We suppose that (u^{0}_{1}(ageq00), (ageq0) is known, we take the solution of (2.7) as history function for (2.8) and we obtain nonnegative solutions if (u^{0}_{1}(a)) is not known.
We take the solution of reduced equations (B) of case (5) of Table 2 in the form F ( xi ) =sum_{i=1}^{m}a_{i} varphi^{i}, qquad H ( xi ) =sum_{i=1}^{m}b_{i} varphi^{i}, (31) where (varphi =varphi ( xi ) ) is a solution of the Riccati equation varphi '=q+varphi^{2}.
We take the solution of the u0 and u1 equations to have the form (41) u 0 = A ^ cos σ, (42) u 1 = m 1 sin 2 σ + m 2 sin 2 σ + m 3, where A = A ^ ϵ from (29) and (31).
Therefore, applying Theorem 2.4 part (d) as in the proof of Theorem 2.1, and taking the solutions of (2.15) as values for parameters, we obtain that there exist positive profiles f i (i = 1, 2,..., k) solving (2.16) and satisfying (2.14).
(II) If α i < 0 and α i + β i > 0 (i = 1, 2,..., k), we can apply Theorem 2.4 part (c) as in the proof of Theorem 2.1 and taking the solutions of (2.15) as the parameters, we obtain that there exist compactly supports profiles f i (i = 1, 2,..., k) solving (2.16) and satisfying the boundary conditions (2.14). .
The idea is to run a distributed optimization routine, taking the solution of (14) as the starting point, to solve for (6).
Proof Firstly, we take the solution ω ( t ) of Eq. (3) and make an appropriate substitution for the variable x.
According to the discrepancy principle, we will take the solution (xi _{max}) of the equation rho xi_{max})=taudelta (3.5) to be the regularization parameter, where (tau>1) is a constant.
Take the solution and allow a couple of drops on each side of the swab.
This method mainly consists of four steps: Step1: Take the complex solutions of (1) in the form u x, t = u ξ, ξ = x - vt, Open image in new window (2).
