Representations of the solutions of the first-order elliptic and hyperbolic systems via harmonic and wave functions respectively

Improvement of Mathematical Model for Sedimentation Process

“…In addition, numerical simulation results can be obtained using the operational matrix based on the Legendre polynomials proposed by Saadatmandi and Dehghan [35]. However, the resulting curves presented in Figure 1 remain unchanged in terms of their approximation by the obtained analytical solution Equation (27).…”

“…For this purpose, the complex parameters θ 1,2 = −α ± j √ n − α 2 are considered (j-imaginary unit: j 2 = -1). The substitution of these parameters to Equation (25), as well as the use of Newton's binomial theorem [26] and properties of hyperbolic functions [27], allows us to obtain the following expressions:…”

“…To analyze the leading orders of the general Equation ( 27), the dimensionless velocity v p (τ) = v p (τ) v p∞ , the dimensionless time τ = nt, and the dimensionless coefficient of the Basset force ε = α √ n are introduced. Accordingly, Equation (27) takes the following dimensionless form:…”

“…It allows us to obtain the velocity variation (34) proportional to the dimensionless parameter ε. This component allows us to reduce the general Equation (27) to the most simplified variation form (33) with respect to dimensionless parameters ε and τ.…”