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Article Contents

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

• * Corresponding author
• 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.

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

 Citation:

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

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

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

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

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

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

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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