in the Middle East Corporate social responsibility fulfilment, product-market competition and debt risk: Evidence from China ARMA analysis of the green innovation technology of core enterprises under the ecosystem – Time series data Reconstruction of multimodal aesthetic critical discourse analysis framework Image design and interaction technology based on Fourier inverse transform What does students’ experience of e-portfolios suggest Research on China interregional industrial transformation slowdown and influencing factors of industrial transformation based on numerical simulation The optimal model of employment and entrepreneurship models in colleges and universities based on probability theory and statistics A generative design method of building layout generated by path Analysis of the causes of the influence of the industrial economy on the social economy based on multiple linear regression equation Research of neural network for weld penetration control Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations Analysing the action techniques of basketball players’ shooting training using calculus method Engineering project management based on multiple regression equation and building information modelling technology Research on predictive control of students’ performance in PE classes based on the mathematical model of multiple linear regression equation Beam control method for multi-array antennas based on improved genetic algorithm The influence of X fuzzy mathematical method on basketball tactics scoring Mathematical model of back propagation for stock price forecasting Application of regression function model based on panel data in bank resource allocation financial risk management Application of Logical Regression Function Model in Credit Business of Commercial Banks Research on aerobics training posture motion capture based on mathematical similarity matching statistical analysis Application of Sobolev-Volterra projection and finite element numerical analysis of integral differential equations in modern art design Research on motion capture of dance training pose based on statistical analysis of mathematical similarity matching Application of B-theory for numerical method of functional differential equations in the analysis of fair value in financial accounting Badminton players’ trajectory under numerical calculation method Research on the influence of fuzzy mathematics simulation model in the development of Wushu market Study on audio-visual family restoration of children with mental disorders based on the mathematical model of fuzzy comprehensive evaluation of differential equation Application of multi-attribute decision-making methods based on normal random variables in supply chain risk management The impact of financial repression on manufacturing upgrade based on fractional Fourier transform and probability AtanK-A New SVM Kernel for Classification Visual positioning system for marine industrial robot assembly based on complex variable function Application of Lane-Emden differential equation numerical method in fair value analysis of financial accounting Regression function model in risk management of bank resource allocation Application of numerical method of functional differential equations in fair value of financial accounting Optimal solution of fractional differential equations in solving the relief of college students’ mental obstacles Risk contagion in financial markets based on copula model Calculating university education model based on finite element fractional differential equations and macro-control analysis Automatic parameter selection ZVD shaping algorithm for crane vibration suppression based on particle swarm optimisation Educational research on mathematics differential equation to simulate the model of children's mental health prevention and control system Analysis of enterprise management technology and innovation based on multilinear regression model Verifying the validity of the whole person model of mental health education activities in colleges based on differential equation RETRACTION NOTE Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Calculation of tourism development income index based on finite element ordinary differential mathematical equation Adoption of deep learning Markov model combined with copula function in portfolio risk measurement Radar system simulation and non-Gaussian mathematical model under virtual reality technology Comparison of compression estimations under the penalty functions of different violent crimes on campus through deep learning and linear spatial autoregressive models Research and application of constructing football training linear programming based on multiple linear regression equation Research on management evaluation of enterprise sales cash flow percentage method based on the application of quadratic linear regression equations Mathematical simulation analysis of optimal detection of shot-putters’ best path Determination of the minimum distance between vibration source and fibre under existing optical vibration signals: a study Mathematical modelling of enterprise financial risk assessment based on risk conduction model Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets Nonlinear Differential Equations in the Teaching Model of Educational Informatisation Mathematical simulation experiment based on optimisation of heat treatment process of aluminium alloy materials The evaluation of college students’ innovation and entrepreneurship ability based on nonlinear model Mathematical model of transforming image elements to structured data based on BP neural network Smart Communities to Reduce Earthquake Damage: A Case Study in Xinheyuan, China Institutional investor company social responsibility report and company performance Mathematical analysis of China's birth rate and research on the urgency of deepening the reform of art education First-principles calculations of magnetic and mechanical properties of Fe-based nanocrystalline alloy Fe _{80}Si_{10}Nb_{6}B_{2}Cu_{2}Educational reform informatisation based on fractional differential equation Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption 3D Mathematical Modelling Technology in Visual Rehearsal System of Sports Dance MCM of Student’s Physical Health Based on Mathematical Cone Attitude control for the rigid spacecraft with the improved extended state observer Sports health quantification method and system implementation based on multiple thermal physiology simulation Research on visual optimization design of machine–machine interface for mechanical industrial equipment based on nonlinear partial equations Informationisation of teaching model for track and field education based on finite element higher-order fractional differential equation Information technology of preschool education reform of fine arts based on fractional differential equation Information Teaching Model of Preschool Art Education in Colleges and Universities Based on Finite Element Higher-Order Fractional Differential Equation Application of artificial intelligence algorithm in mathematical modelling and solving College Students’ Mental Health Climbing Consumption Model Based on Nonlinear Differential Equations Cognitive Computational Model Using Machine Learning Algorithm in Artificial Intelligence Environment Research on tourism income index based on ordinary differential mathematical equation Application of Higher-Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Sports Science Teaching of Athletics Based on Nonlinear Mathematical Equation Informatisation of educational reform based on fractional differential equations Research on the control of quantitative economic management variables under the numerical method based on stochastic ordinary differential equations Network monitoring and processing accuracy of big data acquisition based on mathematical model of fractional differential equation System dynamics model of output of ball mill Communication architecture of power monitoring system based on incidence matrix model Differential equation to verify the validity of the model of the whole-person mental health education activity in Universities Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Analysis and Prediction of College Students’ Mental Health Based on K-means Clustering Algorithm Mathematical function data model analysis and synthesis system based on short-term human movement Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Human gait modelling and tracking based on motion functionalisation Analysis and synthesis of function data of human movement Energy-saving technology of BIM green buildings using fractional differential equation Study on the training model of football movement trajectory drop point based on fractional differential equation Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation User online consumption behaviour based on fractional differential equation Design of Morlet wavelet neural network to solve the non-linear influenza disease system Differential equation model of financial market stability based on Internet big data Multi-attribute Decision Method Based on Normal Random Variable in Economic Management Risk Control Children’s cognitive function and mental health based on finite element nonlinear mathematical model Motion about equilibrium points in the Jupiter-Europa system with oblateness Badminton players’ trajectory under numerical calculation method Optimal preview repetitive control for impulse-free continuous-time descriptor systems Development of main functional modules for MVB and its application in rail transit Value Creation of Real Estate Company Spin-off Property Service Company Listing Selection by differential mortality rates Digital model creation and image meticulous processing based on variational partial differential equation Dichotomy model based on the finite element differential equation in the educational informatisation teaching reform model Nonlinear Dissipative System Mathematical Equations in the Multi-regression Model of Information-based Teaching The modelling and implementation of the virtual 3D animation scene based on the geometric centre-of-mass algorithm The policy efficiency evaluation of the Beijing–Tianjin–Hebei regional government guidance fund based on the entropy method Stock price analysis based on the research of multiple linear regression macroeconomic variables Fractional Linear Regression Equation in Agricultural Disaster Assessment Model Based on Geographic Information System Analysis Technology The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations A review of the treatment techniques of VOC Some classes of complete permutation polynomials in the form of ( x ^{pm}−x +δ )^{s}+ax ^{pm}+bx overF _{p2m}Deformation and stress theory of surrounding rock of shallow circular tunnel based on complex variable function method A mathematical model of the fractional differential method for structural design dynamics simulation of lower limb force movement step structure based on Sanda movement Numerical calculation and study of differential equations of muscle movement velocity based on martial articulation body ligament tension Study on the maximum value of flight distance based on the fractional differential equation for calculating the best path of shot put Sports intensity and energy consumption based on fractional linear regression equation Translog function in government development of low-carbon economy Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Research on the Psychological Distribution Delay of Artificial Neural Network Based on the Analysis of Differential Equation by Inequality Expansion and Contraction Method Study on Establishment and Improvement Strategy of Aviation Equipment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode The Model of Sugar Metabolism and Exercise Energy Expenditure Based on Fractional Linear Regression Equation Constructing Artistic Surface Modeling Design Based on Nonlinear Over-limit Interpolation Equation Numerical Simulation Analysis Mathematics of Fluid Mechanics for Semiconductor Circuit Breaker Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education Human Body Movement Coupling Model in Physical Education Class in the Educational Mathematical Equation of Reasonable Exercise Course The contribution of structural equation model analysis to higher education agglomeration and innovation and entrepreneurship Study on the evolutionary game theory of the psychological choice for online purchase of fresh produce under replicator dynamics formula The influence of X fuzzy mathematics method in basketball tactics scoring Mathematical statistics algorithm in the bending performance test of corroded reinforced concrete beams under fatigue load Nonlinear strategic human resource management based on organisational mathematical model Back propagation mathematical model for stock price prediction Evolutionary game research on the psychological choice of online shopping of fresh agricultural products based on dynamic simulation model Differential equation model of financial market stability based on big data Multi-attribute decision-making methods based on normal random variables in supply chain risk management Linear fractional differential equations in bank resource allocation and financial risk management model Construction and reform of art design teaching mode under the background of the integration of non-linear equations and the internet