This study aims to evaluate a continuous flow model that involves a ramp area at kilometer 31.6 on the highway from Shah Alam to Kuala Lumpur, to analyze the findings of numerical results of instantaneous speed ratios and to observe the convergence patterns for each section. The continuous flow model assumes traffic flow to be similar to the heat equation in regard to the concept of the one-dimensional viscous flow of a compressible fluid. For the methodology, for solving an initial value-boundary value problem, an initial condition together with a set of boundary conditions are required to solve the partial differential equation. The boundary conditions are chosen to assess the suitableness of the design of the entrance ramp in Malaysia, which is for right hand drive traffic. Highway traffic data were collected on the tapered acceleration lane and obtained by the videotaping method. The Maple programming language was used to write a numerical code in order to evaluate the instantaneous speed ratio in terms of a Fourier series. Our results show that the realistic results of instantaneous speed ratios on the ramp at kilometer 31.6 from Shah Alam to Kuala Lumpur are acceptable when compared to the theoretical results. Therefore, a very minimal collision rate is expected due to the well-designed ramp at kilometer 31.6 from Shah Alam to Kuala Lumpur. It is beneficial to study the mathematical model and theories of traffic flows on the merging area to enhance the efficiency of the traffic flowing on highways.

In this paper, we present the distribution of some maximal eigenvalues that are obtained numerically from the discrete Dirichlet Boundary Domain Integro-Differential Equation (BDIDE) operator. We also discuss the convergence of the discrete Dirichlet BDIDE that corresponds with the obtained absolute value of the largest eigenvalues of the discrete BDIDE operator. There are three test domains that are considered in this paper, i.e., a square, a circle, and a parallelogram. In our numerical test, the eigenvalues disperse as the power of the variable coefficient increases. Not only that, we also note that the dispersion of the eigenvalues corresponds with the characteristic size of the test domains. It enables us to predict the convergence of an iterative method. This is an advantage as it enables the use of an iterative method in solving Dirichlet BDIDE as an alternative to the direct methods.