a numerical solution of one class of volterra integral ... - Mathnet.ru

MSC 65R20

DOI: 10.14529/mmp160310

A NUMERICAL SOLUTION OF ONE CLASS OF VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND IN TERMS OF THE MACHINE ARITHMETIC FEATURES Melentiev Energy Systems Institute SB RAS, Irkutsk, Russian Federation, [email protected], I.V. Mokry, Melentiev Energy Systems Institute SB RAS, Irkutsk, Russian Federation, [email protected] S.V. Solodusha,

The research is devoted to a numerical solution of the Volterra equations of the rst kind that were obtained using the Laplace integral transforms for solving the equation of heat conduction. The paper consists of an introduction and two sections. The rst section deals with the calculation of kernels from the respective integral equations at a xed length of the signicand in the oating point representation of a real number. The PASCAL language was used to develop the software for the calculation of kernels, which implements the function of tracking the valid digits of the signicand. The test examples illustrate the typical cases of systematic error accumulation. The second section presents the results obtained from the computational algorithms which are based on the product integration method and the midpoint rule. The results of test calculations are presented to demonstrate the performance of the dierence methods. Keywords: Volterra integral equations of the rst kind; numerical solution; product integration method.

Introduction The paper is devoted to the studies of a special class of Volterra integral equations of the rst kind ∫t KN (t − s)ϕ(s)ds = y(t), (1) 0

KN (t − s) =

N ∑

(−1)q+1 q 2 e−π

2 q 2 (t−s)

.

(2)

q=1

The specic feature of kernel KN (t − s) in (2) lies in that KN = 0 in some neighborhood of zero. The qualitative theory and numerical methods of solving the Volterra equations of the rst kind are dealt with in many studies (for example, [14] and the eqreferences given in them). The goal of this paper is to consider the application of numerical methods for solving the equations of form (1), (2) taking into account the mechanisms of error occurrence in the computer calculations. The work is a continuation of the research started in [5, 6]. The Volterra equation of the convolution type (1), (2) was rst obtained in [7]. The authors of [7] suggest a method of searching for a solution u(1, t) = ϕ(t), t ≥ 0, of an inverse boundary-value problem

ut = uxx ,

x ∈ (0, 1) ,

t > 0,

Âåñòíèê ÞÓðÃÓ. Ñåðèÿ ≪Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå è ïðîãðàììèðîâàíèå≫ (Âåñòíèê ÞÓðÃÓ ÌÌÏ). 2016. Ò. 9, 3. Ñ. 119129

(3) 119

S.V. Solodusha, I.V. Mokry

u (x, 0) = 0, u (0, t) = 0, ux (0, t) = g0 (t)

(4)

by reducing (3), (4) to the Volterra integral equation of convolution type:

∫t K(t − s)ϕ(s)ds = y(t), 0 ≤ s ≤ t ≤ T,

(5)

0

K(t − s) =

∞ ∑

(−1)q+1 q 2 e−π

2 q 2 (t−s)

q=1

, y(t) =

1 g0 (t) . 2π 2

(6)

Instead of g0 (t) we normally know gδ (t): ∥gδ (t) − g0 (t)∥C 6 δ , δ > 0. Problem (3), (4) plays an important part in the applied problems, including those related to the research of non-stationary thermal processes. Solving of the inverse problems is, as a rule, complicated by the instability of these problems with respect to the initial data errors. The application of the methods which employ the Fourier and Laplace transforms in combination with the theory of ill-posed problems found wide application in the construction of stable solutions to the inverse problems of heat conductance. In particular, to solve the problem similar to (3), (4), the authors of [8] used a stabilizing functional after applying the Fourier transform. In [9] to regularize and assess the convergence of the obtained solutions the authors used a method of conjugate gradients. In [10] and [11] the Laplace transform was considered for solving the Cauchy problem. In [12] the Laplace transform was used in a two-dimensional problem. The existing approaches, as a rule, after taking the Laplace transform, apply regularization methods to the obtained equations and then perform the inverse transform. Taking into account the ideas from [13, 14] the authors of [7] approximated ux (0, t) by the sum of N rst summands: ∫t N ∑ 2 2 q+1 2 2 e−π q (t−s) ϕ(s)ds, ux (0, t) = 2π (−1) q q=1

0

where N is a positive integer. Then (5), (6) are reduced to the form (1), (2). The performance of this approach was discussed in [7]. According to the conclusion made by the author, such a way of solving of the inverse problem makes it possible to reduce the initial problem to the Volterra integral equation of the rst kind and exclude the components of the operator calculus from the regularization process. It is known that the Volterra integral equations of the rst kind belong to the class of conditionally-correct problems, and the discretization procedure has a regularizing feature with a regularization parameter, a step of mesh, which is in a certain manner connected to the level of disturbances of the initial data δ . This paper presents an algorithm for numerical solution of (1), (2) at an exactly specied right-hand part. Note that when solving (1), (2) we face three types of errors related rst of all to the approximation of the initial problem (5), (6), secondly to the accuracy of a numerical method, and nally to the computation errors in the machine arithmetic operations with real oating-point numbers. For the research, of greatest interest is the rst of the indicated cases. However, to pass to the problem of assessing the parameter N in (2) it is necessary to develop an algorithm for computation of KNmax , which takes into account the specic features of machine arithmetic and provides the desired (specied) number of valid digits in the signicand. 120

Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming & Computer Software (Bulletin SUSU MMCS), 2016, vol. 9, no. 3, pp. 119129

ÏÐÎÃÐÀÌÌÈÐÎÂÀÍÈÅ

1. The Specic Features of the Numerical Calculations The computational experiment in [5] shows that with an increase in the number of summands in (2) the roots λ∗ of the equation KN (λ) = 0 decrease (at the same time monotonisity is observed only separately in even and odd N ). The values λ∗ will be used further to limit the magnitude of the mesh step h from above, for the value of the mesh h function KN at the rst node to be non-zero. As is known, the condition KN (0) ̸= 0 is 1 ′ necessary for (1), (2) to be correct on the pair (C, C[0,T ] ), where y(0) = 0, y (t) ∈ C[0,T ] . ∗ Note, that there can be computational errors in the calculation of λ , they can be related to the application of a xed mesh in the machine number representation. Let us consider the known cases of systematic error accumulation [15] which appear in the calculation of kernel values (2). Use the system of computer algebra Maple10. Following [16] we will include parameter f in the generally accepted representation of the real number. The parameter is equal to the number of valid digits in the signicand (starting from the left). Assume that the real number x = s · M · 10−L+p is specied by the set (s, M, p, f ), where s ∈ {−1, 0, +1} is the sign of the number, M ∈ {10L−1 , 10L−1 + 1, ..., 10L − 1} ∪ {0} is signicand of the number, L is number of signicand positions, p is exponent part of the number. Let us illustrate the details of the calculations when summing up the numbers with dierent exponent parts in (2), using the example. Example 1.

Let N = 50, λ0 = 10−3 , L ≥ 8. Take

x1 =

34 ∑

(−1)

q+1

2 −π 2 q 2 λ0

q e

, x2 =

q=11

50 ∑

(−1)q+1 q 2 e−π

2 q2 λ

0

q=35

and nd xΣ = x1 + x2 . Assuming

10pΣ −fΣ = 10p1 −f1 + 10p2 −f2 , pΣ ≥ p1 ≥ p2 ,

according to [16], it is easy to obtain that

fΣ ≥ fs = [f1 − lg(1 + 10−p1 +f1 +p2 −f2 )],

(7)

where the symbol [...] means the greatest integer. Table 1 presents the parameters (1, M1 , 2, f1 ), (1, M2 , −2, f2 ) and (1, MΣ , 2, fΣ ), which dene x1 , x2 and xΣ respectively. The last column of the table shows the values of minorant fs , that are calculated using (7). The next example illustrates the situation arising in the calculation of the dierence between the numbers which have coinciding exponent parts and several high-order digits of the signicand. Example 2.

Let N = 50, λ0 = 10−3 , L ≥ 8. Introduce

x3 =

10 ∑ q=1

(−1)

q+1

2 −π 2 q 2 λ0

q e

, x4 =

50 ∑

(−1)q+1 q 2 e−π

2 q2 λ

0

.

q=11

Dene x∆ = |x4 | − |x3 |. Âåñòíèê ÞÓðÃÓ. Ñåðèÿ ≪Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå è ïðîãðàììèðîâàíèå≫ (Âåñòíèê ÞÓðÃÓ ÌÌÏ). 2016. Ò. 9, 3. Ñ. 119129

121


Table 1

Values M and f for x1 , x2 and xΣ

L 8 9 10 11 12 13 14

M1 18652239 186522441 1865224455 18652244592 186522445926 1865224459248 18652244592468

f1 6 8 9 11 11 12 13

M2 f2 MΣ fΣ 44981421 6 18656737 6 449814458 7 186567422 8 4498144699 8 1865674269 8 44981446726 8 18656742737 11 449814466957 10 186567427373 11 4498144669376 10 1865674273715 12 44981446694089 11 18656742737137 13

fs 5 7 8 10 10 11 12

Suppose that

|M4 − M3 | < 10L−1 − 1, p∆ ≤ p3 = p4 and, following [16] use the empiric estimate:

{ f∆ ≥ fr =

0, if fmin − L + lg(λ) ≤ 0, [fmin − L + lg(λ)] , if fmin − L + lg(λ) > 0,

(8)

where λ = |M4 − M3 | + 1, fmin = min{f3 , f4 }. Below are the parameters (−1, M3 , 2, f3 ), (1, M4 , 2, f4 ) and (−1, M∆ , 2, f∆ ), which specify the values x3 , x4 and x∆ . The estimation from below of fr , obtained using (8), is given in the last column of Table 2. Table 2

Values M and f for x3 , x4 and x∆

L 8 9 10 11 12 13 14

M3 18656743 186567428 1865674274 18656742750 186567427505 1865674275054 18656742750534

f3 7 8 9 11 12 12 14

M4 18656737 186567424 1865674268 18656742736 186567427372 1865674273715 18656742737138

f4 6 8 8 10 11 12 13

M∆ f∆ 00000006 0 000000004 0 0000000006 0 00000000014 1 000000000133 3 0000000001339 4 00000000013396 4

fr 0 0 0 0 1 2 3

It is obvious that when several high-order digits are set to zero, there appears the number with lower quantity of signicant digits in the signicand. Figure 1 illustrates an instantaneous loss of high-order digits, which takes place in this situation. The calculations are made using the software developed by the authors in PASCAL. The next section is devoted to the problem of approximate solving of (1), (2) in terms of the specic features of machine arithmetic. 122



. Instantaneous loss of high-order digits for KN (0.001), where N = 12

Fig. 1

2. Results of Solving of

,

(1) (2)

Let us introduce the uniform meshes of nodes ti = ih, ti− 1 = (i − 21 )h, i = 1, n, nh = T 2 in [0, T ] and, by approximating the integral in (1) by the middle rectangle quadrature and product integration method [17], write the corresponding mesh analogues to (1), (2) as

h

N ∑

q+1

(−1)

q

2

q=1

i ∑

e−π

(−1)

ϕhi−j+ 1 = yih 2

j=1

and N ∑

2 q 2 (j− 1 )h 2

q+1

q

2

i ∑

q=1

j=1

∫jh

e−π

ϕhj− 1 2

2 q 2 (ih−s)

ds = yih .

(9)

(10)

(j−1)h

Designate their solutions by ϕˇh and ϕˆh , respectively. Conduct a numerical experiment with the guaranteed number of valid digits in the signicand not less than f5 = 8. Example 3.

Assume ϕ (t) from [18] as the reference function:

1 − e− α t

ϕ (t) =

1 − e− α 1

− t,

where α = 10−1 , 10−2 . Set in (9), (10) N = 2, 5; 10; 15. Table 3, 4 give the results of numerical calculations in the integration interval [0, 1]. Here, the notation is as follows:

γ1 = log2

∥ˇ εh1 ∥Ch ∥ˆ εh1 ∥Ch , γ = log , 2 2 ∥ˇ εh2 ∥Ch ∥ˆ εh2 ∥Ch


123


where h1 = 2h2 , ∥ˇ εh ∥Ch is maximum absolute dierence between the precise solution and the approximate solution at the nodal points (the approximate solution is obtained using the middle rectangle quadrature); ∥ˆ εh ∥Ch is maximum absolute dierence between the accurate solution and the approximate solution at the nodal points (the approximate solution is obtained using the product integration method); symbol ∗ means that the error norm is higher than max |ϕ (t) |. 0≤t≤1

Table 3

Errors in the mesh solution for function ϕ, where α = 10−1

h 1/256 1/512 1/1024 1/2048 h 1/256 1/512 1/1024 1/2048 h 1/256 1/512 1/1024 1/2048

=2 ||ˇ ε||N Ch 0,005001 0,001242 0,000310 0,000078 =4 ||ˇ ε||N Ch 0,065009 0,015090 0,003707 0,000923 =10 ||ˇ ε||N Ch ∗ ∗ 0,137360 0,029650

γ1 2,009 2,002 2,000 1,981 γ1 2,107 2,025 2,006 1,999 γ1 2,212 2,047

=2 ||ˆ ε||N Ch 0,000499 0,000125 0,000031 0,000008 =4 ||ˆ ε||N Ch 0,002056 0,000519 0,000129 0,000032 =10 ||ˆ ε||N Ch 0,009531 0,002845 0,000751 0,000190

γ2 1,996 1,998 1,999 1,999 γ2 1,987 1,996 1,999 2,000 γ2 1,744 1,922 1,979 1,995

=3 ||ˇ ε||N Ch 0,003815 0,000952 0,000238 0,000059 =5 ||ˇ ε||N Ch 0,025682 0,006377 0,001591 0,000398 =15 ||ˇ ε||N Ch ∗ 0,485248 0,109395 0,026724

γ1 2,002 2,000 2,000 1,989 γ1 2,010 2,002 2,000 1,996 γ1 2,149 2,033 2,008


γ2 1,994 1,998 1,999 2,001 γ2 1,973 1,993 1,998 2,000 γ2 1,394 1,719 1,910 1,957 Table 4

Errors in the mesh solution for function ϕ, where α = 10−2

h 1/256 1/512 1/1024 1/2048 h 1/256 1/512 1/1024 1/2048 h 1/256 1/512 1/1024 1/2048 124

=2 ||ˇ ε||N Ch 0,006113 0,001518 0,000379 0,000095 =4 ||ˇ ε||N Ch 0,080125 0,018474 0,004532 0,001128 =10 ||ˇ ε||N Ch ∗ ∗ 0,170835 0,036370

γ1 2,009 2,002 1,999 1,985 γ1 2,117 2,027 2,006 2,000 γ1 2,232 2,052


=3 γ2 ||ˇ ε||N γ1 Ch 1,995 0,007855 2,004 1,998 0,001958 2,001 1,992 0,000489 2,000 1,927 0,000122 1,998 =5 γ2 ||ˇ ε||N γ1 Ch 1,859 0,051629 2,022 1,934 0,012716 2,006 1,968 0,003167 2,001 1,984 0,000791 2,000 =15 γ2 ||ˇ ε||N γ1 Ch 1,584 ∗ 1,849 ∗ 1,945 0,2227532 2,073 1,978 0,0529318 2,018


γ2 1,875 1,939 1,970 1,985 γ2 1,840 1,928 1,966 1,985 γ2 1,214 1,637 1,874 1,959



Tables show that both nite dierence methods have the second order of convergence. Figures 2, and 3 illustrate the behavior of functions |ˇ εhi | = |ϕi− 1 − ϕˇhi− 1 |, |ˆ εhi | = |ϕi− 1 − ϕˆhi− 1 | 2

2

2

2

1 on a unied mesh with the step h = 27 at xed values N = 2, α = 10−1 , T = 1. Plot "Line 1" corresponds to function |ˆ εhi |, plot "Line 2" corresponds to function |ˇ εhi | and plot "Line 3" corresponds to function |εhimin | = min{|ˇ εhi | , |ˆ εhi |}.

. Absolute values of errors in mesh solutions at exactly specied initial data

Fig. 2

Figure 2 was obtained at functions yih accurately specied in (9), (10). Here, the maximum values of errors made up =2 =2 ||ˇ ε||N ε||N C h = 0, 0795, ||ˆ C h = 0, 0424.

. Absolute values of errors in mesh solutions at perturbed initial data

Fig. 3

Figure 3 is obtained at a saw-tooth perturbance of mesh function yih :

y˜(ti ) = y(ti ) + (−1)i 10−2 ,

i = 1, 27, nh = 1.


125


In this case, the maximum values of errors made up =2 N =2 =2 ε||N ||ˇ ε||N C h = 0, 0638, ||ˆ C h = 0, 0609, ||εmin ||C h = 0, 0582.

Step h for the xed level of the initial data perturbances was chosen using the Fibonacci method with ten trials. Remark.

The comparison of ∥ˇ εh ∥Ch and ∥ˆ εh ∥Ch makes it obvious that the use of the product integration method is more preferable. The computational experiment conducted for the given example at α = 10−3 shows the convergence for h < 10−3 . In this connection, further studies are supposed to use the numerical methods of higher order, in particular the third- and fourth-order Runge-Kutta methods. Further, it is planned to construct stability regions of the considered algorithms by the analogy with [19].

Conclusion The paper presents the research of the approximate solution to the Volterra integral equation of the rst kind of convolution type, which occurs in the inverse boundary-value heat conduction problem, by the second-order nite dierence methods. The calculation results obtained using the system Maple 10 are presented. The computational experiment is conducted in terms of the error occurrence mechanism in computer calculations. Typical cases of systematic error accumulation are demonstrated by the test examples. Software for the calculation of kernels, tracking the valid digits in the signicand, is developed in PASCAL. Acknowledgements.

The

work

was

supported

by

the

RFBR,

project

No. 15-01-01425-à.

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ÓÄÊ 517.968

DOI: 10.14529/mmp160310

×ÈÑËÅÍÍÎÅ ÐÅØÅÍÈÅ ÎÄÍÎÃÎ ÊËÀÑÑÀ ÈÍÒÅÃÐÀËÜÍÛÕ ÓÐÀÂÍÅÍÈÉ ÂÎËÜÒÅÐÐÀ I ÐÎÄÀ Ñ Ó×ÅÒÎÌ ÎÑÎÁÅÍÍÎÑÒÅÉ ÌÀØÈÍÍÎÉ ÀÐÈÔÌÅÒÈÊÈ Ñ.Â. Ñîëîäóøà, È.Â. Ìîêðûé

Ñòàòüÿ ïîñâÿùåíà ïðèáëèæåííîìó ðåøåíèþ óðàâíåíèé Âîëüòåððà I ðîäà, ïîëó÷åííûõ â ðåçóëüòàòå ïðèìåíåíèÿ èíòåãðàëüíûõ ïðåîáðàçîâàíèé Ëàïëàñà äëÿ ðåøåíèÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè. Ðàáîòà ñîñòîèò èç ââåäåíèÿ è äâóõ ðàçäåëîâ. Ïåðâûé ðàçäåë ïîñâÿùåí ñïåöèôèêå âû÷èñëåíèÿ ÿäåð Âîëüòåððà èç ñîîòâåòñòâóþùèõ èíòåãðàëüíûõ óðàâíåíèé ïðè ôèêñèðîâàííîé äëèíå ìàíòèññû â ìàøèííîì ïðåäñòàâëåíèè âåùåñòâåííîãî ÷èñëà ñ ïëàâàþùåé òî÷êîé. Íà ÿçûêå PASCAL ðàçðàáîòàíî ïðîãðàììíîå îáåñïå÷åíèå äëÿ âû÷èñëåíèÿ ÿäåð, ðåàëèçóþùåå ôóíêöèþ îòñëåæèâàíèÿ äîñòîâåðíûõ ðàçðÿäîâ ìàíòèññû. Íà òåñòîâûõ ïðèìåðàõ ïðîèëëþñòðèðîâàíû òèïîâûå ñëó÷àè ñèñòåìàòè÷åñêîãî íàêîïëåíèÿ îøèáîê. Âî âòîðîì ðàçäåëå ïðèâåäåíû ðåçóëüòàòû ðàñ÷åòíûõ àëãîðèòìîâ, îñíîâàííûõ íà ìåòîäå èíòåãðèðîâàíèÿ ïðîèçâåäåíèÿ è êâàäðàòóðå ñðåäíèõ ïðÿìîóãîëüíèêîâ. Ñ öåëüþ ñðàâíåíèÿ ýôôåêòèâíîñòè ðàçíîñòíûõ ìåòîäîâ ïðèâåäåíû ðåçóëüòàòû òåñòîâûõ ðàñ÷åòîâ. Êëþ÷åâûå ñëîâà: èíòåãðàëüíûå óðàâíåíèÿ Âîëüòåððà ïåðâîãî ðîäà; ÷èñëåííîå ðåøåíèå; ìåòîä èíòåãðèðîâàíèÿ ïðîèçâåäåíèÿ.

Ëèòåðàòóðà 1. Brunner, H. The Numerical Solution of Volterra Equations / H. Brunner, P.J. van der Houwen. Amsterdam: North-Holland, 1986. 2. Brunner, H. Collocation Methods for Volterra Integral and Related Functional Dierential Equations / H. Brunner. N.Y.: Cambridge Univ. Press, 2004. 3. Âåðëàíü, À.Ô. Èíòåãðàëüíûå óðàâíåíèÿ: ìåòîäû, àëãîðèòìû, ïðîãðàììû / À.Ô. Âåðëàíü, Â.Ñ. Ñèçèêîâ. Êèåâ: Íàóê. äóìêà, 1986. 4. Àïàðöèí, À.Ñ. Íåêëàññè÷åñêèå óðàâíåíèÿ Âîëüòåððà I ðîäà: òåîðèÿ è ÷èñëåííûå ìåòîäû / À.Ñ. Àïàðöèí. Íîâîñèáèðñê: Íàóêà, 1999. 5. Ñîëîäóøà, Ñ.Â. Ïðèìåíåíèå ÷èñëåííûõ ìåòîäîâ äëÿ óðàâíåíèé Âîëüòåððà I ðîäà, âîçíèêàþùèõ â îáðàòíîé ãðàíè÷íîé çàäà÷å òåïëîïðîâîäíîñòè / Ñ.Â. Ñîëîäóøà // Èçâåñòèÿ ÈÃÓ. Ñåðèÿ: Ìàòåìàòèêà. 2015. Ò. 11. Ñ. 96105. 6. Ñîëîäóøà, Ñ.Â. ×èñëåííîå ðåøåíèå îáðàòíîé ãðàíè÷íîé çàäà÷è òåïëîïðîâîäíîñòè ñ ïîìîùüþ óðàâíåíèé Âîëüòåððà I ðîäà / Ñ.Â. Ñîëîäóøà, Í.Ì. ßïàðîâà // Ñèáèðñêèé æóðíàë âû÷èñëèòåëüíîé ìàòåìàòèêè. 2015. Ò. 18, 3. Ñ. 321329. 7. ßïàðîâà, Í.Ì. ×èñëåííîå ìîäåëèðîâàíèå ðåøåíèé îáðàòíîé ãðàíè÷íîé çàäà÷è òåïëîïðîâîäíîñòè / Í.Ì. ßïàðîâà // Âåñòíèê ÞÓðÃÓ. Ñåðèÿ: Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå è ïðîãðàììèðîâàíèå. 2013. Ò. 6, 3. Ñ. 112124. 8. Jonas, P. Approximate Inverse for a One Dimensional Inverse Heat Conduction Problem / P. Jonas, A.K. Louis // Inverse Problems. 2000. V. 16, 1. P. 175185. 9. Prud'homme, M. Fourier Analysis of Conjugate Gradient Method Applied to Inverse Heat Conduction Problems / M. Prud'homme, T.H. Hguyen // International Journal of Heat and Mass Transfer. 1999. V. 42. P. 44474460.

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ÏÐÎÃÐÀÌÌÈÐÎÂÀÍÈÅ 10. Kolodziej, J. Application of the Method of Fundamental Solutions and Radial Basis Functions for Inverse Heat Source Problem in Case of Steady-State / J. Kolodziej, M. Mierzwiczak, M. Cialkowski // International Communications in Heat and Mass Transfer. 2010. V. 37, 2. P. 121124. 11. Cialkowski, M. A Sequential and Global Method of Solving an Inverse Problem of Heat Conduction Equation / M. Cialkowski, K. Grysa // Jornal of Theoretical and Applied Mechanics. 2010. V. 48, 1. P. 111134. 12. An Analytical Solution for Two-Dimensional Inverse Heat Conduction Problems Using Laplace Transform / M. Monde, H. Arima, Wei Liu, Yuhichi Mitutake, J.A. Hammad // International Journal of Heat and Mass Transfer. 2003. V. 46. P. 21352148. 13. Beilina, L. Approximate Global Convergence and Adaptivity for Coecient Inverse Problems / L. Beilina, M.V. Klibanov. N.Y.: Springer, 2012. 14. Kabanikhin, S.I. Inverse and Ill-Posed S.I. Kabanikhin. De Gruyter, 2011.

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15. Êàëèòêèí, Í.Í. ×èñëåííûå ìåòîäû / Í.Í. Êàëèòêèí. Ì.: Íàóêà, 1978. 16. Ìîêðûé, È.Â. Îñíîâíûå ìåõàíèçìû âîçíèêíîâåíèÿ âû÷èñëèòåëüíîé îøèáêè ïðè êîìïüþòåðíûõ ðàñ÷åòàõ / È.Â. Ìîêðûé, Î.Â. Õàìèñîâ, À.Ñ. Öàïàõ // Ìàòåðèàëû IV Âñåðîññ. êîíô. ≪Ïðîáëåìû îïòèìèçàöèè è ýêîíîìè÷åñêèå ïðèëîæåíèÿ≫. Îìñê: Íàñëåäèå, 2009. Ñ. 185. 17. Linz, P. Product Integration Method for Volterra Integral Equations of the First Kind / P. Linz // BIT Numerical Mathematics. 1971. V. 11. P. 413421. 18. Geng, F.Z. Analytical Approximation to Solutions of Singularly Perturbed Boundary Value Problems / F.Z. Geng, M.G. Cui // Bulletin of the Malaysian Mathematical Sciences Society. 2010. V. 33, 2. P. 221232. 19. Áóëàòîâ, Ì.Â. Èññëåäîâàíèå ìíîãîøàãîâûõ ìåòîäîâ äëÿ ðåøåíèÿ èíòåãðîàëãåáðàè÷åñêèõ óðàâíåíèé: ïîñòðîåíèå îáëàñòåé óñòîé÷èâîñòè / Ì.Â. Áóëàòîâ, Î.Ñ. Áóäíèêîâà // Æóðíàë âû÷èñëèòåëüíîé ìàòåìàòèêè è ìàòåìàòè÷åñêîé ôèçèêè. 2013. Ò. 53, 9. Ñ. 14481459.

Ñâåòëàíà Âèòàëüåâíà Ñîëîäóøà, êàíäèäàò ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, çàâåäóþùèé ëàáîðàòîðèåé, ëàáîðàòîðèÿ ≪Íåóñòîé÷èâûå çàäà÷è âû÷èñëèòåëüíîé ìàòåìàòèêè≫, Èíñòèòóò ñèñòåì ýíåðãåòèêè èì. Ë.À. Ìåëåíòüåâà ÑÎ ÐÀÍ (ã. Èðêóòñê, Ðîññèéñêàÿ Ôåäåðàöèÿ), [email protected]. Èãîðü Âëàäèìèðîâè÷ Ìîêðûé, êàíäèäàò òåõíè÷åñêèõ íàóê, ñòàðøèé íàó÷íûé ñîòðóäíèê, ëàáîðàòîðèÿ ≪Ìåòîäû ìàòåìàòè÷åñêîãî ìîäåëèðîâàíèÿ è îïòèìèçàöèè â ýíåðãåòèêå≫, Èíñòèòóò ñèñòåì ýíåðãåòèêè èì. Ë.À. Ìåëåíòüåâà ÑÎ ÐÀÍ (ã. Èðêóòñê, Ðîññèéñêàÿ Ôåäåðàöèÿ), [email protected]. Ïîñòóïèëà â ðåäàêöèþ 3 àâãóñòà 2015 ã.


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