ANN-MoC Approach for Solving First-Order Partial Differential Equations

Abstract

Linear first-order partial differential equations appears in the modeling of many important physical phenomena, such as heat radiative transfer [1] and neutron transport [2]. They have applications in high temperature manufacturing (e.g., class and ceramic manufactures), optical medicine, nuclear energy generation, and many others. In this work, we deal with equations of the following form Ω · ∇u + σt u = f in D, (1) where u = u(xx) ∈ R, x = (x, y) ∈ D = [a, b] × [c, d], σt > 0 and a given direction Ω = (μ, η) in the unitary disc centered at the origin. Incoming boundary condition is assumed as u = uin on Γ− , (2) where uin = uin (xx) is given on Γ− = {x x ∈ ∂D : Ω · n < 0}, with n denoting the outward-pointing normal vector on the boundary. [...]