Numerical solution to an inverse problem on a determination of places and capacities of sources in the hyperbolic systems

Yegana Ashrafova; Kamil Aida-Zade

doi:10.3934/jimo.2019091

Article Contents

2020, Volume 16, Issue 6: 3011-3033. Doi: 10.3934/jimo.2019091

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Numerical solution to an inverse problem on a determination of places and capacities of sources in the hyperbolic systems

Yegana Ashrafova^1,2, , and
Kamil Aida-Zade^2,

1.
Baku State University, Z. Khalilov, 23, Baku, Azerbaijan
2.
Institute of Control Systems of Azerbaijan, National Academy of Sciences, B. Vahabzade, 9, AZ1141 Baku, Azerbaijan

^* Corresponding author
^* Corresponding author

Received: August 31, 2018

Revised: March 31, 2019

Early access: July 2019

Published: October 2020

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Abstract

In the work we investigate the numerical solution to a class of inverse problems with respect to the system of differential equations of hyperbolic type. The specialties of considered problems are: 1) the impulse impacts are present in the system and it is necessary to determine the capacities and the place of their location; 2) the differential equations of the system are only related to boundary values, and arbitrarily; 3) because of the long duration of the object functioning, the exact values of the initial conditions are not known, but a set of possible values is given. The inverse problem under consideration is reduced to the problem of parametric optimal control without initial conditions with non-separated boundary conditions. For the solution it is proposed to use first-order optimization methods. The results of numerical experiments are given on the example of the inverse problem of fluid transportation in the pipeline networks of complex structure. The problem is to determine the locations and the volume of leakage of raw materials based on the results of additional observations of the state of the transportation process at internal points or at the ends of sections of the pipeline network.

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Mathematics Subject Classification: Primary: 35Q35, 65M32; Secondary: 65M55, 49K20.

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Figure 1. The scheme of pipeline network in the problem 1

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Figure 2. Plots of the exact ($ q_{loss} = q^{(1,2)^\ast }(t) = 50 - 10e^{- 0.0003 t} {m^3} /{hour} $ and obtained leak volume functions $ q^{(1,2)^\ast }(t) $ for five various initial approximations $ (\xi ^{(1,2)},\,\,q^{(1,2)}(t))_0^i ,i = 1,..,5. $ for the iteration process (10) in problem 1

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Figure 3. Plots of exact ($ q^{loss*} $) and computed leak volume functions ($ q^*(t) $)for various suspected segments

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Table 1. Results of the solution to the problem 1 regarding leakage in the segment (1, 2) in the presence of noise for various initial states $ (\xi ^{(1,2)},\,\,q^{(1,2)}(t))_0^i ,i = 1,..,5. $

	$ \xi _0^{(1,2)} $	60	20	90	10	45,685
$ \eta $	$ q_0^{(1,2)} (t) $	$ 90 - 10e^{3\alpha t} $	$ 20 -10e^{3\alpha t} $	$ 30-10e^{3\alpha t} $	$ 66 + 20e^{3\alpha t} $	$ 66 + 20e^{3\alpha t} $
	$ \tilde {\Phi _0} $	76.104	16.704	11.664	42.48	57.636
	$ \tilde {\Phi } $	5.73$ \cdot 10^{ - 7} $	1.26$ \cdot 10^{ - 7} $	3.19$ \cdot 10^{ - 6} $	1.85$ \cdot 10^{ - 6} $	7.43$ \cdot 10^{ - 7} $
0%	$ \tilde {\xi }^{(1,2)} $	30.003	29.998	30.008	29.994	29.998
	$ N_{iter} $	6	5	16	14	8
	$ \delta \xi ^{(1,2)} $	0.00009	0.00006	0.0003	0.0002	0.00006
	$ \delta q^{(1,2)} $	0.0003	0. 0006	0.0003	0.0002	0.0003
	$ \tilde {\Phi _0} $	76.352	16.837	11.683	42.148	57.732
	$ \tilde {\Phi } $	0.023	0.014	0.024	0.017	0.020
0.5%	$ \tilde {\xi }^{(1,2)} $	29.841	30.332	30.068	29.654	29.796
	$ N_{iter} $	6	5	14	12	7
	$ \delta \xi ^{(1,2)} $	–0.005	0.011	0.002	–0.011	–0.006
	$ \delta q^{(1,2)} $	0.043	0.030	0.049	0.041	0.042
	$ \tilde{\Phi _0} $	77.119	16.924	11.832	43.744	57.413
	$ \tilde {\Phi } $	0.067	0.071	0.073	0.062	0.065
1%	$ \tilde {\xi }^{(1,2)} $	28.527	29.392	29.923	30.597	29.839
	$ N_{iter} $	6	5	14	12	7
	$ \delta \xi ^{(1,2)} $	–0.049	–0.020	–0.002	0.020	–0.005
	$ \delta q^{(1,2)} $	0.109	0.092	0.107	0.094	0.093

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Table 2. The results of the solution to the problem 1 regarding leakage in the section (1, 2) for different $ \nu $ and for different initial states $ (\xi ^{(1,2)},\,\,q^{(1,2)}(t))_0^i ,i = 1,..,5. $

	$ \xi _0^{(1,2)} $	60	20	90	10	45.685
$ \nu $	$ q_0^{(1,2)} (t) $	$ 90-10e^{3\alpha t} $	$ 20-10e^{3\alpha t} $	$ 30-10e^{3\alpha t} $	$ 66 + 20e^{3\alpha t} $	$ 66 + 20e^{3\alpha t} $
	$ \tilde{\Phi _0} $	155.248	22.323	22.323	84.303	118.809
	$ \tilde {\Phi} $	8.2$ \cdot 10^{ - 4} $	2.7$ \cdot 10^{ - 4} $	1.0$ \cdot 10^{ - 3} $	2.2$ \cdot 10^{ - 4} $	7.4$ \cdot 10^{ - 5} $
3	$ \tilde {\xi }^{(1,2)} $	30.145	29.996	30.190	30.022	30.029
	$ \delta \xi ^{(1,2)} $	0.004	0.0001	0.006	0.0007	0.0005
	$ \delta q^{(1,2)} $	0.0005	0.0004	0.0001	0.0009	0.0005
	$ \tilde{\Phi _0} $	155.390	33.608	22.443	84.086	118.854
	$ \tilde {\Phi } $	1.6$ \cdot 10^{ - 4} $	3.9$ \cdot 10^{ - 4} $	4.3$ \cdot 10^{ - 4} $	2.8$ \cdot 10^{ - 4} $	6.4$ \cdot 10^{ - 5} $
5	$ \tilde {\xi }^{(1,2)} $	29.963	30.093	30.100	29.985	29.995
	$ \delta \xi ^{(1,2)} $	0.001	0.003	0.003	0.0005	0.0001
	$ \delta q^{(1,2)} $	0.0006	0.0003	0.0005	0.0004	0.0002

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Table 3. The results of the solution to the problem 1 regarding leakage in the section (5, 6) for different initial states

$ \xi _0^{(5,6)} $	$ q_0^{(5,6)} (t) $	$ \tilde {\xi }^{(5,6)} $	$ \tilde{\Phi_0} $	$ \tilde {\Phi} $	$ N_{iter} $
25	$ 5+10e^{3\alpha t} $	15.219	2.556	3.55 $ \cdot 10^{ - 5} $	165
10	$ 30 +10e^{3\alpha t} $	14.781	44.615	3.55 $ \cdot 10^{ - 5} $	204
22.56	$ 15+10e^{3\alpha t} $	15.222	3.467	3.59 $ \cdot 10^{ - 5} $	197

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Table 4. Results obtained by solving Problem 2

$ \xi ^{0} $	$ q^{0} (t) $	Segment	$ \tilde{\xi }^{ks} $	$ \Im _{0} $	$ \tilde{\Im } $	Number of
		indices $ (k,s) $				iterations
		(1, 2)	5.991	43.272	39.614	6
		(5, 2)	2.279	139.536	60.386	58
25	$ 15+10e^{-0.0003t} $	(3, 2)	29.974	10.188	1.22$ \cdot 10^{-5} $	12
		(5, 4)	2.244	54.185	37.958	79
		(5, 6)	2.995	72.396	20.621	185
		(1, 2)	5.986	46.836	39.614	5
		(5, 2)	2.305	84.132	60.408	5
5	$ 5+10e^{-0.0003t} $	(3, 2)	29.935	32.148	2.44$ \cdot 10^{-5} $	12
		(5, 4)	1.898	38.628	37.728	150
		(5, 6)	2.947	21.786	20.621	72

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