Journal of Applied & Computational Mathematics

Integrating high-dimensional data is a crucial challenge in modern computational science. As we generate and collect vast amounts of data from diverse sources, the complexity of this task increases exponentially. High-dimensional data sets are characterized by a large number of variables, which often surpass the number of observations. This disparity creates difficulties in data analysis, as traditional statistical methods tend to falter under such conditions. To address these challenges, adaptive algorithms have emerged as powerful tools, offering a computational approach to effectively integrate and analyze high-dimensional data.

Adaptive algorithms are designed to adjust their parameters and structures based on the characteristics of the data they process. This flexibility makes them particularly well-suited for handling high-dimensional data, where the relationships between variables are often complex and not easily discernible. These algorithms are capable of learning and evolving as they interact with the data, allowing for more accurate modeling and integration of high-dimensional datasets.

In our modern era, traffic congestion in big cities has become threat for our daily life. It is one of the greatest problems in many developed countries of the world like Bangladesh. Various mathematical models have been employed to address the issue of traffic congestion. Traffic flow models assume that density and velocity are related. Bruce Greenshields first introduced traffic density-velocity connection. In this paper, we would like to study with linear velocity density relationship (Greenshields Model) and exponential velocity density relationship (Underwood Model) using the model for the flow of traffic based on diffusion. To make a comparative study between Greenshields and Underwood Models we solve the traffic flow model of the diffusion type as analytically as well as numerically. Due to analytical solution complexity, we use the finite difference method to solve the model numerically. We use explicit upwind, explicit centered, and explicit Lax-Wendrooff schemes to solve the model numerically. For each of the schemes, we present a comparison between linear and exponential velocity-density relationships. From this comparison, we can say that the exponential velocity- density relationship is appropriate to overcome the traffic congestion problem for each of the schemes.