Proof By Theorem 4.6, we know that Eq. (1) has a weighted piecewise pseudo almost automorphic mild solution u ( t ), by using integral form of Eq. (1): u ( t ) = T ( t, s 0 ) u ( s 0 ) + ∫ s 0 t T ( t, s ) f ( s, u ( s ) ) ∇ s + ∑ s 0 < t i < t T ( t, t i ) I i ( u ( t i ) ), where t > s 0, s 0 ≠ t i, i ∈ Z.
It uses integral form of wave equation and based on Huygens principle, according to which, the seismic reflector is viewed as if it is composed of closely placed point diffractions.
We are using finite integral form of Gaussian Q-function and the unified MGF-based approach to reach the final expressions.
Proof By Theorem 4.5, we know that Eq. (1.1) has a mild piecewise pseudo almost periodic solution u ( t ), by using the integral form of Eq. (1.1), if t > σ, σ ≠ t i, i ∈ Z, u ( t ) = T ( t − σ ) u + ∫ σ t T ( t − s ) f ( s, u ( s ) ) d s + ∑ σ < t i < t T ( t − t i ) I i ( u ( t i ) ).
Using the integral form of the reverse of inequality from Theorem 2.5 (see [27]) we obtain, for p ∈ ( − 1, 0 ), m ∈ ( − 1, 0 ) and m ≤ p, if f, g : [ a, b ] → R + are two integrable functions on [ a, b ] with g ( x ) > 0, x ∈ [ a, b ] a continuous function on [ a, b ], the inequality M 1 [ f m + 1 g p ] ≤ M 1 m + 1 [ f ] M 1 p [ g ].
In Section 5, the approximate solutions of fuzzy fractional differential equations under Caputo's H-differentiability are obtained using related equivalent integral forms of original FFDE.
For several years now, I have used Inquiry as an integral form of my own practice and am in the process of completing a book on the Transformative power of Inquiry.
Now, we investigate the approximate solutions of fuzzy fractional differential equations under Caputo's H-differentiability by using a corresponding equivalent Volterra integral form of original FFDE.
To calculate the flow field using the dynamic mesh, the integral form of the conservation equation for a general scalar φ on an arbitrary control volume V with moving boundary is employed: (8).
CST Studio Suite 2012 is a general-purpose electromagnetic simulator (used in many electromagnetic problems) based on the integral form of Maxwell's equations.
