Publications

Solving the nonlinear and degenerate p-Laplace equation numerically is highly challenging, yet it has numerous applications across various fields. In this work, we propose and explore numerical schemes, including Picard's Iteration scheme, Crank-Nicolson (CN) method, and Alternating Direction Implicit (ADI) scheme, for solving the p-Laplace equation in ℝ³ for p ∈ [1, 2), aiming to identify the most effective numerical approaches within the framework of finite difference methods. In the existing literature, studies on the parabolic p-Laplace equation primarily consider the Cauchy problem with p ≥ 2. Our study bridges this gap by examining the parabolic p-Laplace equation for p ∈ [1, 2). To our knowledge, this is the first comprehensive study on the three-dimensional (3D) case. Our results demonstrate that the ADI scheme is the most efficient among the proposed 3D numerical schemes as it requires the least computational time while maintaining comparable accuracy in both analytical tests and solvation energy calculations. Moreover, we numerically verify that the solution of the pseudo-time parabolic p-Laplace equation converges to the solution of the corresponding highly nonlinear elliptic p-Laplace equation.

Introduction: Crime and criminal activities have huge influences on society and societal development. The social makeup of the society has a significant impact on the propagation of crime within a population. It is a well-known reality that crime spreads across society like an infectious disease, despite the fact that there are many elements that might affect this dynamic. So, understanding crime and the factors influencing its spread are essential in formulating policies to reduce the prevalence and impacts of crime.

Methods: We formulate a deterministic mathematical model using a system of nonlinear ordinary differential equations incorporating education programs as tools to assess the population-level impact on the spread of crime. The model has a global asymptotically stable crime-free equilibrium whenever a certain criminological threshold, known as the effective reproduction number ℛ_E, is less than unity.

Results and discussion: The model is fitted with prison data reported from July 2021 to June 2022 by the State of Illinois in The United States. The simulations are carried out to assess the population-level impact of the widespread use of the intervention programs and the compliance rate in the State of Illinois. We hypothetically fixed the efficacy of the intervention programs at 25% while varying the compliance rate (by the general public). With no compliance, a high level of active criminal population was recorded. As the compliance rates were significantly improved, the active population level decreased. The global sensitivity analysis is performed primarily to determine the parameters with the most effect on the spread of crime in the State of Illinois. The results demonstrate that the effective community contact rate, β_C, for the criminally active individuals is the main driver of crime in the State of Illinois.