Sentence examples for decaying solutions from inspiring English sources

Thus, for (36) slowly decaying solutions and strongly decaying solutions coexist with oscillatory solutions.

Hence, proper solutions of (3) satisfying (6) are called slowly decaying solutions, and proper solutions satisfying (7) strongly decaying solutions.

Moreover, the same is true for slowly decaying solutions in the sublinear case (alpha>beta); see, e.g., [2, 8].

Does the coexistence between oscillatory solutions and decaying solutions, illustrated in Example 3, occur also when (alphaneq1) (and (beta>alpha))?

Namely, by a linearised analysis we first check that equation (1.4) provides us with a sufficient 'amount' of exponentially decaying solutions at infinity.

According to our analysis above, to get exponentially decaying solutions, the real eigenvalues in (2.8) must satisfy lambda_{beta}>0 quad mbox{for any }beta.

Furthermore, it is easy to verify that the function x t)=frac{sqrt{2}}{3}t^{-2/3} is a slowly decaying solution of (36).

Example 3 suggests that for the existence of at least one slowly decaying solution, the assumption on monotonicity in (17) could be relaxed.

These efficiency gains are obtained, in part through the use of a quadrature-based refinement technique, by integrating Fourier modes exactly and by neglecting the contributions of rapidly decaying solution transients.

Nakao [8] obtained the existence and uniqueness of a global decaying solution for the quasilinear wave equation with Kelvin Voigt dissipation and a derivative nonlinearity.

By an inverse transformation of coordinates, we also obtain the existence of a unique time-decay solution to the original initial-boundary problem with proper initial data.