We can represent these equations in the following matrix form: y = H ⋅ a + n (3).
We can represent these equations in the matrix form as defined by Equation (3) and where the matrix channel is given as follows: H = h 1 ( m, j, p + h 3 m, l, p ⋅ e - j 2 πfτ 0 0 h 2 m, j, p + 1 + h 4 m, l, p + 1 ⋅ e - j 2 πfτ ) (8).
Since the left- and right-hand sides of inequalities (13) and (14) represent linear equations in the variable x, we compare their slopes and the fact that (xgeq0) to conclude frac{v_{y} y)}{K}leq Kv_{y}^(0)quad mbox{and}quad frac{v_{y}^(0)}{K}leq Kv_{y} y) for each (xin mathbb {R}) and almost every (yin 0,infty)).
We argue that such model-based-design is an advantage for representing differential equations in SIMULINK.
For (p = 1) and 2, the above equation represents Burgers' equation in cylindrical and spherical coordinates, respectively.
The univariate searches locate n points on the hipersurface that represents each equation in the n-dimensional space.
By substituting Equation 6 into Equation 5 followed by integration by parts, the equation of motion represented in Equation 5 becomes the Duffing equation [39 41] as follows μ ∂ t 2 z t + α z t + λ z t 3 = p 0 cos Ω t (8).
Expanding the EME with respect to the plane waves 〈z|K〉 means representing this equation in terms of the Bloch function r → | j m j k → + K e → z.
We use Table 2 results to build the Mills ratio, and estimate the performance equation, as represented in Equation (4a).
Therefore, a simplified approximation of Equation 6 is represented in Equation 7. d [ ⋅ OH ] dt = K p y r [ pyr ] - k Phe [ Phe ] o [ ⋅ OH ] = 0 (7).
