Based on such a compact exact solution, a new gas-kinetic scheme is constructed for the compressible Navier Stokes equations.
Based on Equation (33), Equation (30) can be obtained.
We therefore utilized the method of modified scaled distances (SDs) based on empirical equations considering such factors as blasting parameters and geological parameters of rock mass for bench blasting in quarries.
This means if there exist bounds M2l, M2h (based on equation (3)) such that M2l ≤ M2 ≤ M2h, then the stability condition (3) would be equivalent to (8) Condition (8) represents an analytical relationship between the feedback strength indicators K1 and K2 of the loops in action, and f1(M2l, K2) and f1(M2h, K2) are the bifurcation points of K1.
Our model is based on three coupled equations: Poisson's equation, continuity equation, and transport equation.
Hirota discretized various soliton equations such as the KdV, the mKdV, and the sine-Gordon equations based on the bilinear equations [16 20], Ablowitz and Ladik proposed a method of integrable discretizations of soliton equations, including the nonlinear Schrödinger equation and the modified KdV (mKdV) equation, based on the Ablowitz-Kaup-Newell-Segur (AKNS) form [1 5].
The resulting expansions can be used to appropriately choose the ansatz in the design of high-frequency numerical solvers, such as those based on integral equations, in order to produce more accurate approximation of the solutions around the shadow and the deep shadow regions than the ones based on the usual ansatz.
Most mathematical models used in the natural sciences and engineering are based on ordinary differential equations, partial differential equations, and integral equations.
Our results are based on regression equations in which other influences, such as demographic variables and income, are held constant.
