Sentence examples for formula of solutions from inspiring English sources
Meanwhile, they introduced the correct formula of solutions for an impulsive Cauchy problem with the Caputo fractional derivative.
For the basis of our study, we construct the fundamental solution and establish variations of constant formula of solutions for the linear systems.
(ii) Impulsive conditions make the formula of solutions to fractional differential equations more complex due to the memory property of the fractional derivative. .
Impulsive conditions make the formula of solutions to fractional differential equations more complex due to the memory property of the fractional derivative.
Recently, Wang et al. [1] presented a counterexample to show an error formula of solutions to the traditional boundary value problem for impulsive differential equations with fractional derivative in [2 5].
We deal with the well-posedness for solutions of nonlinear integrodifferential equations of second-order in Hilbert spaces by converting the problem into the contraction mapping principle with more general conditions on the principal operators and the nonlinear terms and obtain a variation of constant formula of solutions of the given nonlinear equations.
We employ some elementary results of semi-group theory to present the formula of solution, then show the instability cause.
From formula of solution representation (1.3), it is clear that property (1.4) (1.5) is true if all elements of the matrices and are nonnegative.
From formula of solution's representation (1.5) it follows that this property is reduced to sign-constancy of all elements standing only in the r th row of Green's matrix.
From formula of solution's representation (1.3), it follows that this property is reduced to sign-constancy of all elements standing only in the th row of Green's matrix.
From formula of solution representation (1.5) it is clear that property (1.6) ⇒ (1.7) with lx ≡ x 0)−x is true if and only if all elements of the matrices G t, s) and X t) are non-negative.
