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Word series high-order averaging of highly oscillatory differential equations with delay

J. M. Sanz-Serna
J. M. Sanz-Serna
 and 
Beibei Zhu
Beibei Zhu
   | Dec 20, 2019
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Published Online: Dec 20, 2019
Page range: 445 - 454
Received: Jun 17, 2019
Accepted: Jul 22, 2019
DOI: https://doi.org/10.2478/AMNS.2019.2.00042
Keywords
© 2019 J. M. Sanz-Serna and Beibei Zhu, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 Public License.

Fig. 1

The solution x of the delay oscillatory problem (8)–(9) (top subplot) is represented by an array (x(0), …, x(L)) of functions defined in the interval 0 ≤ t ≤ τ; the four bottom subplots depict these in the case L = 3. Averaging the ordinary differential system without delay satisfied by the (x(0), …, x(L)) leads to a differential system without delay for the averaged solution (X(0), …, X(L)). The functions X(ℓ) (dotted lines) are patched together to get the solution X of the averaged delay problem that we wish to find.
The solution x of the delay oscillatory problem (8)–(9) (top subplot) is represented by an array (x(0), …, x(L)) of functions defined in the interval 0 ≤ t ≤ τ; the four bottom subplots depict these in the case L = 3. Averaging the ordinary differential system without delay satisfied by the (x(0), …, x(L)) leads to a differential system without delay for the averaged solution (X(0), …, X(L)). The functions X(ℓ) (dotted lines) are patched together to get the solution X of the averaged delay problem that we wish to find.

Fig. 2

The left (respectively right) panels correspond to the u (respectively v) component of the solution. The top panels plot the true oscillatory solution as a function of t. Due to the fast oscillations of large amplitude, the graph for u appears as a solid band. The oscillations in v have a smaller amplitude; the differential equation for v does no include any periodic forcing and therefore the fast oscillations in v are only due to its coupling to u. The bottom panels give the second order averaged solution (discontinuous line) and third order averaged solution (solid line). In the bottom panels, the true solution (circles) is represented only at stroboscopic times. Clearly the third order system reproduces accurately the behaviour of the true solution (the circles are on the solid line). That is not the case if averaging is only carried out to second order.
The left (respectively right) panels correspond to the u (respectively v) component of the solution. The top panels plot the true oscillatory solution as a function of t. Due to the fast oscillations of large amplitude, the graph for u appears as a solid band. The oscillations in v have a smaller amplitude; the differential equation for v does no include any periodic forcing and therefore the fast oscillations in v are only due to its coupling to u. The bottom panels give the second order averaged solution (discontinuous line) and third order averaged solution (solid line). In the bottom panels, the true solution (circles) is represented only at stroboscopic times. Clearly the third order system reproduces accurately the behaviour of the true solution (the circles are on the solid line). That is not the case if averaging is only carried out to second order.

Maximum errors in u, with respect to the true solution, for the second-order and the third-order averaged solutions in the interval 0 ≤ t ≤ 2. The notation 3.31(–4) means 3.31 × 10–4, etc.

Ω = 16π Ω = 32π Ω = 64π Ω = 128π Ω = 256π Ω = 512π
AS2 3.31(-4) 8.83(-5) 2.28(-5) 5.77(-6) 1.46(-6) 3.66(-7)
AS3 1.04(-4) 1.28(-5) 1.59(-6) 1.96(-7) 2.07(-8) 2.30(-9)
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