American Institute of Mathematical Sciences

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November  2020, 16(6): 3011-3033. doi: 10.3934/jimo.2019091

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

Received  September 2018 Revised  April 2019 Published  November 2020 Early access  July 2019

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.

Keywords: Hyperbolic system of equations, inverse problem, non-separated boundary conditions, non-given initial conditions, gradient of function, hydraulic network, non-stationary mode, place of leakage, volume of leakage.
Mathematics Subject Classification: Primary: 35Q35, 65M32; Secondary: 65M55, 49K20.
Citation: Yegana Ashrafova, Kamil Aida-Zade. Numerical solution to an inverse problem on a determination of places and capacities of sources in the hyperbolic systems. Journal of Industrial & Management Optimization, 2020, 16 (6) : 3011-3033. doi: 10.3934/jimo.2019091
References:
[1]

A. Adamkowski, Analysis of transient flow in pipes with expanding or contracting sections, J. Fluids Eng., 125 (2003), 716-722.   Google Scholar

[2]

K. R. Aida-zade, Computational problems on hydraulic networks, Computational Math. and Math. Physics, 29 (1989), 125-132.  doi: 10.1016/0041-5553(89)90056-6.  Google Scholar

[3]

K. R. Aida-zade and J. A. Asadova, Study of transients in oil pipelines, Aut. and Remote Control, 72 (2011), 2563-2577.   Google Scholar

[4]

K. R. Aida-zade and E. R. Ashrafova, Localization of the points of leakage in an oil main pipeline under non-stationary conditions, J. of Eng. Phys. and Therm., 85 (2012), 1148-1156.   Google Scholar

[5]

K. R. Aida-zade and Y. R. Ashrafova, Calculation of transient fluid flow regimes in pipeline networks, Sib. Zh. Ind. Mat., 18 (2015), 12-23.  doi: 10.17377/sibjim.2015.18.202.  Google Scholar

[6]

K. R. Aida-zade and Y. R. Ashrafova, Optimal control of sources on some classes of functions, Optimization: A J. of Mathem. Prog. and Oper. Research., 63 (2014), 1135-1152.  doi: 10.1080/02331934.2012.711831.  Google Scholar

[7]

K. R. Aida-zade and Y. R. Ashrafova, Solving systems of differential equations of block structure with nonseparated boundary conditions, J. of Appl. and Industrial Math., 9 (2015), 1-10.  doi: 10.1134/S1990478915010019.  Google Scholar

[8]

K. R. Aida-zade and A. V. Handzel, An approach to lumped control synthesis in distributed systems, Appl. and Computational Math., 6 (2007), 69-79.   Google Scholar

[9]

A. M. Al-Khomairi, Leak detection in long pipelines using the least squares method, J. Hydraulic Res., 46 (2008), 392-401.   Google Scholar

[10]

Y. R. Ashrafova, Numerical investigation of the duration of the effect exerted by initial regimes on the process of liquid motion in a pipeline, J. of Eng. Physics and Therm., 88 (2015), 1-9.   Google Scholar

[11]

K. R. Ayda-zade and V. M. Abdullayev, Numerical solution of optimal control problems with unseparated conditions on phase state, Appl. and Computational Math.: An Int. Journal, 4 (2005), 165-177.   Google Scholar

[12]

A. Bergant, A. R. Simpson and J. P. Vitkovsky, Developments in unsteady pipe flow friction modeling, J. Hydraulic Res. IAHR, 39 (2001), 249-257.   Google Scholar

[13]

A. Q. Butkovskii and L. M. Pustilnikov, Theory of moving control of systems with distributed parameters, in Series in Theoretical Foundations of Engineering Cybernetics, Moscow, 1980.  Google Scholar

[14]

X. Chao, D. Yimeng, R. Zhigang, J. Huachen and Y. Xin, Sensor deployment for pipeline leakage detection via optimal boundary control strategies, J. of Indust. and Mgmt. Optimization, 11 (2015), 199-216.  doi: 10.3934/jimo.2015.11.199.  Google Scholar

[15]

I. A. Charniy, Unsteady motion of a real fluid in pipes, Moskva, Nedra, 1975. Google Scholar

[16]

M. H. Chaudhry, Applied hydraulic transients, Van Nostrand Reinhold, New York, 1987. Google Scholar

[17]

T. Chen, Z. Ren, C. Xu and R. Loxton, Optimal boundary control for water hammer suppression in fluid transmission pipelines, Comp. & Math. with Appl., 69 (2015), 275-290.  doi: 10.1016/j.camwa.2014.11.008.  Google Scholar

[18]

T. Chen, C. Xu, Q. Lin, et al., Water hammer mitigation via PDE-constrained optimization, J. Control Engineering Practice, 45 (2015), 54-63. Google Scholar

[19]

A. F. Colombo, P. Lee and B. W. Karney, A selective literature review of transient-based leak detection methods, J. Hydro-environment Res., 2 (2009), 212-227.   Google Scholar

[20]

S. Datta and S. Sarkar, A review on different pipeline fault detection methods, J. of Loss Prevention in the Process Indust., 41 (2016), 97-106.   Google Scholar

[21]

M. Ferrante, B. Brunone, S. Meniconi, B. W. Karney and and C. Massari, Leak Size, Detectability and Test Conditions in Pressurized Pipe Systems, Water Resources Mgmt., 28 (2014), 4583-4598.   Google Scholar

[22]

Z. Guanghui, N. Qin and Z. Meilan, A scaled conjugate gradient method with moving asymptotes for unconstrained optimization problems, J. of Indust. and Mgmt. Optimization, 13 (2017), 595-608.  doi: 10.3934/jimo.2016034.  Google Scholar

[23]

M. A. Huseynzade and V. A. Yufin, Unsteady oil and gas movement in main pipelines, M.: Nedra, 1981,232p. Google Scholar

[24]

Z. S. Kapelan, D. A. Savic and G. A. Walters, A hybrid inverse transient model for leakage detection and roughness calibration in pipe networks, J. Hydraulic Res., 41 (2003), 481-492.   Google Scholar

[25]

O. A. Ladijenskaya, Boundary-value probems of mathematical physics, Moskva, Nauka, 1973. Google Scholar

[26]

H. G. Lee and J. Kim, Regularized Dirac delta functions for phase field models, Int. J. for Num. Methods in Engineering, 91 (2012), 269-288.   Google Scholar

[27]

K. F. Oyedeko and H. A. Balogun, Modeling and simulation of a leak detection for oil and gas pipelines via transient model: A case study of the niger delta, J. of Energy Tech. and Policy, 5 (2015). Google Scholar

[28]

A. A. Samarskii, Theory of difference schemes, Moskow, Nauka, 1989.  Google Scholar

[29]

A. A. Samarskii and P. N. Vabishevich, Numerical method to solve inverse problems of mathematical physics, Moskva, LKI, 2009. Google Scholar

[30]

H. Shamloo and A. Haghighi, Leak detection in pipelines by inverse backward transient analysis, J. Hydraulic Res., 47 (2009), 311-318.   Google Scholar

[31]

H. Shamloo and A. Haghighi, Optimum leak detection and calibration of pipe networks by inverse transient analysis, J. Hydraulic Res., 48 (2010), 371-376.  doi: 10.1080/00221681003726304.  Google Scholar

[32]

R. Wichowski, Hydraulic transients analysis in pipe networks by the method of characteristics (MOC), Archives of Hydro-Engineering and Environ. Mech., 53 (2006), 267-291.   Google Scholar

[33]

E. B. Wylie and V. L. Streeter, Fluid Transients, McGraw-Hill International Book Co., New York, 1978. Google Scholar

show all references

References:
[1]

A. Adamkowski, Analysis of transient flow in pipes with expanding or contracting sections, J. Fluids Eng., 125 (2003), 716-722.   Google Scholar

[2]

K. R. Aida-zade, Computational problems on hydraulic networks, Computational Math. and Math. Physics, 29 (1989), 125-132.  doi: 10.1016/0041-5553(89)90056-6.  Google Scholar

[3]

K. R. Aida-zade and J. A. Asadova, Study of transients in oil pipelines, Aut. and Remote Control, 72 (2011), 2563-2577.   Google Scholar

[4]

K. R. Aida-zade and E. R. Ashrafova, Localization of the points of leakage in an oil main pipeline under non-stationary conditions, J. of Eng. Phys. and Therm., 85 (2012), 1148-1156.   Google Scholar

[5]

K. R. Aida-zade and Y. R. Ashrafova, Calculation of transient fluid flow regimes in pipeline networks, Sib. Zh. Ind. Mat., 18 (2015), 12-23.  doi: 10.17377/sibjim.2015.18.202.  Google Scholar

[6]

K. R. Aida-zade and Y. R. Ashrafova, Optimal control of sources on some classes of functions, Optimization: A J. of Mathem. Prog. and Oper. Research., 63 (2014), 1135-1152.  doi: 10.1080/02331934.2012.711831.  Google Scholar

[7]

K. R. Aida-zade and Y. R. Ashrafova, Solving systems of differential equations of block structure with nonseparated boundary conditions, J. of Appl. and Industrial Math., 9 (2015), 1-10.  doi: 10.1134/S1990478915010019.  Google Scholar

[8]

K. R. Aida-zade and A. V. Handzel, An approach to lumped control synthesis in distributed systems, Appl. and Computational Math., 6 (2007), 69-79.   Google Scholar

[9]

A. M. Al-Khomairi, Leak detection in long pipelines using the least squares method, J. Hydraulic Res., 46 (2008), 392-401.   Google Scholar

[10]

Y. R. Ashrafova, Numerical investigation of the duration of the effect exerted by initial regimes on the process of liquid motion in a pipeline, J. of Eng. Physics and Therm., 88 (2015), 1-9.   Google Scholar

[11]

K. R. Ayda-zade and V. M. Abdullayev, Numerical solution of optimal control problems with unseparated conditions on phase state, Appl. and Computational Math.: An Int. Journal, 4 (2005), 165-177.   Google Scholar

[12]

A. Bergant, A. R. Simpson and J. P. Vitkovsky, Developments in unsteady pipe flow friction modeling, J. Hydraulic Res. IAHR, 39 (2001), 249-257.   Google Scholar

[13]

A. Q. Butkovskii and L. M. Pustilnikov, Theory of moving control of systems with distributed parameters, in Series in Theoretical Foundations of Engineering Cybernetics, Moscow, 1980.  Google Scholar

[14]

X. Chao, D. Yimeng, R. Zhigang, J. Huachen and Y. Xin, Sensor deployment for pipeline leakage detection via optimal boundary control strategies, J. of Indust. and Mgmt. Optimization, 11 (2015), 199-216.  doi: 10.3934/jimo.2015.11.199.  Google Scholar

[15]

I. A. Charniy, Unsteady motion of a real fluid in pipes, Moskva, Nedra, 1975. Google Scholar

[16]

M. H. Chaudhry, Applied hydraulic transients, Van Nostrand Reinhold, New York, 1987. Google Scholar

[17]

T. Chen, Z. Ren, C. Xu and R. Loxton, Optimal boundary control for water hammer suppression in fluid transmission pipelines, Comp. & Math. with Appl., 69 (2015), 275-290.  doi: 10.1016/j.camwa.2014.11.008.  Google Scholar

[18]

T. Chen, C. Xu, Q. Lin, et al., Water hammer mitigation via PDE-constrained optimization, J. Control Engineering Practice, 45 (2015), 54-63. Google Scholar

[19]

A. F. Colombo, P. Lee and B. W. Karney, A selective literature review of transient-based leak detection methods, J. Hydro-environment Res., 2 (2009), 212-227.   Google Scholar

[20]

S. Datta and S. Sarkar, A review on different pipeline fault detection methods, J. of Loss Prevention in the Process Indust., 41 (2016), 97-106.   Google Scholar

[21]

M. Ferrante, B. Brunone, S. Meniconi, B. W. Karney and and C. Massari, Leak Size, Detectability and Test Conditions in Pressurized Pipe Systems, Water Resources Mgmt., 28 (2014), 4583-4598.   Google Scholar

[22]

Z. Guanghui, N. Qin and Z. Meilan, A scaled conjugate gradient method with moving asymptotes for unconstrained optimization problems, J. of Indust. and Mgmt. Optimization, 13 (2017), 595-608.  doi: 10.3934/jimo.2016034.  Google Scholar

[23]

M. A. Huseynzade and V. A. Yufin, Unsteady oil and gas movement in main pipelines, M.: Nedra, 1981,232p. Google Scholar

[24]

Z. S. Kapelan, D. A. Savic and G. A. Walters, A hybrid inverse transient model for leakage detection and roughness calibration in pipe networks, J. Hydraulic Res., 41 (2003), 481-492.   Google Scholar

[25]

O. A. Ladijenskaya, Boundary-value probems of mathematical physics, Moskva, Nauka, 1973. Google Scholar

[26]

H. G. Lee and J. Kim, Regularized Dirac delta functions for phase field models, Int. J. for Num. Methods in Engineering, 91 (2012), 269-288.   Google Scholar

[27]

K. F. Oyedeko and H. A. Balogun, Modeling and simulation of a leak detection for oil and gas pipelines via transient model: A case study of the niger delta, J. of Energy Tech. and Policy, 5 (2015). Google Scholar

[28]

A. A. Samarskii, Theory of difference schemes, Moskow, Nauka, 1989.  Google Scholar

[29]

A. A. Samarskii and P. N. Vabishevich, Numerical method to solve inverse problems of mathematical physics, Moskva, LKI, 2009. Google Scholar

[30]

H. Shamloo and A. Haghighi, Leak detection in pipelines by inverse backward transient analysis, J. Hydraulic Res., 47 (2009), 311-318.   Google Scholar

[31]

H. Shamloo and A. Haghighi, Optimum leak detection and calibration of pipe networks by inverse transient analysis, J. Hydraulic Res., 48 (2010), 371-376.  doi: 10.1080/00221681003726304.  Google Scholar

[32]

R. Wichowski, Hydraulic transients analysis in pipe networks by the method of characteristics (MOC), Archives of Hydro-Engineering and Environ. Mech., 53 (2006), 267-291.   Google Scholar

[33]

E. B. Wylie and V. L. Streeter, Fluid Transients, McGraw-Hill International Book Co., New York, 1978. Google Scholar

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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$ \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
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$ \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
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$ \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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$ \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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