Solvability and numerical solutions of systems of nonlinear Volterra integral equations of the first kind with piecewise continuous kernels Текст научной статьи по специальности «Математика»

MSC 45D05

DOI: 10.14529/ mmp 160111

SOLVABILITY AND NUMERICAL SOLUTIONS OF SYSTEMS OF NONLINEAR VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND WITH PIECEWISE CONTINUOUS KERNELS

I.R. Muftahov, Irkutsk National Research Technical University, Irkutsk, Russian Federation, contact.dns@gmail.com

D.N. Sidorov, Melentiev Energy Systems Institute, Siberian Branch of Russian Academy of Sciences; Irkutsk National Research Technical University; Irkutsk State University, Irkutsk, Russian Federation, contact.dns@gmail.com

The existence theorem for systems of nonlinear Volterra integral equations kernels of the first kind with pieeewise continuous is proved. Such equations model evolving dynamical systems. A numerical method for solving nonlinear Volterra integral equations of the first kind with pieeewise continuous kernels is proposed using midpoint quadrature rule. Also numerical method for solution of systems of linear Volterra equations of the first kind is described. The examples demonstrate efficiency of proposed algorithms. The accuracy of proposed numerical methods is O(N-1).

Keywords: Volterra integral equations; discontinuous kernel; ill-posed problem; evolving dynamical systems; quadrature; Dekker - Brent method.

1. Problem Statement and Existence Theorem. In this paper we study systems of nonlinear Volterra integral equations of the first kind with pieeewise continuous kernels using our previous results [1, 2, 3, 5], where theory of linear scalar Volterra integral equations of the first kind with pieeewise continuous kernels have been addressed.

Consider the following system of nonlinear Volterra equations of the first kind

[ K(t,s,x(s)) ds = f (t), 0 ^ s ^ t ^ T, f (0) = 0. (1)

J 0

We make the following assumptions:

A) vector-function K(t, s, x(s)) is defined for < x < 0 ^ s ^ t ^ T and has a discontinuities of the first kind at the curves s = ai(t), i = 1,... ,n — 1, i. e.

K(t, s, x(s))

Ki(t,s)Gi(s,x(s)), t,s e Dl, Kn(t,s)Gn(s,x(s)), t,s e Dn,

where Di = {t,s | ai-l(t) < s < ai(t)}, i = 1,...,n, a0(t) = 0,an(t) = t, f(t) = (fl(t),..., fm(t))', x(t) = (xl(t),... ,xm(t))', m x m matrices Ki are defined, continuous

t Di

fj (t),j = 1,... ,m, and ai(t), i = 1,... ,n — 1 have continuous derivatives, 0 < al(0) ^ ... ^ a'n-l(0) < 1 fj(0) = 0,ai(0) = 0, 0 < ai(t) < a2(t) < ... < an-i(t) < t for t e (0,T];

B) matrices Ki,i = 1,..., n have continuously differentiate with respect to t continuation into compact {0 ^ s ^ t ^ T}.

For the theory of systems of linear integral equations with piecewise continuous kernels readers may refer to monograph [5]. The objective of this paper is to generalize this theory on nonlinear case and suggest algorithm for numerical solution for such systems.

It is to be noted that Volterra equations with piecewise continuous kernels are employed in energetics and in other fields for evolving dynamical systems modeling. Here readers may refer to bibliography in [2, 4].

Let us first formulate the sufficient conditions for existence and uniqueness of the solution of system (1).

Theorem 1. Let conditions A) and B) be fulfilled for 0 ^ s ^ t ^ T. Let Lipehitz condition ||Gj(s, xi(s)) — Gi(s,x2(s)) — (xi(s) — x2(s))|| ^ qi ||xi — x2\\ , Vxi,x2 £ Rn,

n— 1

be also fulfilled and qn + a{(0) \\Kn(0, 0)—1(Ki(0,0) — Ki+i(0, 0))\\ (1 + qi) < 1. Then

i=1

3r > 0 such that system (1) has unique local solution in C[0,T]. Moreover, if min^(t —

an—i(t)) = h > 0 then such local solution can be continuously extended to the whole domain [t, T] using combination of the method of steps and successive approximations.

Proof of this theorem is similar to the proof of Theorem 3.2 in [2] using the Lipehitz condition.

2. Numerical Solution. In this section we construct an algorithm based on of the midpoint quadrature for numerical solution of nonlinear Volterra integral equations (1). We also included the numerical examples to demonstrate the efficiency of the proposed scheme.

For sake of clarity we first consider the scalar case. To construct a numerical solution of equation (1) on the interval [0,T] introduce the mesh (not necessarily uniform)

0 = to <ti <t2 < ...<tN = T, h = max(ti — t—i) = O(N—i). (2)

i=i,N

Following our paper [1] we search an approximate solution of equation (1) as following piecewise constant function

XN(t) = ^Xi6i(t), t £ (0,T], 6i(t) = {1,1111 £ ^ = (ti—iM (3)

i=i ^

with undetermined yet coefficients Xi, i = 1, N.

In order to find x0 let us differentiate both sides of equation (1) with respect to t

f (*) = >;( Г Gi(s, x(s)) ds+

/

i=i V Ja

at-l(t) dt

+ai(t)Ki(t,ai(t))Gi(ai(t),x(ai(t))) - a{_i(t)Ki(t,ai-i(t))Gi(ai-i(t),x(ai-i(t)))

)

Then f'(0) = J2(ai(0)Ki(0, 0)Gi(0,xo) - a— (0)Ki(0, 0)Gi(0,xo)). The desired

i=i

coefficient x0 is included in nonlinearly in the right hand side. We use the Van Wijngaarden

- Dekker - Brent method to find this coefficient x0 (see e.g. [6]). Below we shall use the notationfk = f (tk), k = 1,..., N, and vij denotes the index of mesh's segment (2) that contains the value ai(tj), i.e. ai(tj) G Avij. Obviously vij < j for i = 0,n — 1, j = 1,N. Let us now assume that the coefficients x0,xi,... ,xk-i of approximate solution are

n r ai(tk)

known. Equation (1) in the knot t = tk is ^^ / Ki(tk,s)Gi(s,x(s)) ds = fk, and it

i=i Jat-i(tk)

can be presented as Ii(tk) + I2(tk) + ■ ■ ■ + In(tk) = fk, where

vi k-1 rtj ra i(tk)

Ii(tk) = / Ki(tk,s)Gi(s,x(s)) ds + / Ki(tk,s)Gi(s,x(s)) ds,

j=i Jtj-1 ^»1 ,k-1

ft»n-1 k * fj

In(tk )=l Kn(tk ,s)Gn(s,x(s)) ds + / Kn(tk ,s)Gn (s,x(s)) ds.

Jan-l(tk ) j=vn_i k k + iJ tj-1

1. If vp-i,k = vPkk, p = 2,... ,n — 1 then

,.t Vp, k-i

ft»p-1, k

Ip(tk )=l Kp(tk ,s)Gp(s,x(s)) ds + / Kp(tk ,s)Gp(s,x(s)) ds+

JaP-1(tk) j=Vp-1,k+iJ tj-1

faP(tk)

+ / Kp(tk,s)Gp(s,x(s)) ds. ^tvp, k-i

rap (tk)

2. If Vp-ik = Vpk ,p = 2,... ,n — 1 then Ip(tk) = / Kp(tk, s)Gp (s, x(s)) ds.

J ap-1 (tk)

It is to be noted that the number of terms in each row of this formula depends on array vij, which can be determined based on input data: functions ai(t), i = 1,n — 1 and fixed N

Each integral in the last equation can be approximated using the midpoint quadrature rule, i.e.

i-ap(tk)

/ Kp(tk, s)Gp(s, x(s)) ds « (ap(tk) — tm) Kp (tk, sk) Gp (sk,xN (sk)) ,

J tm

where m = vp, k — 1, sk = ap(tk2^+im. Moreover, on the intervals where the desired function is determined, we select xN(t) (i.e. t ^ tk-i). On the rest of the intervals the unknown

xk

the analysis of array v^. In order to find xk we use the Van Wijngaarden - Dekker -Brent method and proceed with computation of xk+i. The accuracy of proposed method is e = max \x(ti) — xN(ti) \ , where x(ti^d xN(ti) are exact and approximate solution in

the point ti correspondingly. The method has order of O (D-

Let us now consider the numerical solution of the following system of linear integral equations

[ K(t, s)x(s) ds = f (t), 0 ^ s ^ t ^ T, f (0) = 0, (4)

Jo

where m x m matrix kernel K(t, s) on the compact 0 ^ s ^ t ^ T has 1st kind discontinuities on the curves s = ai(t),i = 1,... ,n — 1.

An approximate solution of system (4) is searched again as a piecewise constant function xN(t) = (x[N\t),... ,x(N\t))', where

xf'(t) = t xj % If), t € (0,T], j (t) = { 0' £ t € j ^(5)

3=1 ^ ' T 3

with coefficients x[N\ i = 1,m, j = 1,N.

Let us introduce the following mesh of knots

0 = t0 <t1 <t2 < ...<tN = T, h = max(ti — ti-1) = O(N-1) (6)

i=l,N

for numerical solution of system (4) on the interval [0,T]. Make the following notation fi,,k = fi(tk), k = 1,..., N, K(i'p~)(tk, s) is (i, p)-th entry of matrix K(tk, s), i = 1,m, p = 1, m. This is the function with 1st kind discontinuities on curves a1(tk) < a2(tk) < ... < an-i(tk), i.e.

Гк K(i,p)(tk,s)xp(s) ds = " " K(Î'p)(tk,s)xp(s) ds. (7)

0 q=i aq-! (tk)

t xN(0)

f'(t) = Y YA ' ^ ' ) xP(s) ds+

m n / „

^ Ц

p=i q=i \ Ja

-l(t) dt

+ai(t)K(i,p)(t,ai(t))xp(ai(t)) - a[_i(t)K(i'p)(t,ai-i(t))xp(ai-i(t))\.

)

Since all the components xp(0) of the desired vector x(0) can be assumed constant, we gain the following system

f'(0) = K^(0, 0)(a' (0) — a'_ i(0)))Y xN (0). (8)

nm

£ K(i,p)(0, 0)(a'q (0) - a'q-i(0))\Y', q=i\ ' p=i

Solve the system of linear algebraic equations (8) and determine x^^O(0) for p = l,m

r(N) p,k—i

Let coefficients xpO ¡xpi,..., xpNj}-i be known. Rewrite system (4) in the point t = tk.

For zth row we have

m rtk m rtk-i

Y / K^(tk, s)xp(s) ds = fi(tk) ~Y K{i'P)(tk> sXS ds-

p=1 ^ tk-l P=1 t0

Taking into account (5) we obtain

m rtk m k-1 rtj

Y K(ip)(tk,s) dsx(N = fi,k-YY K(ip)(tk,s) dsxPN. p=iJ tk-1 p=i j=i J tj-1

Finally taking into account (7) we obtain the following system of linear algebraic equations

m k-1 n raq (tk)

__" ¡-a„ (tk) m k—l n ra„ (tk)

EE/ K^ s) dsxpN = UK^(tk,s) dsxpN \ (9)

p=l q=\Jaq-1(tk) p=l j = l q=\Jaq-1(tk)

which is solved to determine coefficients x^ for p = 1,m. 3. Numerical Examples

1.

nt/8 rt/4 nt

/ (1 + t — s) sinx(s) ds + / (t — 1)2 sinx(s) ds + / (—1)(sin2 x(s) + 1) ds = JO Jt/8 Jt/4

1 7t t , t t 1 t 1 r

= —2-1 + ^cos- + (1 + 2t)sin- — 2t sin — — sin — + -sin2t, t G [0, 2]. 8 8 8 8 4 4 2 4

The exact solution is x(t) = t. Tab. 1 demonstrates errors e for various step sizes.

Table 1

Errors for various h (example of nonlinear equation)

h 1/32 1/64 1/128 1/256 1/512 1/1024 1/2048

£ 0,345567 0,178316 0,091509 0,047107 0,023892 0,012031 0,006037

2.

rt/4

(

2 + ts, 1 - ts 1+ t + s, 1 - ts

ft/2

)& ds + , , .

(

1 + t + s, t + s

l+t, t + s

x1(s)

) ' \x2(s)J

+

t/2

(

1 + t + s, -1 1,

a) • (Xfflds=U

t6 + ll29t5 + 2027t4 55t3 40960 + 20480 + 24576 192

- 17t

6

+

4943t5 | 210714 55t3

t+l \ 20480 61440 1 24576 192

+

ds+

,

where exact solution is x(t) SN),

t E [0, 2]. Tab. 2 demonstrates errors £i

max \xi(tj) — x(N)(tj)\, i = 1, 2, where xi(tj) and x(N\tj) are corresponding exact

O^j^N

tj

Table 2

h

h 1/32 1/64 1/128 1/256 1/512 1/1024 1/2048

£i 0,054205 0,028218 0,014392 0,007267 0,003652 0,001830 0,000916

£2 0,127717 0,064389 0,032327 0,016196 0,008106 0,004055 0,002028

0

t

Conslusion. In this brief paper we continue our studies of Volterra integral equations of the first kind with discontinuous kernels [1-5]. We further develop the numerical methods [1] and suggest methods for solutions of systems of such linear equations and nonlinear equations. The midpoint quadrature rule is employed and the error order is O(1/N). The

T

[7] of the sequence {xN(ti) to solution x(t) with rate O(1/N).

Acknowledgment. The authors are thankfull to Dr. A.N. Tynda for valuable comments and discussions of the results presented in this article. The second author is partly supported by the International science and technology cooperation program of China and Russia under Grant No. 2015DFA70580.

References

1. Sidorov D.N., Tynda A.N., Muftahov I.R. Numerical Solution of Volterra Integral Equations of the First Kind with Piecewise Continuous Kernel. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 3, pp. 107-115. (in Russian) DOI: 10.14529/mmpl40311

2. Sidorov D.N. Integral Dynamical Models: Singularities, Signals and Control. World Scientific Series on Nonlinear Science, ser. A, vol. 87, Singapore, World Sc. PubL, 2015. 260 p.

3. Sidorov D.N. Solution to the Volterra Integral Equations of the First Kind with Discontinuous Kernels. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 18 (277), pp. 44-52. (in Russian)

4. Markova E.V., Sidorov D.N. On One Integral Volterra Model of Developing Dynamical Systems. Automation and Remote Control, 2014, vol. 75, no 3, pp. 413-421. DOI: 10.1134/S0005117914030011

5. Sidorov D.N. Solution to Systems of Volterra Integral Equations of the First Kind with Piecewise Continuous Kernels. Russian Mathematics, 2013, vol. 57, pp. 62-72.

DOI: 10.3103/S1066369X13010064

6. Brent R.P. An Algorithm with Guaranteed Convergence for Finding a Zero of a Function. The Computer Journal, 1971, vol. 14, no 4, pp. 422-425. DOI: 10.1093/comjnl/14.4.422

7. Trenogin V.A. Funktsional'nyy analiz [Functional Analysis]. Moscow, Fizmatlit, 2002. (in Russian)

Received November 21, 2015

УДК 517.968 Б01: 10.14529/ттр160111

РАЗРЕШИМОСТЬ И АЛГОРИТМ ЧИСЛЕННОГО РЕШЕНИЯ СИСТЕМЫ НЕЛИНЕЙНЫХ ИНТЕГРАЛЬНЫХ УРАВНЕНИЙ ВОЛЬТЕРРА I РОДА С КУСОЧНО-НЕПРЕРЫВНЫМИ ЯДРАМИ

И. Р. Муфтахов, Д.Н. Сидоров

Доказана теорема существования и разработан численный метод решения систем нелинейных интегральных уравнений Вольтерра первого рода с кусочно-непрерывными ядрами, возникающих в моделировании развивающихся динамических

систем. В качестве квадратурной формулы используется метод средних прямоугольников, при этом решение ищется в виде кусочно-постоянной функции. Для решения нелинейного уравнения использован комбинированный метод Дэккера и Брэнта. Приведены результаты расчетов для скалярного нелинейного уравнения и для систем линейных уравнений. Точность предложенных численных методов O(1/N).

Ключевые слова: интегральные уравнения Вольтeppa I рода; развивающиеся системы,; разрывное ядро; нелинейные системы; численные методы; метод Дэккера и Брэнта; квадратурной формулы.

Литература

1. Сидоров, Д.Н. Численное решение интегральных уравнений Вольтерра I рода с кусочно-непрерывными ядрами / Д.Н. Сидоров, А.Н. Тында, И.Р. Муфтахов // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2014. - Т. 7, № 3. -С. 107-115.

2. Sidorov, D. Integral Dynamical Models: Singularities, Signals and Control. World Scientific Series on Nonlinear Science Series A: V. 87 / D. Sidorov. - Singapore: World Sc. PubL, 2015. - 260 p.

3. Сидоров, Д.Н. О семействах решений интегральных уравнений Вольтерры первого рода с разрывными ядрами / Д.Н. Сидоров // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2012. - №18 (277), вып. 12. - С. 44-52.

4. Markova, E.V. On One Integral Volterra Model of Developing Dynamical Systems / E.V. Markova, D.N. Sidorov // Automation and Remote Control. - 2014. - V. 75, № 3. - P. 413-421.

5. Sidorov, D.N. Solution to Systems of Volterra Integral Equations of the First Kind with Piecewise Continuous Kernels / D.N. Sidorov // Russian Mathematics. - 2013. - V. 57. -P. 62-72.

6. Brent, R.P. An Algorithm with Guaranteed Convergence for Finding a Zero of a Function / R.P. Brent // The Computer Journal. - 1971. - V. 14, № 4. - P. 422-425.

7. Треногин, В.А. Функциональный анализ / В.А. Треногин. - М: Физматлит, 2002.

Ильдар Ринатович Муфтахов, аспирант кафедры «Вычислительная техника:», Иркутский национальный исследовательский технический университет (г. Иркутск, Российская Федерация).

Денис Николаевич Сидоров, доктор физико-математических наук, профессор кафедры «Вычислительная техника», Иркутский национальный исследовательский технический университет; Иркутский государственный университет, Институт систем энергетики им. Л.А. Мелентьева СО РАН (г. Иркутск, Российская Федарация), contact.dns@gmail.com.

Поступила в редакцию 21 ноября 2015 г.