Sentence examples for inner solutions from inspiring English sources

Leading and first order asymptotic inner solutions in the temporal domain (early growth) are presented for the case of instantaneous diffusion when the fluid volume surrounding the bubble is large compared to the bubble volume.

In this method we compare the inner expansion of the outer solutions with the outer expansion of the inner solutions to obtain the unknown constants (K_{1} - K_{5}).

According to the boundary conditions (19), we assume the inner solutions near the wall to be f tau)=1+sum_{i=1}^{infty} varepsilon^{i}phi_{i} tau),qquad g tau)=sum _{i=1}^{infty}varepsilon^{i} psi_{i} tau).

The first-order corrections to the usual quasi-steady state (zeroth-order outer) solution, consisting of the zeroth-order inner solutions at both ends of the interval of operation and the first-order outer solutions, are calculated.

Initial conditions for this outer solution are identified through matching with the asymptotic inner solution.

In order to obtain the inner solution in the viscous layer, we introduce the stretching transformation (tau= 1-eta)/varepsilon).

Coated wire electrodes were statistically shown to behave in the same way (selectivity and sensitivity) as inner solution electrodes.

To avoid grafting in the inner core of CNTs, CNT membranes were placed in U-tube fittings under a 2-cm inner solution column pressure.

Then the first two inner solution of g can be expressed as follows: g tau)=-varepsilonfrac{2N_{1}{KA_{{0}}bigl(e^{-A_{0}tau}-1 bigr -varepsilon Omega_{1}bigl(e^{-B_{0}tau}-1bigr).

The boundary conditions corresponding to the inner solution are phi_{i}(0)=0,qquad phi_{i}'(0)=0, qquad psi_{i}(0)=0,quad i=1,2,3,ldots.

The study presents by far the most complete solutions to these problems, with the outer solution up to O ε5) and the inner solution up to O ε2).