Tjang Daniel Chandra, Farah Anandya Ramadhani
The linear Boltzmann Equation is a type of Kinetic Equation, derived from a modification of the Boltzmann Equation (which is non-linear). The equation can model particle behavior over time, where collisions and external forces on a system (generally on a gas system) become linear. The linear Boltzmann equation is relatively difficult to find its solution, including analytical solutions, which is caused by the complexity of its differential integral. The equation requires a simplification process, so that it can be easier to analyze and solve it, which can use the Fourier transform as an alternative solution method. This is also supported by the linear nature of the equation, which is one of the requirements of an equation that can be solved by the Fourier transform. The purpose of this research is to find the analytical solution of the linear Boltzmann Equation in one, two, and three dimensions, using the Fourier transform and to find out the characteristics (graph) of the solution. This research uses the literature study method, so that the entire process in finding analytical solutions using Fourier transform refers to previous research. After transforming to convolution, the analytic solution of the linear Boltzmann Equation in one, two, and three dimensions is obtained, and it can be seen that the analytic solutions of the three dimensions have similar patterns and shapes. However, it has differences in the space domain (x) and velocity (v), which also affects the frequency domain (ω). The use of the Fourier transform approach in obtaining a three-dimensional analytical solution of the linear Boltzmann Equation can also be implemented in solving various dimensions. This study presents an analytical and comprehensive framework, which can improve understanding the characteristics (graphs) of the solution. This is different from previous studies that discuss numerical methods on other equations as well. Furthermore, from the analytical solution obtained, it can be represented in the form of a solution graph with the help of Matlab R2013b software and by giving some assumptions on the variables Q, t, v1, v2, v3, ω1, ω2, ω3, x1, x2, x3, y, and the function h(y). The shape of the resulting solution graph depends on the assumptions of the variables and functions. The solution graph in this study shows that the particles around the central region are denser and closer together when t and v are constant. Meanwhile, when v is constant as time t progresses, it applies that the smaller the value of t, the more centered the distribution of the function, and the opposite applies. Furthermore, the validity test is carried out by substituting the initial value f(0, v, x) = h(x) in the analytical solution before convolution, followed by substituting the analytical solution after convolution in the boundary condition lim f(t, v, x) = 0, so that the analytical solution obtained previously is proven x→±∞ correct and satisfies the initial value and boundary condition. In addition to being a medium for simplifying the solution of complex linear Boltzmann Equations, the implementation of Fourier transforms and analytical approaches can be factors that allow graphical representations of solutions to be found. Through this, the characteristics (graphs) and also the behavior of particle distribution over time t can be known. Thus, the results of this study can be used as a reference in analyzing more complex kinetic theories, as well as systems involving external forces and boundary conditions in further research. © 2025 American Institute of Physics Inc.. All rights reserved.
Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Negeri Malang, Malang, Indonesia
