A numerical solution of one class of Volterra integral equations of the first kind in terms of the machine arithmetic features Текст научной статьи по специальности «Математика»

MSC 65R20

DOI: 10.14529/mmp 160310

A NUMERICAL SOLUTION OF ONE CLASS OF VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND IN TERMS OF THE MACHINE ARITHMETIC FEATURES

S. V. Solodusha, Melentiev Energy Systems Institute SB EAS, Irkutsk, Russian Federation, solodusha@isem.irk.ru,

I. V. Mokry, Melentiev Energy Systems Institute SB EAS, Irkutsk, Russian Federation, ygr@isem.irk.ru

The research is devoted to a numerical solution of the Volterra equations of the first 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 first section deals with the calculation of kernels from the respective integral equations at a fixed length of the significand in the floating 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 significand. 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 difference methods.

Keywords: Volterra integral equations of the first kind; numerical solution; product integration method.

Introduction

The paper is devoted to the studies of a special class of Volterra integral equations of the first kind

t

J KN(t — s)$(s)ds = y(t), (1)

0

N

Kn(t — s) = J2 (—l)q+1 q2e-n2q2(t~s). (2)

q=1

The specific 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 first kind are dealt with in many studies (for example, [1-4] 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 first obtained in [7]. The authors of [7] suggest a method of searching for a solution u(1,t) = $(t), t > 0, of an inverse boundary-value problem

ut = Uxx, x £ (0,1), t > 0, (3)

u (x, 0) = 0, u (0, t) = 0, ux (0, t) = go (t) (4)

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

t

\K(t — s)0(s)ds = y(t), 0 < s < t < T, (5)

K(t— s) = YJ(-i)q+1 q2e-n2*(t-'\ y(t) = ngo(t). (6)

q=i

Instead of g0(t) we normally know gs(t): \\gs(t) — g0(t)\\C ^ 5, 5 > 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 first summands:

N lf.

Ux(0,t) = (—l)q+l q2 e-n2q2(t-s)0(s)ds,

q=1 o

N

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 first kind and exclude the components of the operator calculus from the regularization process.

It is known that the Volterra integral equations of the first 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 5.

This paper presents an algorithm for numerical solution of (1), (2) at an exactly specified right-hand part. Note that when solving (1), (2) we face three types of errors related first of all to the approximation of the initial problem (5), (6), secondly to the accuracy of a numerical method, and finally to the computation errors in the machine arithmetic operations with real floating-point numbers. For the research, of greatest interest is the first 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 specific features of machine arithmetic and provides the desired (specified) number of valid digits in the significand.

1. The Specific Features of the Numerical Calculations

The computational experiment in [5] shows that with an increase in the number of summands in (2) the roots A* of the equation KN(A) = 0 decrease (at the same time monotonisity is observed only separately in even and odd N). The values A* will be used further to limit the magnitude of the mesh step h from above, for the value of the mesh function KN at the first node to be non-zero. As is known, the condition KN (0) = 0 is

necessary for (1), (2) to be correct on the pair (C, C^ Tj), where y(0) = 0 y'(t) £ C[0)Tj.

A*

related to the application of a fixed 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 MaplelO. 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 significand (starting from the left). Assume that the real number x = s • M • 10-L+p is specified 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} U {0} is significand of the number, L is number of significand positions, p is

exponent part of the number.

Let us illustrate the details of the calculations when summing up the numbers with different exponent parts in (2), using the example.

Example 1. Let N = 50, Ao = 10-3, L > 8. Take

34 50

xi = £ (-1)q+1 q2e-n2i2x°, x2 (-1)q+1 q2e

q=11 q=35

and find xs = x1 + x2. Assuming

10p£-/E = 10pi-fl + 10P2-/2, ps > p1 > p2,

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

fs > fs = [fi — lg(1 + 10-Pl+/l+P2-/2)], (7)

where the symbol [...] means the greatest integer. Table 1 presents the parameters (1,Mi, 2, fi), (1,M2, —2, f2^d (1, Ms, 2, fs), which define xl5 x^d xs respectively.

fs

The next example illustrates the situation arising in the calculation of the difference between the numbers which have coinciding exponent parts and several high-order digits of the significand.

Example 2. Let N = 50, A0 = 10-3, L > 8. Introduce

10 50

хз = £ (-1)q+1 q2e-n2i2x°, X4 (-1)9+1 q2e

q=1 q=11

Define хд = |x4| — |x3|.

Table 1

Values M and f for xl5 i^d xs

L M1 fi M2 f2 Ms fs fs

8 18652239 6 44981421 6 18656737 6 5

9 186522441 8 449814458 7 186567422 8 7

10 1865224455 9 4498144699 8 1865674269 8 8

11 18652244592 11 44981446726 8 18656742737 11 10

12 186522445926 11 449814466957 10 186567427373 11 10

13 1865224459248 12 4498144669376 10 1865674273715 12 11

14 18652244592468 13 44981446694089 11 18656742737137 13 12

Suppose that

IM4 - Ms| < 10L-1 - 1, Pa < Ps = P4 and, following [16] use the empiric estimate:

{

f I 0, if fmin - L + lg(A) < 0,

fA г fr = 1 [fmin - L + lg(A)j , if fmn - L + lg(A) > 0, ^

where A = IM4 - Ms| + 1, fmin = mm{f3, /4}.

Below are the parameters (-1,Ms, 2,fs), (1,M4, 2, f4) and (-1,MA, 2, fA), which specify the values xs, x^d xA. 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 xA

L M3 fs M4 f4 ma /a fr

8 18656743 7 18656737 6 00000006 0 0

9 186567428 8 186567424 8 000000004 0 0

10 1865674274 9 1865674268 8 0000000006 0 0

11 18656742750 11 18656742736 10 00000000014 1 0

12 186567427505 12 186567427372 11 000000000133 3 1

13 1865674275054 12 1865674273715 12 0000000001339 4 2

14 18656742750534 14 18656742737138 13 00000000013396 4 3

It is obvious that when several high-order digits are set to zero, there appears the number with lower quantity of significant digits in the significand. 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 specific features of machine arithmetic.

2. Results of Solving of (1), (2)

Let us introduce the uniform meshes of nodes ti = ih, ti-1 = (i — 2)h, i = l,n, nh = T 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

N i

h£ (—lf+1 ?2E 1 )h2 = yh (9)

q=1 j=1 2

and

¿("1)9+1 j 1 / e-n2q2(ih-s)ds = yh. (10)

q=1 j=1 \j-i)h

Designate their solutions by <j)h and 4>h, respectively. Conduct a numerical experiment with the guaranteed number of valid digits in the significand not less than f5 = 8.

Example 3. Assume ф (t) from [18] as the reference function:

-, -t — 1 — e a

ф (t) = --TT - t,

1 — e a

where a = 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:

! ¥hl\\ch л \\êhl\\ch

Ъ = g2 W4Ch, Ъ = l0g2 Jhûh,

where h1 = 2h2, eiC is maximum absolute difference between the precise solution

and the approximate solution at the nodal points (the approximate solution is obtained

using the middle rectangle quadrature); He^lC is maximum absolute difference 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 0 (t) I.

0<t<1

Table 3

Errors in the mesh solution for function 0, where a = 10 1

h M^llN=2 \\t\\ch Yi WfiWN=2 \ \ * \ \ Ch Y2 \\xl\N=3 \ \ * \ \ Ch Yi ||-||N=3 \ \ * \ \ Ch Y2

1/256 0,005001 2,009 0,000499 1,996 0,003815 2,002 0,001171 1,994

1/512 0,001242 2,002 0,000125 1,998 0,000952 2,000 0,000294 1,998

1/1024 0,000310 2,000 0,000031 1,999 0,000238 2,000 0,000074 1,999

1/2048 0,000078 1,981 0,000008 1,999 0,000059 1,989 0,000018 2,001

h 1 Ici|N=4 \\* \\ch Yi 1 Ici|N=4 \ \ * \ \ Ch Y2 M^llN=5 \ \ * \ \ Ch Yi llcl|N=5 \ \ * \ \ Ch Y2

1/256 0,065009 2,107 0,002056 1,987 0,025682 2,010 0,003130 1,973

1/512 0,015090 2,025 0,000519 1,996 0,006377 2,002 0,000797 1,993

1/1024 0,003707 2,006 0,000129 1,999 0,001591 2,000 0,000200 1,998

1/2048 0,000923 1,999 0,000032 2,000 0,000398 1,996 0,000050 2,000

h H* \\W=10 Yi 1 Ici|N=10 \ \ * \ \ ch Y2 \ \ * \ \ Nh=15 Yi 1 Ici |N =i5 \ \ * \ \ Ch Y2

1/256 * _ 0,009531 1,744 * _ 0,013378 1,394

1/5)12 * _ 0,002845 1,922 0,485248 2,149 0,005092 1,719

1/1024 0,137360 2,212 0,000751 1,979 0,109395 2,033 0,001547 1,910

1/2048 0,029650 2,047 0,000190 1,995 0,026724 2,008 0,000411 1,957

Table 4

Errors in the mesh solution for function 0, where a =10 2

h llcllN=2 \ \ * \ \ Ch Yi llcllN=2 \ \ * \ \ Ch Y2 llcllN=3 \ \ * \ \ Ch Yi 1 Ici |N=3 \ \ * \ \ Ch Y2

1/256 0,006113 2,009 0,000402 1,995 0,007855 2,004 0,006159 1,875

1/512 0,001518 2,002 0,000101 1,998 0,001958 2,001 0,001679 1,939

1/1024 0,000379 1,999 0,000025 1,992 0,000489 2,000 0,000438 1,970

1/2048 0,000095 1,985 0,000006 1,927 0,000122 1,998 0,000112 1,985

h llcllN=4 \ \ * \ \ Ch Yi llêllN=4 \ \ * \ \ Ch Y2 llcllN=5 \ \ * \ \ Ch Yi 1 Ici |N=5 \ \ * \ \ Ch Y2

1/256 0,080125 2,117 0,014295 1,859 0,051629 2,022 0,024140 1,840

1/5)12 0,018474 2,027 0,003940 1,934 0,012716 2,006 0,006744 1,928

1/1024 0,004532 2,006 0,001031 1,968 0,003167 2,001 0,001772 1,966

1/2048 0,001128 2,000 0,000264 1,984 0,000791 2,000 0,000453 1,985

h \ \ * \ \ Nh=i0 Yi 1 Ici |N =i0 \ \ * \ \ Ch Y2 \ \ * \ \ Nh=i5 Yi 1 Ici |N=i5 \ \ * \ \ Ch Y2

1/256 * _ 0,081670 1,584 * _ 0,114747 1,214

1/512 * _ 0,027232 1,849 * _ 0,049456 1,637

1/1024 0,170835 2,232 0,007556 1,945 0,2227532 2,073 0,015899 1,874

1/2048 0,036370 2,052 0,001962 1,978 0,0529318 2,018 0,004338 1,959

Tables show that both finite difference methods have the second order of convergence. Figures 2, and 3 illustrate the behavior of functions \£h\ = \ 1 —<j)h_k |, \£h\ = \ 1 — <h_

2 ■ г-2 " ' ■" 2 ' г-2

on a unified mesh with the step h = ^ at fixed values N = 2, a = 10_1, T = 1. Plot "Line 1" corresponds to function \êh\, ^^^t ^^^^^e 2" corresponds to function \ëh\ and plot "Line 3" corresponds to function \£h . \ = min{\éhl\ , \êh\}.

Fig. 2. Absolute values of errors in mesh solutions at exactly specified initial data

Figure 2 was obtained at functions yh accurately specified in (9), (10). Here, the maximum values of errors made up

\\£\\N=2 = 0, 0795, \\£\\N=2 = 0, 0424.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

—■— Line 1 -A- Line 2 - Line 3

Fig. 3. Absolute values of errors in mesh solutions at perturbed initial data

Figure 3 is obtained at a saw-tooth perturbance of mesh function yh: y(ti) = y(ti) + (-1)* 10"2, i = 1727; nh = 1.

In this case, the maximum values of errors made up

\\£\\N=2 = 0,0638, \\£\\N=2 = 0,0609, \\emin\\N=2 = 0, 0582.

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

The comparison of ||£h||cfe and ||£h||cfe makes it obvious that the use of the product integration method is more preferable. The computational experiment conducted for the given example at a = 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 Eunge-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 first kind of convolution type, which occurs in the inverse boundary-value heat conduction problem, by the second-order finite difference 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 significand, is developed in PASCAL.

Acknowledgements. The work was supported by the RFBR, project No. 15-01-01425-a.

References

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12. Monde M., Arima H., Liu Wei, Mitutake Yuhichi, Hammad J.A. An Analytical Solution for Two-Dimensional Inverse Heat Conduction Problems Using Laplace Transform. International Journal of Heat and Mass Transfer, 2003, vol. 46, pp. 2135-2148. DOI: 10.1016/S0017-9310(02)00510-0

13. Beilina L., Klibanov M.V. Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems. N.Y., Springer, 2012. DOI: 10.1007/978-1-4419-7805-9

14. Kabanikhin S.I. Inverse and Ill-Posed Problems. Theory and Applications. De Gruyter, 2011. DOI: 10.1515/9783110224016

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Received August 3, 2015

УДК 517.968 DOI: 10.14529/mmpl60310

ЧИСЛЕННОЕ РЕШЕНИЕ ОДНОГО КЛАССА ИНТЕГРАЛЬНЫХ УРАВНЕНИЙ ВОЛЬТЕРРА I РОДА С УЧЕТОМ ОСОБЕННОСТЕЙ МАШИННОЙ АРИФМЕТИКИ

C.B. Солодуша, И.В. Мокрый

Статья посвящена приближенному решению уравнений Вольтерра I рода, полученных в результате применения интегральных преобразований Лапласа для решения уравнения теплопроводности. Работа состоит из введения и двух разделов. Первый раздел посвящен специфике вычисления ядер Вольтерра из соответствующих интегральных уравнений при фиксированной длине мантиссы в машинном представлении вещественного числа с плавающей точкой. На языке PASCAL разработано программное обеспечение для вычисления ядер, реализующее функцию отслеживания достоверных разрядов мантиссы. На тестовых примерах проиллюстрированы типовые случаи систематического накопления ошибок. Во втором разделе приведены результаты расчетных алгоритмов, основанных на методе интегрирования произведения и квадратуре средних прямоугольников. С целью сравнения эффективности разностных методов приведены результаты тестовых расчетов.

Ключевые слова: интегральные уравнения Вольтерра первого рода; численное решение; метод интегрирования произведения.

Литература

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 Differential Equations / H. Brunner. - N.Y.: Cambridge Univ. Press, 2004.

3. Верлань, А.Ф. Интегральные уравнения: методы, алгоритмы, программы / А.Ф. Вер-лань, B.C. Сизиков. - Киев: Наук, думка, 1986.

4. Аиарцин, A.C. Неклассические уравнения Вольтерра I рода: теория и численные методы / A.C. Апарцин. - Новосибирск: Наука, 1999.

5. Солодуша, C.B. Применение численных методов для уравнений Вольтерра I рода, возникающих в обратной граничной задаче теплопроводности / C.B. Солодуша // Известия ИГУ. Серия: Математика. - 2015. - Т. 11. - С. 96-105.

6. Солодуша, C.B. Численное решение обратной граничной задачи теплопроводности с помощью уравнений Вольтерра I рода / C.B. Солодуша, Н.М. Япарова // Сибирский журнал вычислительной математики. - 2015. - Т. 18, № 3. - С. 321-329.

7. Япарова, Н.М. Численное моделирование решений обратной граничной задачи теплопроводности / Н.М. Япарова // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2013. - Т. 6, № 3. - С. 112-124.

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. 175-185.

9. Prud'homme, M. Fourier Analysis of Conjugate Gradient Method Applied to Inverse Heat Conduction Problems / M. Prud'homme, Т.Н. Hguyen // International Journal of Heat and Mass Transfer. - 1999. - V. 42. - P. 4447-4460.

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. 121-124.

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. 111-134.

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. 2135-2148.

13. Beilina, L. Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems / L. Beilina, M.V. Klibanov. - N.Y.: Springer, 2012.

14. Kabanikhin, S.I. Inverse and Ill-Posed Problems. Theory and Applications / S.I. Kabanikhin. - De Gruyter, 2011.

15. Калиткин, H.H. Численные методы / H.H. Калиткин. - М.: Наука, 1978.

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Светлана Витальевна Солодуша, К&НДИДсХТ физико-математических наук, заведующий лабораторией, лаборатория «Неустойчивые задачи вычислительной мате:

Российская Федерация), solodusha@isem.irk.ru.

Игорь Владимирович Мокрый, кандидат технических наук, старший научный сотрудник, лаборатория «Методы математического моделирования и оптимизации в

:

Российская Федерация), ygr@isem.irk.ru.

Поступила в редакцию 3 августа 2015 г.