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Research ArticleOn Some Classes of Linear Volterra Integral Equations
Anatoly S Apartsyn
Melentiev Energy Systems Institute SB RAS Irkutsk Russia
Correspondence should be addressed to Anatoly S Apartsyn apartsynisemseiirkru
Received 19 December 2013 Revised 29 May 2014 Accepted 29 May 2014 Published 6 July 2014
Academic Editor Hossein Jafari
Copyright copy 2014 Anatoly S ApartsynThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The sufficient conditions are obtained for the existence and uniqueness of continuous solution to the linear nonclassical Volterraequation that appears in the integral models of developing systems The Volterra integral equations of the first kind with piecewisesmooth kernels are considered Illustrative examples are presented
1 Introduction
Volterra integral equations of the first kind with variableupper and lower limits of integrationwere studied byVolterrahimself [1] The publications on this topic in the first half ofthe 20th century were reviewed in [2] and later studies werediscussed in [3ndash5]
A noticeable impetus to the development of this areais related to the research [6] which suggested a macroe-conomic two-sector integral model The Glushkovrsquos modelsof developing systems were further extended in [7 8] andused in many applications (see [9] and references therein)In particular a one-sector version of the Glushkovrsquos modelapplied to the power engineering problems was considered in[10ndash12] In the recent years the researchers have got attractedby the equation (see [13] and references therein) that in ageneral case has the following form
kernels 119870119894and right-hand side 119910(119905) are given and 119909(119905) is an
unknown desired solutionAt 119899 = 1 the problems of the existence and uniqueness
of solution to (1) in the space 119862[0119879]
as well as the numerical
methods are studied in detail in [5] In this paper we will beinterested in the same problems for (1) at 119899 gt 1 Further forsimplicity we will consider only the case 119899 = 2 since manyresults are easily generalized for the case 119899 gt 2
2 Sufficient Conditions for the Correctness of
(1) at 119899=2 in Pair (119862[0119879]
∘
119862(1)
[0119879])
For convenience present (1) with 119899 = 2 in operator form
so that for the right-hand side of (20) 119910(119905) = 119910(119905) = 119905119896119896119896 = 1 2 3 from (33) we obtain
119909 (119905) = 119905119896minus1
1 + 120572119896minus1 119896 = 1 2 (34)
In conclusion of this section it should be noted thatinequalities (8) and (17) can be interpreted as constraints onthe value 119879 which guarantee at given 119870
1(119905 119904) 119870
2(119905 119904) and
1198861(119905) the correct solvability of (3) in 119862
[0119879] Since all param-
eters in the left-hand side of (8) and (17) are nondecreasingfunctions of119879 and the right-hand side of (8) and (17) at 119871
1= 0
(1
= 0) on the contrary monotonously decreases then thereal positive root of corresponding nonlinear equation thatgives a guaranteed lower-bound estimate of 119879 exists and isunique if 1198861015840(0) is sufficiently small In some special casesthis root can be found analytically in terms of the Lambertfunction119882 [15 16]
In [17ndash22] the authors studied the characteristic of con-tinuous solution locality and the role of the Lambert functionas applied to the polynomial (multilinear) Volterra equationsof the first kind The calculations of the test examples showthat the locality feature of the solution to the linear equation(3) is not the result of the inaccuracy of estimates (8) and (17)and reflects the specifics of the considered class of problemsIn this paper we do not dwell on the problem of numericallysolving (3) It is of independent interest and deserves specialconsideration
3 The Volterra Integral Equations of the FirstKind with Discontinuous Kernels
Equation (2) can be written in the form of Volterra integralequation of the first kind
119894 = 2 119899 minus 1(119870119894(119905 119904) + 119870
119894+1(119905 119904))
2 119904 = 119886
119894(119905) 119894 = 1 119899 minus 1
119870119899(119905 119904) 0 le 119904 lt 119886
119899minus1(119905)
(36)
To illustrate the fundamental difference between (35)(36) and classical Volterra equation of the first kind withsmooth kernel we confine ourselves to (20) that has the formof (35) at
119870 (119905 119904) =
1 120572119905 lt 119904 le 1199051 + 1205982
119904 = 120572119905120598 0 le 119904 lt 120572119905
(37)
where 120598 = 0 1 and 120572 isin (0 1) In particular at 120572 = 12 120598 = minus1
119870 (119905 119904) = sign(119904 minus 1199052) =
1 119904 gt 1199052
0 119904 = 1199052
minus1 119904 lt 1199052
(38)
For this case the solution to (35) with 119910(119905) = 119905 given in[23] is
If119870(119905 119904) is continuous in arguments and continuously differ-entiable with respect to 119905 inΔ then condition (40)means that(35) is Volterra integral equation of the third kind
The theory (whose foundation was laid by Volterra (see[24 pages 104ndash106])) of such equations is developed in theresearch done by Magnitsky [25ndash28]
In particular the author of [25ndash28] studies the structureof one- or many-parameter family of solutions to (35)
If 119870(119905 119904) is discontinuous then the solution to (35) maybe nonunique even if 119870(119905 119905) = 0 forall119905 ge 0
For example if120572 = 12 and 120598 = 1minus(1120572) = minus1 the solutionto equation
so that condition119870(119905 119905) equiv 1 is trueWe prove that solutions to (35) (37) and (35) (43)
coincide It suffices to show that the equivalent functionalequations for (35) (37) and (35) (43) coincide Recall that for(35) (37) the equivalent functional equation is (24)
Theorem 3 The equivalent functional equations for (35) (37)and (35) (43) coincide
Proof Let us represent (43) by
119870 (119905 119904) equiv 1 + (120598 minus 1) 119890 (120572119905 minus 119904) (44)
where 119890(sdot)ndash is a Heaviside function
119890 (]) = 1 ] ge 00 ] lt 0
(45)
Substitution of (44) in (35) gives
int119905
0
119909 (119904) 119889119904 + (120598 minus 1) int119905
0
119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = 119910 (119905)
119905 isin [0 119879] (46)
Transform the second integral Let ] = 120572119905 minus 119904 Then
int119905
0
119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = int120572119905
(120572minus1)119905
119890 (]) 119909 (120572119905 minus ]) 119889]
= int120572119905
0
119909 (120572119905 minus ]) 119889]
(47)
By virtue of (47) differentiation of (46) results in
119909 (119905) + (120598 minus 1) 120572119909 (0) + (120598 minus 1) int120572119905
0
1199091015840119905(120572119905 minus ]) 119889] = 1199101015840 (119905)
(48)
But
1199091015840119905(120572119905 minus ]) = minus1205721199091015840] (120572119905 minus ]) (49)
By virtue of (49) we have
119909 (119905) + (120598 minus 1) 120572119909 (0) minus (120598 minus 1) 120572 [119909 (120572119905 minus ])|1205721199050] = 1199101015840 (119905)
The solution to (35) (43) in the class of piecewisecontinuous functions with a jump on line 119904 = 120572119905 is interestingfrom the application perspective
It is easy to see that this solution is
119909 (119905 119904) =
1199101015840 (119904) 119904 ge 12057211990511205981199101015840 (119904) 119904 lt 120572119905
(52)
At last consider the concept of 120572-convolution Volterraintegral equations of convolution type
As is mentioned in the introduction the main results of thisstudy can be easily applied to the case 119899 gt 2 in (1) Theequations of type (1) not only are of theoretical interest butalso play an important role in the mathematical modeling ofdeveloping dynamic systemsMoreover by119910(119905) we canmeansome criterion that characterizes the level of developmentof the system as a whole and the 119894th term in (1) representsa contribution of the system components 119909(119904) of the 119894thage group whose operation is reflected by the efficiencycoefficient 119870
119894(119905 minus 119904) As a rule 119870
1ge sdot sdot sdot ge 119870
119899ge 0
Such an approach is implemented for instance in [29 30]in the problem of the analysis of strategies for the long-termexpansion of the Russian electric power system with theconsideration of aging of the power plants equipment
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author wishes to thank the reviewers for their helpfulnotes The study is supported by the Russian Foundation forBasic Research Grant no 12-01-00722a
References
[1] V Volterra ldquoSopra alcune questioni di inversione di integralidefinitirdquo Annali di Matematica Pura ed Applicata Series 2 vol25 no 1 pp 139ndash178 1897
[2] H Brunner ldquo1896ndash1996 One hundred years of Volterra integralequations of the first kindrdquoAppliedNumericalMathematics vol24 no 2-3 pp 83ndash93 1997
[3] H Brunner and P J van der HouwenTheNumerical Solution ofVolterra Equations vol 3 of CWI Monographs North-HollandAmsterdam The Netherlands 1986
[4] H Brunner Collocation Methods for Volterra Integral andRelated Functional Differential Equations vol 15 of CambridgeMonographs on Applied and Computational Mathematics Cam-bridge University Press Cambridge Mass USA 2004
[5] A S Apartsyn Nonclassical Linear Volterra Equations of theFirst Kind VSP Utrecht The Netherlands 2003
[6] V M Glushkov ldquoOn one class of dynamic macroeconomicmodelsrdquo Upravlyayushchiye Sistemy I Mashiny no 2 pp 3ndash61977 (Russian)
[7] V M Glushkov V V Ivanov and V M Yanenko Modeling ofDeveloping Systems Nauka Moscow Russia 1983 (Russian)
[8] Y P Yatsenko Integral Models of Systems with ControlledMemory Naukova Dumka Kiev Ukraine 1991 (Russian)
[9] N Hritonenko and Y Yatsenko Applied Mathematical Mod-elling of Engineering Problems vol 81 of Applied OptimizationKluwer Academic Publishers Dordrecht The Netherlands2003
[10] A S Apartsyn E V Markova and V V Trufanov IntegralModels of Electric Power System Development Energy SystemsInstitute SB RAS Irkutsk Russia 2002 (Russian)
6 Abstract and Applied Analysis
[11] D V Ivanov V Karaulova E VMarkova V V Trufanov andOV Khamisov ldquoControl of power grid development numericalsolutionsrdquo Automation and Remote Control vol 65 no 3 pp472ndash482 2004
[12] A S Apartsyn I V Karaulova E V Markova and V VTrufanov ldquoApplication of the Volterra integral equations for themodeling of strategies of technical re-equipment in the electricpower industryrdquo Electrical Technology Russia no 10 pp 64ndash752005 (Russian)
[13] E Messina E Russo and A Vecchio ldquoA stable numericalmethod for Volterra integral equations with discontinuouskernelrdquo Journal of Mathematical Analysis and Applications vol337 no 2 pp 1383ndash1393 2008
[14] L V Kantorovich and G P Akilov Functional Analysis NaukaMoscow Russia 1977 (Russian)
[15] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert 119882 functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996
[16] R M Corless G H Gonnet D E G Hare and D JJeffrey ldquoLambertrsquos 119882 function in Maplerdquo The Maple TechnicalNewsletter no 9 pp 12ndash22 1993
[17] A S Apartsyn ldquoMultilinear Volterra equations of the first kindrdquoAutomation and Remote Control vol 65 no 2 pp 263ndash2692004
[18] A S Apartsyn ldquoPolilinear integral Volterra equations of thefirst kind the elements of the theory and numeric meth-odsrdquo Izvestiya Irkutskogo Gosudarstvennogo Universiteta SeriesMathematics no 1 pp 13ndash41 2007
[19] A S Apartsin ldquoOn the convergence of numerical methodsfor solving a Volterra bilinear equations of the first kindrdquoComputational Mathematics and Mathematical Physics vol 47no 8 pp 1323ndash1331 2007
[20] A S Apartsin ldquoMultilinear Volterra equations of the firstkind and some problems of controlrdquo Automation and RemoteControl vol 69 no 4 pp 545ndash558 2008
[21] A S Apartsyn ldquoUnimprovable estimates of solutions for someclasses of integral inequalitiesrdquo Journal of Inverse and Ill-PosedProblems vol 16 no 7 pp 651ndash680 2008
[22] A S Apartsyn ldquoPolynomial Volterra integral equations of thefirst kind and the Lambert functionrdquo Proceedings of the Instituteof Mathematics and Mechanics Ural Branch of RAS vol 18 no1 pp 69ndash81 2012 (Russian)
[23] D N Sidorov ldquoOn parametric families of solutions of Volterraintegral equations of the first kind with piecewise smoothkernelrdquo Differential Equations vol 49 no 2 pp 210ndash216 2013
[24] V Volterra Theory of Functionals and of Integral and Integro-Differential Equations Nauka Moscow Russia 1982 (Russian)
[25] N A Magnitsky ldquoThe existence of multiparameter familiesof solutions of a Volterra integral equation of the first kindrdquoReports of theUSSRAcademy of Sciences vol 235 no 4 pp 772ndash774 1977 (Russian)
[26] N A Magnitsky ldquoLinear Volterra integral equations of the firstand third kindsrdquo Computational Mathematics and Mathemati-cal Physics vol 19 no 4 pp 970ndash988 1979 (Russian)
[27] N A Magnitsky ldquoThe asymptotics of solutions to the Volterraintegral equation of the first kindrdquoReports of the USSRAcademyof Sciences vol 269 no 1 pp 29ndash32 1983 (Russian)
[28] N A Magnitsky Asymptotic Methods for Analysis of Non-Stationary Controlled Systems Nauka Moscow Russia 1992(Russian)
[29] A S Apartsyn ldquoOn one approach to modeling of developingsystemsrdquo in Proceedings of the 6th International WorkshopldquoGeneralized Statments and Solutions of Control Problemsrdquo pp32ndash35 Divnomorskoe Russia 2012
[30] A S Apartsin and I V Sidler ldquoUsing the nonclassical Volterraequations of the first kind to model the developing systemsrdquoAutomation and Remote Control vol 74 no 6 pp 899ndash9102013
so that for the right-hand side of (20) 119910(119905) = 119910(119905) = 119905119896119896119896 = 1 2 3 from (33) we obtain
119909 (119905) = 119905119896minus1
1 + 120572119896minus1 119896 = 1 2 (34)
In conclusion of this section it should be noted thatinequalities (8) and (17) can be interpreted as constraints onthe value 119879 which guarantee at given 119870
1(119905 119904) 119870
2(119905 119904) and
1198861(119905) the correct solvability of (3) in 119862
[0119879] Since all param-
eters in the left-hand side of (8) and (17) are nondecreasingfunctions of119879 and the right-hand side of (8) and (17) at 119871
1= 0
(1
= 0) on the contrary monotonously decreases then thereal positive root of corresponding nonlinear equation thatgives a guaranteed lower-bound estimate of 119879 exists and isunique if 1198861015840(0) is sufficiently small In some special casesthis root can be found analytically in terms of the Lambertfunction119882 [15 16]
In [17ndash22] the authors studied the characteristic of con-tinuous solution locality and the role of the Lambert functionas applied to the polynomial (multilinear) Volterra equationsof the first kind The calculations of the test examples showthat the locality feature of the solution to the linear equation(3) is not the result of the inaccuracy of estimates (8) and (17)and reflects the specifics of the considered class of problemsIn this paper we do not dwell on the problem of numericallysolving (3) It is of independent interest and deserves specialconsideration
3 The Volterra Integral Equations of the FirstKind with Discontinuous Kernels
Equation (2) can be written in the form of Volterra integralequation of the first kind
119894 = 2 119899 minus 1(119870119894(119905 119904) + 119870
119894+1(119905 119904))
2 119904 = 119886
119894(119905) 119894 = 1 119899 minus 1
119870119899(119905 119904) 0 le 119904 lt 119886
119899minus1(119905)
(36)
To illustrate the fundamental difference between (35)(36) and classical Volterra equation of the first kind withsmooth kernel we confine ourselves to (20) that has the formof (35) at
119870 (119905 119904) =
1 120572119905 lt 119904 le 1199051 + 1205982
119904 = 120572119905120598 0 le 119904 lt 120572119905
(37)
where 120598 = 0 1 and 120572 isin (0 1) In particular at 120572 = 12 120598 = minus1
119870 (119905 119904) = sign(119904 minus 1199052) =
1 119904 gt 1199052
0 119904 = 1199052
minus1 119904 lt 1199052
(38)
For this case the solution to (35) with 119910(119905) = 119905 given in[23] is
If119870(119905 119904) is continuous in arguments and continuously differ-entiable with respect to 119905 inΔ then condition (40)means that(35) is Volterra integral equation of the third kind
The theory (whose foundation was laid by Volterra (see[24 pages 104ndash106])) of such equations is developed in theresearch done by Magnitsky [25ndash28]
In particular the author of [25ndash28] studies the structureof one- or many-parameter family of solutions to (35)
If 119870(119905 119904) is discontinuous then the solution to (35) maybe nonunique even if 119870(119905 119905) = 0 forall119905 ge 0
For example if120572 = 12 and 120598 = 1minus(1120572) = minus1 the solutionto equation
so that condition119870(119905 119905) equiv 1 is trueWe prove that solutions to (35) (37) and (35) (43)
coincide It suffices to show that the equivalent functionalequations for (35) (37) and (35) (43) coincide Recall that for(35) (37) the equivalent functional equation is (24)
Theorem 3 The equivalent functional equations for (35) (37)and (35) (43) coincide
Proof Let us represent (43) by
119870 (119905 119904) equiv 1 + (120598 minus 1) 119890 (120572119905 minus 119904) (44)
where 119890(sdot)ndash is a Heaviside function
119890 (]) = 1 ] ge 00 ] lt 0
(45)
Substitution of (44) in (35) gives
int119905
0
119909 (119904) 119889119904 + (120598 minus 1) int119905
0
119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = 119910 (119905)
119905 isin [0 119879] (46)
Transform the second integral Let ] = 120572119905 minus 119904 Then
int119905
0
119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = int120572119905
(120572minus1)119905
119890 (]) 119909 (120572119905 minus ]) 119889]
= int120572119905
0
119909 (120572119905 minus ]) 119889]
(47)
By virtue of (47) differentiation of (46) results in
119909 (119905) + (120598 minus 1) 120572119909 (0) + (120598 minus 1) int120572119905
0
1199091015840119905(120572119905 minus ]) 119889] = 1199101015840 (119905)
(48)
But
1199091015840119905(120572119905 minus ]) = minus1205721199091015840] (120572119905 minus ]) (49)
By virtue of (49) we have
119909 (119905) + (120598 minus 1) 120572119909 (0) minus (120598 minus 1) 120572 [119909 (120572119905 minus ])|1205721199050] = 1199101015840 (119905)
The solution to (35) (43) in the class of piecewisecontinuous functions with a jump on line 119904 = 120572119905 is interestingfrom the application perspective
It is easy to see that this solution is
119909 (119905 119904) =
1199101015840 (119904) 119904 ge 12057211990511205981199101015840 (119904) 119904 lt 120572119905
(52)
At last consider the concept of 120572-convolution Volterraintegral equations of convolution type
As is mentioned in the introduction the main results of thisstudy can be easily applied to the case 119899 gt 2 in (1) Theequations of type (1) not only are of theoretical interest butalso play an important role in the mathematical modeling ofdeveloping dynamic systemsMoreover by119910(119905) we canmeansome criterion that characterizes the level of developmentof the system as a whole and the 119894th term in (1) representsa contribution of the system components 119909(119904) of the 119894thage group whose operation is reflected by the efficiencycoefficient 119870
119894(119905 minus 119904) As a rule 119870
1ge sdot sdot sdot ge 119870
119899ge 0
Such an approach is implemented for instance in [29 30]in the problem of the analysis of strategies for the long-termexpansion of the Russian electric power system with theconsideration of aging of the power plants equipment
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author wishes to thank the reviewers for their helpfulnotes The study is supported by the Russian Foundation forBasic Research Grant no 12-01-00722a
References
[1] V Volterra ldquoSopra alcune questioni di inversione di integralidefinitirdquo Annali di Matematica Pura ed Applicata Series 2 vol25 no 1 pp 139ndash178 1897
[2] H Brunner ldquo1896ndash1996 One hundred years of Volterra integralequations of the first kindrdquoAppliedNumericalMathematics vol24 no 2-3 pp 83ndash93 1997
[3] H Brunner and P J van der HouwenTheNumerical Solution ofVolterra Equations vol 3 of CWI Monographs North-HollandAmsterdam The Netherlands 1986
[4] H Brunner Collocation Methods for Volterra Integral andRelated Functional Differential Equations vol 15 of CambridgeMonographs on Applied and Computational Mathematics Cam-bridge University Press Cambridge Mass USA 2004
[5] A S Apartsyn Nonclassical Linear Volterra Equations of theFirst Kind VSP Utrecht The Netherlands 2003
[6] V M Glushkov ldquoOn one class of dynamic macroeconomicmodelsrdquo Upravlyayushchiye Sistemy I Mashiny no 2 pp 3ndash61977 (Russian)
[7] V M Glushkov V V Ivanov and V M Yanenko Modeling ofDeveloping Systems Nauka Moscow Russia 1983 (Russian)
[8] Y P Yatsenko Integral Models of Systems with ControlledMemory Naukova Dumka Kiev Ukraine 1991 (Russian)
[9] N Hritonenko and Y Yatsenko Applied Mathematical Mod-elling of Engineering Problems vol 81 of Applied OptimizationKluwer Academic Publishers Dordrecht The Netherlands2003
[10] A S Apartsyn E V Markova and V V Trufanov IntegralModels of Electric Power System Development Energy SystemsInstitute SB RAS Irkutsk Russia 2002 (Russian)
6 Abstract and Applied Analysis
[11] D V Ivanov V Karaulova E VMarkova V V Trufanov andOV Khamisov ldquoControl of power grid development numericalsolutionsrdquo Automation and Remote Control vol 65 no 3 pp472ndash482 2004
[12] A S Apartsyn I V Karaulova E V Markova and V VTrufanov ldquoApplication of the Volterra integral equations for themodeling of strategies of technical re-equipment in the electricpower industryrdquo Electrical Technology Russia no 10 pp 64ndash752005 (Russian)
[13] E Messina E Russo and A Vecchio ldquoA stable numericalmethod for Volterra integral equations with discontinuouskernelrdquo Journal of Mathematical Analysis and Applications vol337 no 2 pp 1383ndash1393 2008
[14] L V Kantorovich and G P Akilov Functional Analysis NaukaMoscow Russia 1977 (Russian)
[15] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert 119882 functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996
[16] R M Corless G H Gonnet D E G Hare and D JJeffrey ldquoLambertrsquos 119882 function in Maplerdquo The Maple TechnicalNewsletter no 9 pp 12ndash22 1993
[17] A S Apartsyn ldquoMultilinear Volterra equations of the first kindrdquoAutomation and Remote Control vol 65 no 2 pp 263ndash2692004
[18] A S Apartsyn ldquoPolilinear integral Volterra equations of thefirst kind the elements of the theory and numeric meth-odsrdquo Izvestiya Irkutskogo Gosudarstvennogo Universiteta SeriesMathematics no 1 pp 13ndash41 2007
[19] A S Apartsin ldquoOn the convergence of numerical methodsfor solving a Volterra bilinear equations of the first kindrdquoComputational Mathematics and Mathematical Physics vol 47no 8 pp 1323ndash1331 2007
[20] A S Apartsin ldquoMultilinear Volterra equations of the firstkind and some problems of controlrdquo Automation and RemoteControl vol 69 no 4 pp 545ndash558 2008
[21] A S Apartsyn ldquoUnimprovable estimates of solutions for someclasses of integral inequalitiesrdquo Journal of Inverse and Ill-PosedProblems vol 16 no 7 pp 651ndash680 2008
[22] A S Apartsyn ldquoPolynomial Volterra integral equations of thefirst kind and the Lambert functionrdquo Proceedings of the Instituteof Mathematics and Mechanics Ural Branch of RAS vol 18 no1 pp 69ndash81 2012 (Russian)
[23] D N Sidorov ldquoOn parametric families of solutions of Volterraintegral equations of the first kind with piecewise smoothkernelrdquo Differential Equations vol 49 no 2 pp 210ndash216 2013
[24] V Volterra Theory of Functionals and of Integral and Integro-Differential Equations Nauka Moscow Russia 1982 (Russian)
[25] N A Magnitsky ldquoThe existence of multiparameter familiesof solutions of a Volterra integral equation of the first kindrdquoReports of theUSSRAcademy of Sciences vol 235 no 4 pp 772ndash774 1977 (Russian)
[26] N A Magnitsky ldquoLinear Volterra integral equations of the firstand third kindsrdquo Computational Mathematics and Mathemati-cal Physics vol 19 no 4 pp 970ndash988 1979 (Russian)
[27] N A Magnitsky ldquoThe asymptotics of solutions to the Volterraintegral equation of the first kindrdquoReports of the USSRAcademyof Sciences vol 269 no 1 pp 29ndash32 1983 (Russian)
[28] N A Magnitsky Asymptotic Methods for Analysis of Non-Stationary Controlled Systems Nauka Moscow Russia 1992(Russian)
[29] A S Apartsyn ldquoOn one approach to modeling of developingsystemsrdquo in Proceedings of the 6th International WorkshopldquoGeneralized Statments and Solutions of Control Problemsrdquo pp32ndash35 Divnomorskoe Russia 2012
[30] A S Apartsin and I V Sidler ldquoUsing the nonclassical Volterraequations of the first kind to model the developing systemsrdquoAutomation and Remote Control vol 74 no 6 pp 899ndash9102013
so that for the right-hand side of (20) 119910(119905) = 119910(119905) = 119905119896119896119896 = 1 2 3 from (33) we obtain
119909 (119905) = 119905119896minus1
1 + 120572119896minus1 119896 = 1 2 (34)
In conclusion of this section it should be noted thatinequalities (8) and (17) can be interpreted as constraints onthe value 119879 which guarantee at given 119870
1(119905 119904) 119870
2(119905 119904) and
1198861(119905) the correct solvability of (3) in 119862
[0119879] Since all param-
eters in the left-hand side of (8) and (17) are nondecreasingfunctions of119879 and the right-hand side of (8) and (17) at 119871
1= 0
(1
= 0) on the contrary monotonously decreases then thereal positive root of corresponding nonlinear equation thatgives a guaranteed lower-bound estimate of 119879 exists and isunique if 1198861015840(0) is sufficiently small In some special casesthis root can be found analytically in terms of the Lambertfunction119882 [15 16]
In [17ndash22] the authors studied the characteristic of con-tinuous solution locality and the role of the Lambert functionas applied to the polynomial (multilinear) Volterra equationsof the first kind The calculations of the test examples showthat the locality feature of the solution to the linear equation(3) is not the result of the inaccuracy of estimates (8) and (17)and reflects the specifics of the considered class of problemsIn this paper we do not dwell on the problem of numericallysolving (3) It is of independent interest and deserves specialconsideration
3 The Volterra Integral Equations of the FirstKind with Discontinuous Kernels
Equation (2) can be written in the form of Volterra integralequation of the first kind
119894 = 2 119899 minus 1(119870119894(119905 119904) + 119870
119894+1(119905 119904))
2 119904 = 119886
119894(119905) 119894 = 1 119899 minus 1
119870119899(119905 119904) 0 le 119904 lt 119886
119899minus1(119905)
(36)
To illustrate the fundamental difference between (35)(36) and classical Volterra equation of the first kind withsmooth kernel we confine ourselves to (20) that has the formof (35) at
119870 (119905 119904) =
1 120572119905 lt 119904 le 1199051 + 1205982
119904 = 120572119905120598 0 le 119904 lt 120572119905
(37)
where 120598 = 0 1 and 120572 isin (0 1) In particular at 120572 = 12 120598 = minus1
119870 (119905 119904) = sign(119904 minus 1199052) =
1 119904 gt 1199052
0 119904 = 1199052
minus1 119904 lt 1199052
(38)
For this case the solution to (35) with 119910(119905) = 119905 given in[23] is
If119870(119905 119904) is continuous in arguments and continuously differ-entiable with respect to 119905 inΔ then condition (40)means that(35) is Volterra integral equation of the third kind
The theory (whose foundation was laid by Volterra (see[24 pages 104ndash106])) of such equations is developed in theresearch done by Magnitsky [25ndash28]
In particular the author of [25ndash28] studies the structureof one- or many-parameter family of solutions to (35)
If 119870(119905 119904) is discontinuous then the solution to (35) maybe nonunique even if 119870(119905 119905) = 0 forall119905 ge 0
For example if120572 = 12 and 120598 = 1minus(1120572) = minus1 the solutionto equation
so that condition119870(119905 119905) equiv 1 is trueWe prove that solutions to (35) (37) and (35) (43)
coincide It suffices to show that the equivalent functionalequations for (35) (37) and (35) (43) coincide Recall that for(35) (37) the equivalent functional equation is (24)
Theorem 3 The equivalent functional equations for (35) (37)and (35) (43) coincide
Proof Let us represent (43) by
119870 (119905 119904) equiv 1 + (120598 minus 1) 119890 (120572119905 minus 119904) (44)
where 119890(sdot)ndash is a Heaviside function
119890 (]) = 1 ] ge 00 ] lt 0
(45)
Substitution of (44) in (35) gives
int119905
0
119909 (119904) 119889119904 + (120598 minus 1) int119905
0
119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = 119910 (119905)
119905 isin [0 119879] (46)
Transform the second integral Let ] = 120572119905 minus 119904 Then
int119905
0
119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = int120572119905
(120572minus1)119905
119890 (]) 119909 (120572119905 minus ]) 119889]
= int120572119905
0
119909 (120572119905 minus ]) 119889]
(47)
By virtue of (47) differentiation of (46) results in
119909 (119905) + (120598 minus 1) 120572119909 (0) + (120598 minus 1) int120572119905
0
1199091015840119905(120572119905 minus ]) 119889] = 1199101015840 (119905)
(48)
But
1199091015840119905(120572119905 minus ]) = minus1205721199091015840] (120572119905 minus ]) (49)
By virtue of (49) we have
119909 (119905) + (120598 minus 1) 120572119909 (0) minus (120598 minus 1) 120572 [119909 (120572119905 minus ])|1205721199050] = 1199101015840 (119905)
The solution to (35) (43) in the class of piecewisecontinuous functions with a jump on line 119904 = 120572119905 is interestingfrom the application perspective
It is easy to see that this solution is
119909 (119905 119904) =
1199101015840 (119904) 119904 ge 12057211990511205981199101015840 (119904) 119904 lt 120572119905
(52)
At last consider the concept of 120572-convolution Volterraintegral equations of convolution type
As is mentioned in the introduction the main results of thisstudy can be easily applied to the case 119899 gt 2 in (1) Theequations of type (1) not only are of theoretical interest butalso play an important role in the mathematical modeling ofdeveloping dynamic systemsMoreover by119910(119905) we canmeansome criterion that characterizes the level of developmentof the system as a whole and the 119894th term in (1) representsa contribution of the system components 119909(119904) of the 119894thage group whose operation is reflected by the efficiencycoefficient 119870
119894(119905 minus 119904) As a rule 119870
1ge sdot sdot sdot ge 119870
119899ge 0
Such an approach is implemented for instance in [29 30]in the problem of the analysis of strategies for the long-termexpansion of the Russian electric power system with theconsideration of aging of the power plants equipment
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author wishes to thank the reviewers for their helpfulnotes The study is supported by the Russian Foundation forBasic Research Grant no 12-01-00722a
References
[1] V Volterra ldquoSopra alcune questioni di inversione di integralidefinitirdquo Annali di Matematica Pura ed Applicata Series 2 vol25 no 1 pp 139ndash178 1897
[2] H Brunner ldquo1896ndash1996 One hundred years of Volterra integralequations of the first kindrdquoAppliedNumericalMathematics vol24 no 2-3 pp 83ndash93 1997
[3] H Brunner and P J van der HouwenTheNumerical Solution ofVolterra Equations vol 3 of CWI Monographs North-HollandAmsterdam The Netherlands 1986
[4] H Brunner Collocation Methods for Volterra Integral andRelated Functional Differential Equations vol 15 of CambridgeMonographs on Applied and Computational Mathematics Cam-bridge University Press Cambridge Mass USA 2004
[5] A S Apartsyn Nonclassical Linear Volterra Equations of theFirst Kind VSP Utrecht The Netherlands 2003
[6] V M Glushkov ldquoOn one class of dynamic macroeconomicmodelsrdquo Upravlyayushchiye Sistemy I Mashiny no 2 pp 3ndash61977 (Russian)
[7] V M Glushkov V V Ivanov and V M Yanenko Modeling ofDeveloping Systems Nauka Moscow Russia 1983 (Russian)
[8] Y P Yatsenko Integral Models of Systems with ControlledMemory Naukova Dumka Kiev Ukraine 1991 (Russian)
[9] N Hritonenko and Y Yatsenko Applied Mathematical Mod-elling of Engineering Problems vol 81 of Applied OptimizationKluwer Academic Publishers Dordrecht The Netherlands2003
[10] A S Apartsyn E V Markova and V V Trufanov IntegralModels of Electric Power System Development Energy SystemsInstitute SB RAS Irkutsk Russia 2002 (Russian)
6 Abstract and Applied Analysis
[11] D V Ivanov V Karaulova E VMarkova V V Trufanov andOV Khamisov ldquoControl of power grid development numericalsolutionsrdquo Automation and Remote Control vol 65 no 3 pp472ndash482 2004
[12] A S Apartsyn I V Karaulova E V Markova and V VTrufanov ldquoApplication of the Volterra integral equations for themodeling of strategies of technical re-equipment in the electricpower industryrdquo Electrical Technology Russia no 10 pp 64ndash752005 (Russian)
[13] E Messina E Russo and A Vecchio ldquoA stable numericalmethod for Volterra integral equations with discontinuouskernelrdquo Journal of Mathematical Analysis and Applications vol337 no 2 pp 1383ndash1393 2008
[14] L V Kantorovich and G P Akilov Functional Analysis NaukaMoscow Russia 1977 (Russian)
[15] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert 119882 functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996
[16] R M Corless G H Gonnet D E G Hare and D JJeffrey ldquoLambertrsquos 119882 function in Maplerdquo The Maple TechnicalNewsletter no 9 pp 12ndash22 1993
[17] A S Apartsyn ldquoMultilinear Volterra equations of the first kindrdquoAutomation and Remote Control vol 65 no 2 pp 263ndash2692004
[18] A S Apartsyn ldquoPolilinear integral Volterra equations of thefirst kind the elements of the theory and numeric meth-odsrdquo Izvestiya Irkutskogo Gosudarstvennogo Universiteta SeriesMathematics no 1 pp 13ndash41 2007
[19] A S Apartsin ldquoOn the convergence of numerical methodsfor solving a Volterra bilinear equations of the first kindrdquoComputational Mathematics and Mathematical Physics vol 47no 8 pp 1323ndash1331 2007
[20] A S Apartsin ldquoMultilinear Volterra equations of the firstkind and some problems of controlrdquo Automation and RemoteControl vol 69 no 4 pp 545ndash558 2008
[21] A S Apartsyn ldquoUnimprovable estimates of solutions for someclasses of integral inequalitiesrdquo Journal of Inverse and Ill-PosedProblems vol 16 no 7 pp 651ndash680 2008
[22] A S Apartsyn ldquoPolynomial Volterra integral equations of thefirst kind and the Lambert functionrdquo Proceedings of the Instituteof Mathematics and Mechanics Ural Branch of RAS vol 18 no1 pp 69ndash81 2012 (Russian)
[23] D N Sidorov ldquoOn parametric families of solutions of Volterraintegral equations of the first kind with piecewise smoothkernelrdquo Differential Equations vol 49 no 2 pp 210ndash216 2013
[24] V Volterra Theory of Functionals and of Integral and Integro-Differential Equations Nauka Moscow Russia 1982 (Russian)
[25] N A Magnitsky ldquoThe existence of multiparameter familiesof solutions of a Volterra integral equation of the first kindrdquoReports of theUSSRAcademy of Sciences vol 235 no 4 pp 772ndash774 1977 (Russian)
[26] N A Magnitsky ldquoLinear Volterra integral equations of the firstand third kindsrdquo Computational Mathematics and Mathemati-cal Physics vol 19 no 4 pp 970ndash988 1979 (Russian)
[27] N A Magnitsky ldquoThe asymptotics of solutions to the Volterraintegral equation of the first kindrdquoReports of the USSRAcademyof Sciences vol 269 no 1 pp 29ndash32 1983 (Russian)
[28] N A Magnitsky Asymptotic Methods for Analysis of Non-Stationary Controlled Systems Nauka Moscow Russia 1992(Russian)
[29] A S Apartsyn ldquoOn one approach to modeling of developingsystemsrdquo in Proceedings of the 6th International WorkshopldquoGeneralized Statments and Solutions of Control Problemsrdquo pp32ndash35 Divnomorskoe Russia 2012
[30] A S Apartsin and I V Sidler ldquoUsing the nonclassical Volterraequations of the first kind to model the developing systemsrdquoAutomation and Remote Control vol 74 no 6 pp 899ndash9102013
119894 = 2 119899 minus 1(119870119894(119905 119904) + 119870
119894+1(119905 119904))
2 119904 = 119886
119894(119905) 119894 = 1 119899 minus 1
119870119899(119905 119904) 0 le 119904 lt 119886
119899minus1(119905)
(36)
To illustrate the fundamental difference between (35)(36) and classical Volterra equation of the first kind withsmooth kernel we confine ourselves to (20) that has the formof (35) at
119870 (119905 119904) =
1 120572119905 lt 119904 le 1199051 + 1205982
119904 = 120572119905120598 0 le 119904 lt 120572119905
(37)
where 120598 = 0 1 and 120572 isin (0 1) In particular at 120572 = 12 120598 = minus1
119870 (119905 119904) = sign(119904 minus 1199052) =
1 119904 gt 1199052
0 119904 = 1199052
minus1 119904 lt 1199052
(38)
For this case the solution to (35) with 119910(119905) = 119905 given in[23] is
If119870(119905 119904) is continuous in arguments and continuously differ-entiable with respect to 119905 inΔ then condition (40)means that(35) is Volterra integral equation of the third kind
The theory (whose foundation was laid by Volterra (see[24 pages 104ndash106])) of such equations is developed in theresearch done by Magnitsky [25ndash28]
In particular the author of [25ndash28] studies the structureof one- or many-parameter family of solutions to (35)
If 119870(119905 119904) is discontinuous then the solution to (35) maybe nonunique even if 119870(119905 119905) = 0 forall119905 ge 0
For example if120572 = 12 and 120598 = 1minus(1120572) = minus1 the solutionto equation
so that condition119870(119905 119905) equiv 1 is trueWe prove that solutions to (35) (37) and (35) (43)
coincide It suffices to show that the equivalent functionalequations for (35) (37) and (35) (43) coincide Recall that for(35) (37) the equivalent functional equation is (24)
Theorem 3 The equivalent functional equations for (35) (37)and (35) (43) coincide
Proof Let us represent (43) by
119870 (119905 119904) equiv 1 + (120598 minus 1) 119890 (120572119905 minus 119904) (44)
where 119890(sdot)ndash is a Heaviside function
119890 (]) = 1 ] ge 00 ] lt 0
(45)
Substitution of (44) in (35) gives
int119905
0
119909 (119904) 119889119904 + (120598 minus 1) int119905
0
119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = 119910 (119905)
119905 isin [0 119879] (46)
Transform the second integral Let ] = 120572119905 minus 119904 Then
int119905
0
119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = int120572119905
(120572minus1)119905
119890 (]) 119909 (120572119905 minus ]) 119889]
= int120572119905
0
119909 (120572119905 minus ]) 119889]
(47)
By virtue of (47) differentiation of (46) results in
119909 (119905) + (120598 minus 1) 120572119909 (0) + (120598 minus 1) int120572119905
0
1199091015840119905(120572119905 minus ]) 119889] = 1199101015840 (119905)
(48)
But
1199091015840119905(120572119905 minus ]) = minus1205721199091015840] (120572119905 minus ]) (49)
By virtue of (49) we have
119909 (119905) + (120598 minus 1) 120572119909 (0) minus (120598 minus 1) 120572 [119909 (120572119905 minus ])|1205721199050] = 1199101015840 (119905)
The solution to (35) (43) in the class of piecewisecontinuous functions with a jump on line 119904 = 120572119905 is interestingfrom the application perspective
It is easy to see that this solution is
119909 (119905 119904) =
1199101015840 (119904) 119904 ge 12057211990511205981199101015840 (119904) 119904 lt 120572119905
(52)
At last consider the concept of 120572-convolution Volterraintegral equations of convolution type
As is mentioned in the introduction the main results of thisstudy can be easily applied to the case 119899 gt 2 in (1) Theequations of type (1) not only are of theoretical interest butalso play an important role in the mathematical modeling ofdeveloping dynamic systemsMoreover by119910(119905) we canmeansome criterion that characterizes the level of developmentof the system as a whole and the 119894th term in (1) representsa contribution of the system components 119909(119904) of the 119894thage group whose operation is reflected by the efficiencycoefficient 119870
119894(119905 minus 119904) As a rule 119870
1ge sdot sdot sdot ge 119870
119899ge 0
Such an approach is implemented for instance in [29 30]in the problem of the analysis of strategies for the long-termexpansion of the Russian electric power system with theconsideration of aging of the power plants equipment
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author wishes to thank the reviewers for their helpfulnotes The study is supported by the Russian Foundation forBasic Research Grant no 12-01-00722a
References
[1] V Volterra ldquoSopra alcune questioni di inversione di integralidefinitirdquo Annali di Matematica Pura ed Applicata Series 2 vol25 no 1 pp 139ndash178 1897
[2] H Brunner ldquo1896ndash1996 One hundred years of Volterra integralequations of the first kindrdquoAppliedNumericalMathematics vol24 no 2-3 pp 83ndash93 1997
[3] H Brunner and P J van der HouwenTheNumerical Solution ofVolterra Equations vol 3 of CWI Monographs North-HollandAmsterdam The Netherlands 1986
[4] H Brunner Collocation Methods for Volterra Integral andRelated Functional Differential Equations vol 15 of CambridgeMonographs on Applied and Computational Mathematics Cam-bridge University Press Cambridge Mass USA 2004
[5] A S Apartsyn Nonclassical Linear Volterra Equations of theFirst Kind VSP Utrecht The Netherlands 2003
[6] V M Glushkov ldquoOn one class of dynamic macroeconomicmodelsrdquo Upravlyayushchiye Sistemy I Mashiny no 2 pp 3ndash61977 (Russian)
[7] V M Glushkov V V Ivanov and V M Yanenko Modeling ofDeveloping Systems Nauka Moscow Russia 1983 (Russian)
[8] Y P Yatsenko Integral Models of Systems with ControlledMemory Naukova Dumka Kiev Ukraine 1991 (Russian)
[9] N Hritonenko and Y Yatsenko Applied Mathematical Mod-elling of Engineering Problems vol 81 of Applied OptimizationKluwer Academic Publishers Dordrecht The Netherlands2003
[10] A S Apartsyn E V Markova and V V Trufanov IntegralModels of Electric Power System Development Energy SystemsInstitute SB RAS Irkutsk Russia 2002 (Russian)
6 Abstract and Applied Analysis
[11] D V Ivanov V Karaulova E VMarkova V V Trufanov andOV Khamisov ldquoControl of power grid development numericalsolutionsrdquo Automation and Remote Control vol 65 no 3 pp472ndash482 2004
[12] A S Apartsyn I V Karaulova E V Markova and V VTrufanov ldquoApplication of the Volterra integral equations for themodeling of strategies of technical re-equipment in the electricpower industryrdquo Electrical Technology Russia no 10 pp 64ndash752005 (Russian)
[13] E Messina E Russo and A Vecchio ldquoA stable numericalmethod for Volterra integral equations with discontinuouskernelrdquo Journal of Mathematical Analysis and Applications vol337 no 2 pp 1383ndash1393 2008
[14] L V Kantorovich and G P Akilov Functional Analysis NaukaMoscow Russia 1977 (Russian)
[15] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert 119882 functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996
[16] R M Corless G H Gonnet D E G Hare and D JJeffrey ldquoLambertrsquos 119882 function in Maplerdquo The Maple TechnicalNewsletter no 9 pp 12ndash22 1993
[17] A S Apartsyn ldquoMultilinear Volterra equations of the first kindrdquoAutomation and Remote Control vol 65 no 2 pp 263ndash2692004
[18] A S Apartsyn ldquoPolilinear integral Volterra equations of thefirst kind the elements of the theory and numeric meth-odsrdquo Izvestiya Irkutskogo Gosudarstvennogo Universiteta SeriesMathematics no 1 pp 13ndash41 2007
[19] A S Apartsin ldquoOn the convergence of numerical methodsfor solving a Volterra bilinear equations of the first kindrdquoComputational Mathematics and Mathematical Physics vol 47no 8 pp 1323ndash1331 2007
[20] A S Apartsin ldquoMultilinear Volterra equations of the firstkind and some problems of controlrdquo Automation and RemoteControl vol 69 no 4 pp 545ndash558 2008
[21] A S Apartsyn ldquoUnimprovable estimates of solutions for someclasses of integral inequalitiesrdquo Journal of Inverse and Ill-PosedProblems vol 16 no 7 pp 651ndash680 2008
[22] A S Apartsyn ldquoPolynomial Volterra integral equations of thefirst kind and the Lambert functionrdquo Proceedings of the Instituteof Mathematics and Mechanics Ural Branch of RAS vol 18 no1 pp 69ndash81 2012 (Russian)
[23] D N Sidorov ldquoOn parametric families of solutions of Volterraintegral equations of the first kind with piecewise smoothkernelrdquo Differential Equations vol 49 no 2 pp 210ndash216 2013
[24] V Volterra Theory of Functionals and of Integral and Integro-Differential Equations Nauka Moscow Russia 1982 (Russian)
[25] N A Magnitsky ldquoThe existence of multiparameter familiesof solutions of a Volterra integral equation of the first kindrdquoReports of theUSSRAcademy of Sciences vol 235 no 4 pp 772ndash774 1977 (Russian)
[26] N A Magnitsky ldquoLinear Volterra integral equations of the firstand third kindsrdquo Computational Mathematics and Mathemati-cal Physics vol 19 no 4 pp 970ndash988 1979 (Russian)
[27] N A Magnitsky ldquoThe asymptotics of solutions to the Volterraintegral equation of the first kindrdquoReports of the USSRAcademyof Sciences vol 269 no 1 pp 29ndash32 1983 (Russian)
[28] N A Magnitsky Asymptotic Methods for Analysis of Non-Stationary Controlled Systems Nauka Moscow Russia 1992(Russian)
[29] A S Apartsyn ldquoOn one approach to modeling of developingsystemsrdquo in Proceedings of the 6th International WorkshopldquoGeneralized Statments and Solutions of Control Problemsrdquo pp32ndash35 Divnomorskoe Russia 2012
[30] A S Apartsin and I V Sidler ldquoUsing the nonclassical Volterraequations of the first kind to model the developing systemsrdquoAutomation and Remote Control vol 74 no 6 pp 899ndash9102013
The solution to (35) (43) in the class of piecewisecontinuous functions with a jump on line 119904 = 120572119905 is interestingfrom the application perspective
It is easy to see that this solution is
119909 (119905 119904) =
1199101015840 (119904) 119904 ge 12057211990511205981199101015840 (119904) 119904 lt 120572119905
(52)
At last consider the concept of 120572-convolution Volterraintegral equations of convolution type
As is mentioned in the introduction the main results of thisstudy can be easily applied to the case 119899 gt 2 in (1) Theequations of type (1) not only are of theoretical interest butalso play an important role in the mathematical modeling ofdeveloping dynamic systemsMoreover by119910(119905) we canmeansome criterion that characterizes the level of developmentof the system as a whole and the 119894th term in (1) representsa contribution of the system components 119909(119904) of the 119894thage group whose operation is reflected by the efficiencycoefficient 119870
119894(119905 minus 119904) As a rule 119870
1ge sdot sdot sdot ge 119870
119899ge 0
Such an approach is implemented for instance in [29 30]in the problem of the analysis of strategies for the long-termexpansion of the Russian electric power system with theconsideration of aging of the power plants equipment
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author wishes to thank the reviewers for their helpfulnotes The study is supported by the Russian Foundation forBasic Research Grant no 12-01-00722a
References
[1] V Volterra ldquoSopra alcune questioni di inversione di integralidefinitirdquo Annali di Matematica Pura ed Applicata Series 2 vol25 no 1 pp 139ndash178 1897
[2] H Brunner ldquo1896ndash1996 One hundred years of Volterra integralequations of the first kindrdquoAppliedNumericalMathematics vol24 no 2-3 pp 83ndash93 1997
[3] H Brunner and P J van der HouwenTheNumerical Solution ofVolterra Equations vol 3 of CWI Monographs North-HollandAmsterdam The Netherlands 1986
[4] H Brunner Collocation Methods for Volterra Integral andRelated Functional Differential Equations vol 15 of CambridgeMonographs on Applied and Computational Mathematics Cam-bridge University Press Cambridge Mass USA 2004
[5] A S Apartsyn Nonclassical Linear Volterra Equations of theFirst Kind VSP Utrecht The Netherlands 2003
[6] V M Glushkov ldquoOn one class of dynamic macroeconomicmodelsrdquo Upravlyayushchiye Sistemy I Mashiny no 2 pp 3ndash61977 (Russian)
[7] V M Glushkov V V Ivanov and V M Yanenko Modeling ofDeveloping Systems Nauka Moscow Russia 1983 (Russian)
[8] Y P Yatsenko Integral Models of Systems with ControlledMemory Naukova Dumka Kiev Ukraine 1991 (Russian)
[9] N Hritonenko and Y Yatsenko Applied Mathematical Mod-elling of Engineering Problems vol 81 of Applied OptimizationKluwer Academic Publishers Dordrecht The Netherlands2003
[10] A S Apartsyn E V Markova and V V Trufanov IntegralModels of Electric Power System Development Energy SystemsInstitute SB RAS Irkutsk Russia 2002 (Russian)
6 Abstract and Applied Analysis
[11] D V Ivanov V Karaulova E VMarkova V V Trufanov andOV Khamisov ldquoControl of power grid development numericalsolutionsrdquo Automation and Remote Control vol 65 no 3 pp472ndash482 2004
[12] A S Apartsyn I V Karaulova E V Markova and V VTrufanov ldquoApplication of the Volterra integral equations for themodeling of strategies of technical re-equipment in the electricpower industryrdquo Electrical Technology Russia no 10 pp 64ndash752005 (Russian)
[13] E Messina E Russo and A Vecchio ldquoA stable numericalmethod for Volterra integral equations with discontinuouskernelrdquo Journal of Mathematical Analysis and Applications vol337 no 2 pp 1383ndash1393 2008
[14] L V Kantorovich and G P Akilov Functional Analysis NaukaMoscow Russia 1977 (Russian)
[15] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert 119882 functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996
[16] R M Corless G H Gonnet D E G Hare and D JJeffrey ldquoLambertrsquos 119882 function in Maplerdquo The Maple TechnicalNewsletter no 9 pp 12ndash22 1993
[17] A S Apartsyn ldquoMultilinear Volterra equations of the first kindrdquoAutomation and Remote Control vol 65 no 2 pp 263ndash2692004
[18] A S Apartsyn ldquoPolilinear integral Volterra equations of thefirst kind the elements of the theory and numeric meth-odsrdquo Izvestiya Irkutskogo Gosudarstvennogo Universiteta SeriesMathematics no 1 pp 13ndash41 2007
[19] A S Apartsin ldquoOn the convergence of numerical methodsfor solving a Volterra bilinear equations of the first kindrdquoComputational Mathematics and Mathematical Physics vol 47no 8 pp 1323ndash1331 2007
[20] A S Apartsin ldquoMultilinear Volterra equations of the firstkind and some problems of controlrdquo Automation and RemoteControl vol 69 no 4 pp 545ndash558 2008
[21] A S Apartsyn ldquoUnimprovable estimates of solutions for someclasses of integral inequalitiesrdquo Journal of Inverse and Ill-PosedProblems vol 16 no 7 pp 651ndash680 2008
[22] A S Apartsyn ldquoPolynomial Volterra integral equations of thefirst kind and the Lambert functionrdquo Proceedings of the Instituteof Mathematics and Mechanics Ural Branch of RAS vol 18 no1 pp 69ndash81 2012 (Russian)
[23] D N Sidorov ldquoOn parametric families of solutions of Volterraintegral equations of the first kind with piecewise smoothkernelrdquo Differential Equations vol 49 no 2 pp 210ndash216 2013
[24] V Volterra Theory of Functionals and of Integral and Integro-Differential Equations Nauka Moscow Russia 1982 (Russian)
[25] N A Magnitsky ldquoThe existence of multiparameter familiesof solutions of a Volterra integral equation of the first kindrdquoReports of theUSSRAcademy of Sciences vol 235 no 4 pp 772ndash774 1977 (Russian)
[26] N A Magnitsky ldquoLinear Volterra integral equations of the firstand third kindsrdquo Computational Mathematics and Mathemati-cal Physics vol 19 no 4 pp 970ndash988 1979 (Russian)
[27] N A Magnitsky ldquoThe asymptotics of solutions to the Volterraintegral equation of the first kindrdquoReports of the USSRAcademyof Sciences vol 269 no 1 pp 29ndash32 1983 (Russian)
[28] N A Magnitsky Asymptotic Methods for Analysis of Non-Stationary Controlled Systems Nauka Moscow Russia 1992(Russian)
[29] A S Apartsyn ldquoOn one approach to modeling of developingsystemsrdquo in Proceedings of the 6th International WorkshopldquoGeneralized Statments and Solutions of Control Problemsrdquo pp32ndash35 Divnomorskoe Russia 2012
[30] A S Apartsin and I V Sidler ldquoUsing the nonclassical Volterraequations of the first kind to model the developing systemsrdquoAutomation and Remote Control vol 74 no 6 pp 899ndash9102013
[11] D V Ivanov V Karaulova E VMarkova V V Trufanov andOV Khamisov ldquoControl of power grid development numericalsolutionsrdquo Automation and Remote Control vol 65 no 3 pp472ndash482 2004
[12] A S Apartsyn I V Karaulova E V Markova and V VTrufanov ldquoApplication of the Volterra integral equations for themodeling of strategies of technical re-equipment in the electricpower industryrdquo Electrical Technology Russia no 10 pp 64ndash752005 (Russian)
[13] E Messina E Russo and A Vecchio ldquoA stable numericalmethod for Volterra integral equations with discontinuouskernelrdquo Journal of Mathematical Analysis and Applications vol337 no 2 pp 1383ndash1393 2008
[14] L V Kantorovich and G P Akilov Functional Analysis NaukaMoscow Russia 1977 (Russian)
[15] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert 119882 functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996
[16] R M Corless G H Gonnet D E G Hare and D JJeffrey ldquoLambertrsquos 119882 function in Maplerdquo The Maple TechnicalNewsletter no 9 pp 12ndash22 1993
[17] A S Apartsyn ldquoMultilinear Volterra equations of the first kindrdquoAutomation and Remote Control vol 65 no 2 pp 263ndash2692004
[18] A S Apartsyn ldquoPolilinear integral Volterra equations of thefirst kind the elements of the theory and numeric meth-odsrdquo Izvestiya Irkutskogo Gosudarstvennogo Universiteta SeriesMathematics no 1 pp 13ndash41 2007
[19] A S Apartsin ldquoOn the convergence of numerical methodsfor solving a Volterra bilinear equations of the first kindrdquoComputational Mathematics and Mathematical Physics vol 47no 8 pp 1323ndash1331 2007
[20] A S Apartsin ldquoMultilinear Volterra equations of the firstkind and some problems of controlrdquo Automation and RemoteControl vol 69 no 4 pp 545ndash558 2008
[21] A S Apartsyn ldquoUnimprovable estimates of solutions for someclasses of integral inequalitiesrdquo Journal of Inverse and Ill-PosedProblems vol 16 no 7 pp 651ndash680 2008
[22] A S Apartsyn ldquoPolynomial Volterra integral equations of thefirst kind and the Lambert functionrdquo Proceedings of the Instituteof Mathematics and Mechanics Ural Branch of RAS vol 18 no1 pp 69ndash81 2012 (Russian)
[23] D N Sidorov ldquoOn parametric families of solutions of Volterraintegral equations of the first kind with piecewise smoothkernelrdquo Differential Equations vol 49 no 2 pp 210ndash216 2013
[24] V Volterra Theory of Functionals and of Integral and Integro-Differential Equations Nauka Moscow Russia 1982 (Russian)
[25] N A Magnitsky ldquoThe existence of multiparameter familiesof solutions of a Volterra integral equation of the first kindrdquoReports of theUSSRAcademy of Sciences vol 235 no 4 pp 772ndash774 1977 (Russian)
[26] N A Magnitsky ldquoLinear Volterra integral equations of the firstand third kindsrdquo Computational Mathematics and Mathemati-cal Physics vol 19 no 4 pp 970ndash988 1979 (Russian)
[27] N A Magnitsky ldquoThe asymptotics of solutions to the Volterraintegral equation of the first kindrdquoReports of the USSRAcademyof Sciences vol 269 no 1 pp 29ndash32 1983 (Russian)
[28] N A Magnitsky Asymptotic Methods for Analysis of Non-Stationary Controlled Systems Nauka Moscow Russia 1992(Russian)
[29] A S Apartsyn ldquoOn one approach to modeling of developingsystemsrdquo in Proceedings of the 6th International WorkshopldquoGeneralized Statments and Solutions of Control Problemsrdquo pp32ndash35 Divnomorskoe Russia 2012
[30] A S Apartsin and I V Sidler ldquoUsing the nonclassical Volterraequations of the first kind to model the developing systemsrdquoAutomation and Remote Control vol 74 no 6 pp 899ndash9102013