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This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm.

The authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations.

Approaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approach’s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software.

The convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement.

There is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively.

This work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution.

To the best of the authors’ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.

The purpose of this study is to explore the coopetition relationships between platform owners and complementors in complementary product markets. Drawing on the coopetition theory, the authors examined the evolutionary trends of the coopetition relationships between platform owners and complementors and explore the main influence factors.

The authors used Lotka–Volterra model to analyze the coopetition relationship between platform owners and complementors, including the evolutionary trends as well as the results. Considering the feasibility of sample data collection, simulation is used to verify the effects of different factors on the evolution of coopetition relationships.

The results show that there are four possible results of the competition in the complementary products market. That comprises “winner-take-all for platform owners,” “winner-take-all for complementors,” “stable competitive coexistence” and “unstable competitive coexistence,” where “stable competitive coexistence” is the optimal evolutionary state. Moreover, the results of competitive evolution are determined by innovation subjects’ interaction parameters. However, the natural growth rate, the initial market benefits of the two innovators and the overall benefits of the complementary product markets influence the time to reach a steady state.

The study provides new insights into the entry of platform owners into complementary markets, and the findings highlight the fact that in complementary product markets, platform owners and complementors should seek “competitive coexistence” rather than “winner-takes-all.” Moreover, the authors also enrich the coopetition theory by revealing the core factors that influence the evolution of coopetition relationships, which further enhance the analysis of the evolutionary process of coopetition relationships.

Online question-and-answer (Q&A) communities serve as important channels for knowledge diffusion. The purpose of this study is to investigate the dynamic development process of online knowledge systems and explore the final or progressive state of system development. By measuring the nonlinear characteristics of knowledge systems from the perspective of complexity science, the authors aim to enrich the perspective and method of the research on the dynamics of knowledge systems, and to deeply understand the behavior rules of knowledge systems.

The authors collected data from the programming-related Q&A site Stack Overflow for a ten-year period (2008–2017) and included 48,373 tags in the analyses. The number of tags is taken as the time series, the correlation dimension and the maximum Lyapunov index are used to examine the chaos of the system and the Volterra series multistep forecast method is used to predict the system state.

There are strange attractors in the system, the whole system is complex but bounded and its evolution is bound to approach a relatively stable range. Empirical analyses indicate that chaos exists in the process of knowledge sharing in this social labeling system, and the period of change over time is about one week.

This study contributes to revealing the evolutionary cycle of knowledge stock in online knowledge systems and further indicates how this dynamic evolution can help in the setting of platform mechanics and resource inputs.