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17
INTERVAL AND TWO-SIDED METHODS FOR SOLVING BOUNDARY VALUE PROBLEMS AND INTEGRAL EQUATIONS
IN POTENTIAL THEORY
Tigran R. A^ty^^ DOI 10.24411/2072-8735-2018-10088
Moscow Technical University of communications
and Informatics (MTUSI), Moscow, Russia, rob57@mail.ru Sergej A. Nekrasov,
Department of Applied Mathematics of the South-Russian State Polytechnical University (NPI), Novocherkassk, Russia, nekrasoff_novoch@mail.ru
Keywords: boundary value problem, boundary integral equations, double layer potential, interval method, two-way method.
The article deals with finding solutions to initial and boundary value problems with guaranteed accuracy. A two-sided firing method for calculating the alternating electric field in the dielectric layer is described and substantiated. The corresponding two-point boundary value problem is considered. The advantage of the proposed method of firing in comparison with other methods of integration of the boundary value problem is a slightly lower degree of dependence on the permanent Lipschitz on the right side of the system. An example of application of a two-way method for calculation of an alternating electric field in a dielectric layer is considered. In the second part of the article the methods of solving the boundary integral equations of the potential theory with guaranteed accuracy are investigated. In implementing this approach, the transformation of the problem to integral equations with kernels having regular properties is used. A two-sided method for solving boundary value problems for the Poisson equation based on MBIE in the case of a smooth boundary is substantiated.
The proposed method does not require setting the initial approximation to the solution, and can also be applied even in cases where the density and the free term have an integrable singularity. An effective two-way method for solving the corresponding system based on a posteriori error estimation is considered. The described method has the first order of accuracy. If the kernel of the integral equation is differentiable, then interval estimates can be computed for the derivatives of the corresponding orders from the density function. Using similar relations, interval estimates are calculated for the values of the potential and its partial derivatives by coordinates. An example of calculation with guaranteed accuracy of the electrostatic field created by two charged strands in the presence of an electro-neutral cylindrical conductor with an elliptical cross-section is solved. A two-sided solution of the BIE and the DLP is found in the case of a non-smooth boundary. Justified interval method of quadratures with the evaluation of the remainder term, and highlights a singular point. The cases when the norm of BIE kernel is greater than or equal to one and the case of irregular kernel are considered. The results of computational experiments have shown that in the implementation of the above described method of reducing the core norm, the main difficulties are related to the presence of the kernel function features, whereby the approximation error increases significantly. For this reason, the use of even more accurate spline approximation may not be sufficient to reduce the norm of the BIE kernel. As an example, we consider the case when the angle between tangents at the angular point is 90 degrees. The regulator operator is constructed using the Mellin integral transformation. An example of calculation of an electrostatic field is solved for the case when the conductor cross-section contour is a square with a side of a unit length. The two-sided method using regularization is substantiated.
Information about authors:
Tigran R. Arutyunjan, master, Moscow Technical University of communications and Informatics (MTUd), Moscow, Russia
Sergej A. Nekrasov, doctor of technical Sciences, Professor of applied mathematicsDepartment of Applied Mathematics of the South-Russian State
Polytechnical University (NPI), Novocherkassk, Russia
Для цитирования:
Арутюнян Т.Р., Некрасов С.А. Интервальные и двусторонние методы решения краевых задач и интегральных уравнений теории потенциала // T-Comm: Телекоммуникации и транспорт. 2018. Том 12. №5. С. 59-69.
For citation:
Arutyunjan T.R., Nekrasov S.A. (2018). Interval and two-sided methods for solving boundary value problems and integral equations in potential theory. T-Comm, vol. 12, no.5, pр. 59-69.
introduction
One of the urgent problems of computational mathematics is finding solutions to initial and boundary value problems with guaranteed accuracy [1-12]. Universal interval and bilateral methods for solving a number of important systems of differential equations of mathematical physics (Maxwell's system, equations of elasticity theory, etc.) have not been developed yet.
The first approach to the construction of interval and two-sided methods for solving such systems is based on their transformation to second-order differential equations. The corresponding methods are developed for the case of internal linear boundary value problems for scalar elliptic, parabolic and hyperbolic equations of order 2 [1-4],
The second approach is based on the use of the boundary integral equation method (HM), which has a number of advantages in solv ing external boundary-value problems in the ease of complex geometry of the domain boundary, the large dimension of the problem and therefore is of interest to applications [4].
An important problem in implementing this approach is to obtain integral equations with operators and kernels having regular properties as a result of reduction of the boundary value problem. This circumstance is connected with the peculiarities of interval and two-sided methods, the effectiveness of which often strongly depends on the properties of the problem operator,
1. Two-way firing method for calculation of alternating electric Held in dielectric layer
/. I. Method description
Consider the following two-point boundary value problem: dx/dt =/(*,/),< e(0,l),x= {])
(*,,;t3 ),/ = (_/;, _/;);*,((}) = *,(!)= 0.
Map (1) the Cauehy problem that depends on the parameter; dxjdt = f(xa,t\t e (0.1), = /2)
0.*aj(0)= tga, ae(/?,/)c(-,r/2,/r/2).
Denote by .y*(/,a) two-sided estimates of the solution Xir(t)
problem (2) obtained by methods of [1-4, 8-12j, .v,'(0,«) = 0,x2'(0.a) = tga. We introduce also the notation:
V>(a) =*,'{[,a).
The right parts in (1), (2) and solutions of problems are assumed to be continuous and limited, then it is fair
Approval 1. The two-sided solution of the boundary value problem (1) can be reduced to finding the equation roots by bisection method (p(a) =0.
Proof. Eiy assumption on the interval (ft,y)cz(-,T/2,tt/2) functions tp(a) and tp'(a) continuous. Exact solutions of the equations (pz(a) =0 denote by a". Method bisection find their rating a- ■¿a'- <a! ■
c mm MUX
Because <p(a')<q>"(a ) = 0,0 (a ) <<p(a "), and function <p(a) under the assumption it is continuous, then on the segment [a,,«,], where i2] - mm\anin, a*,nn}, % = »*«{» m„.<(1„}, function </}(a) Know a, and according to 11-4], two-sided estimates of the exact solution can be found (1) x(t) = x^ (I) on the whole interval (0,1):
Convergence to the exact solution follows directly from the continuous function dependency *from initial conditions, i.e. from parameters « and a., as well as the convergence
of the corresponding bilateral methods for solving the Cauchy problem.
The advantage of the firing method in comparison with other methods of integration of the boundary value problem is a slightly low er degree of dependence on the permanent Lipschitz on the right side of the system. The disadvantage is the uneven-ness of the global error distribution over the integration interval (0,1).
It is not difficult to see that the algorithm is applicable to other varieties of boundary conditions in (1),
1.2. An example of applying a bilateral approach
for the calculation of the electric field in the dielectric layer
Let us consider the case when the alternating electric field vector E has one component other than zero E. = EAx), values
of which are set at layer boundaries:
£Д0) = £^,£.(я) = £;,, (3)
where a — the thickness of the dielectric Esome
complex values.
After a series of standard transformations from the Maxwell system under conditions /.t = const,y= Q,s = s(x), где e(x)-скалярмая вещественная функция переменной х, получается следующее уравнение для функции Е:
d2E. I dx1 = -к~Е_, 0 < х <а, (4)
где к2 = //£(л-)«:, ц - magnetic, а е - permittivity of medium, СО - the angular frequency of the field. As a parameter к accepts only valid values, then due to the linearity of the equation and the boundary conditions of the real and imaginary parts of the function E.(x) are solutions of boundary value problems
similar to (3)-(4).
As a result of replacements ^ = x/a,ut=E^,u1=dEIld^ (3)-{4) is given to the:
du, idc = u2,du2 / dt = -А:щ,с e {0,1);
iii (0) = E_t,ut (l) = ЕфЛ2 -k'a2.
Results of the model problem solution in £ = Ez2 = —Is exp(3^) firing method based on interval method [10,11] of the first order of accuracy in increments h = 0.001 presented in table. 1, 2 and Fig. 1. Interval estimates of unknown parameter value ¿1 = г/,(0) calculated by solving the equation tii(]) =Elt
by the method of half-dividing and have a look;
p e [-3.940.-3.938]./Ге [-4.097,-4.094],/; e [-4.097,-3.938].
Error of the parameter estimates p and p* the accuracy of calculations in the bisection method is determined, and the error of the final interval estimation of the desired parameter also depends on the accuracy of the Cauchy problem integration method. In calculations, it should be borne in mind that it is advisable to agree on the order of accuracy in the bisection method and numerical integration procedures.
Fig, 1. Graphs ofthe solution and coefficient of the boundary value problem
Results of the solution of the ODE system at the value of p+ = -4.097
Table I
£ Щ + u\ u2 Uj
0,1 -1.40343 -1.40325 -3,95568 -3.95478
0.2 -1.78764 -1.78705 -3.70341 -3.70109
0.3 -2.13892 -2.13758 -3.28483 -3.28030
0.4 -2.43718 -2.43457 -2.62680 -2.61888
0.5 -2.65413 -2.64949 -1.63916 -1,62605
0.6 -2.75196 -2,74422 -0,22457 -0.20349
0.7 -2.68386 -2.67149 1.69277 1.72723
0.8 -2.39886 -2.37949 4.10528 4,16486
0.9 -1.85517 -1.82500 6.81185 6.91845
1.0 -1.04731 -1.00005 9.25303 9,44810
Table 2
Results of the solution of the ODE system at the value of p* = -4.094
£ Щ Щ*
0.1 -1.40319 -1.40300 -3.95326 -3.95235
0.2 -1.78716 -1.78657 -3.70104 -3.69872
0.3 -2.13821 -2.13686 -3.28259 -3.27806
0.4 -2,43625 -2.43364 -2.62480 -2.61688
0.5 -2.65302 -2.64838 -1,63755 -1.62445
0.6 -2.75072 -2.74298 -0.22358 -0.20251
0.7 -2.68256 -2.67020 1.69286 1.72731
0.8 -2.39761 -2.37825 4.10415 4.16371
0.9 -1.85410 -1.82395 6.80924 6,91579
1.0 -1.04658 -0.99934 9.2489 9.44388
2. Solution of boundary integral equations of potential theory with guaranteed accuracy
As it was noted in the solution of external boundary-value problems, in the case of complex geometry of the domain boundary and large dimension of the problem a number of advantages are provided by MBIE [5-7]. One of the problems in the
implementation of this approach is to obtain as a result of reduction of the problem of integral equations with kernels having regular properties.
2.I. A two-sided method for solving boundary value problems for the Poisson equation based on MBIE in the case of a smooth boundary
We consider the outer edge Dirichlet problem for the Poisson equation on a plane:
A<p ~-f,Me CS';<p = g,M = M(x,y) € T, (5)
where x,y - cartesian coordinates, r- a smooth closed curve, bounding a region S, S'=S\j T, CS' - addition S' to the plane, the function g continuous, function f finite, absolutely integrable on the set CS* and vanishes in the neighborhood P, n the potential is limited at infinity, the total charge in the system is zero. We will look for a solution in the form of:
<p(M)= + |/l(A')5(,V)^rv A'/eCS1',
where the first term represents the potential of the double layer (PDL) with a density N - N(£,ij) eI", <p„(M,N) ~ fundamental solution of the Laplace equation on the plane:
<p„(M,N) = \H2n)ln(\lr{M,N)),
¡f(N)<p¿M,N)dSN,
(6) (7)
where r(M,N) - distance between points M h N : r(M,N) = l(x~<02+0-7)2f\ HK) ~ some given differentiate function. The expression (6), according to known properties of the potentials [5-7], in pixels CS* satisfies the Poisson equation in (6). In points r equation (7) has the form [5-7]:
<p( M ) = - ,9( M) + J,9( JV ) —( Ai, N)dTfl +
2 f dv
\l{N)8{N)dTN + y/{M),M ь Г.
(8)
Due to (8) and boundary conditions in (5) density 9{M) satisfy the boundary integral equation of type 11:
2 f. dv
¡Á(N)&(N)drN =g(M)-y(M),M e Г.
m
Equation (9) given the parametric representation of the curve r x = £a(t),y - % (/),/ e (0, ) is given to mind:
-S(t) + ¡U(t,T)S(T)dT =a(t), t ep,tf),
(10)
where
U(t,r) = \l{2ir)[dnJdT{Q(t) -C„(r)) -dQ/МпЛО -ЧЩ/
ЩМ <IOa>
W = M^rXq^mdCJ dry +{drjJ drff1, (10b)
ш =жмлm -fftao,%('))■ o°c>
Function /^(r) in (10b) it is advisable to choose from the minimum norm of the integral operator equation (8).
Because the curve Г smooth, then the core is bounded and continuous. If the norm of the integral operator in (8) in the space of continuous functions is less than one, and the free term
is continuous, then iterative interval as well as bilateral methods described in ¡ 1 -121 are applicable tor its solution.
Consider a method that has a number of differences and advantages over those described in [1, 2]. This method does not require setting the initial approximation to the solution, and can also be applied even in cases where the density and the free term have an integrable singularity. This method requires a sign density constant over the integration interval, w hich in the case of a bounded solution can be easily achieved by replacing the view = &(t) + const ■
Let o =t9<t. <...<tm = tf partition of the interval (0,ff). Denote /?,
\9(t)dt, er = j- jcr{t)dtil-\,m. ' v-i ''M
Ratio (8) after integration by splitting intervals T = [/( „/,],( = I, ...,m> taking into account the average theorem,
it is converted to the following system of equations:
-p,+YJPjU§,Cj)hJ=erl, i=!,m,
/-i
(11)
t, eTi,tl€7},(J = ],m.
Because UtfiJj)eU(TnTj),at 6 а(ТЦ,Ц= l^n, to find an interval solution, it is enough to solve the corresponding (11) system of linear algebraic equations (SLAE) with intervals given by a matrix and a free term.
We consider an effective two-way method for solving the corresponding system based on a posteriori error estimation. We write corresponding to (11) linear algcbraic equation in the form:
J)а9Р/=^=и>, <]2)
i-1
where n= in \ — the vector of unknowns, (а \ и
r 11 1 / I.....m 7 ' >.,' I.....til
{b }M - approximately given coefficients of matrix and free term: aIJ<£A.i,bleBJJ= 1 ,...,,„.
Denote ait" = mid(A^) — the middle of the interval A.., wv" = width(Ay) - the width of the interval ,,, likewise,
b" = mid(B, Iw" = width(Й,),i,j =
Assume that you have a matrix a" = , nondegen-
erate, then there exists an inverse matrix to it C- Denote p" - C'b", then (12) can be rewritten in the form:
p = pu + C(,b-b") -C(A-A*)p.
Матрица, состоящая из абсолютных значений матричных коэффициентов С, denote by D. By virtue of the above defined conditions and properties of interval operations f 1, 2]: pep" +1 / 2(Dwh + fli/i-)[-l,l]c
p" +1 / 2 (Dw1, + Dw"v" + Dw'z) [-1.1] = = p°+ [~U]z, where
.....'v>=\p'\ - i'0=i р>ш= 1.-Л
z - a vector with nonnegative coordinates. From (13) follows
z = \! 2Dw"z + 1 / 2{Dwh + £>n>V'). (14)
(13)
The system (12) under appropriate conditions can be solved by Gauss method. Since all solutions (12) turned out to be nonnegative (which is always the case for sufficiently small values || w" | )> that is proven
Statement 2. Under certain conditions, the desired interval estimates of the solutions of the original systems (9)-( l0) can be found in the form of:
pep0 + [-1,1]*. (i5>
After you have calculated the estimates (!3), you can define the interval to which the density belongs 3{t )Vf e [0,fF]:
9(t)z-2Y,U{tJiXpai+zj[-\,\\)hj+2<T{t). (i6>
As follows from the obtained relations, the described method has the first order of accuracy. If the kernel of the integral equation is differentiable, then interval estimates of the form (14) can be calculated for the derivatives of the corresponding orders from the density function. These estimates of derivatives can be used in two-sided methods of higher orders of accuracy based on quadrature formulas [1,2],
Using relations similar to (14), interval estimates are calculated for the values of the potential and its partial derivatives by coordinates.
Example I. Calculate with guaranteed accuracy the electrostatic field created by two charged strands in the presence of an electro-neutral cylindrical conductor with an elliptical cross-section. Threads and forming cylinders are parallel (Fig.2). The values of the specific charges of the threads have different signs and the same absolute values. At an infinite distance from the electrode potential is limited.
-0 -q
Fig. 2
Decision. The field potential satisfies the boundary value problem (1), w here
/ = qd(x-c,y-d) -qS(x-c,y + d), <JU,.y) - two-dimensional Delta function, the value of the potential of the conductor g, nostiano but a priori unknown. We present the solution of the problem (5) in the form of p = u + v, where An = - ftMeCS';u = 0,M er;Ai> = O.jW s CS',v = g,M e E. Functions « and v uniformly limited. Decision v = g, obviously satisfies the task for the function r, therefore, <p = u + g. In [4] it is shown that the value g can be chosen arbitrary.
For definiteness, take the value g equal to unity. Note that the function u tends to zero at a distance from the electrode boundary, so the potential at infinity is equal to the potential on the surface of the cylinder. To calculate the potential of the field, you can use the ratios (2) and (7). Parametric representation of the boundary r have the form:
£(r) = acas(/),77(/) = bsin{t),t e (-.r.irj. According to (4), (8) the kernel and the free term in (7) have the form:
U(t,r) = K(t + r) +Ax>(t),K(') -
= -ab / (4jt) / [a-sin\t 12) + b2cos2 (I / 2)], <T{t) = I -qH2n)ln{rJr,),,f =
=(ae<M(r)-c) + (bsin(t)-df,
r= (¿TCiijfr) —c)* + + .
Value ^(r) significantly affects the value of the norm of the equation operator and the efficiency of the interval method. Selected value /¡,t(r) = i/(4ff) from the condition of equality to
zero ofthe mean integral value of the kernel U(I,t).
The results of solving the problem on a PC of the Pentium type with the following source data: m= 50,\,b- 1.25,c = 2,t/ = \,q- 5 presented in Fig.3.4. Graphs of curves in Fig.3 are derived in normalized form: I - schedule of interval solving systems of linear equations (9)-(I0) (p°+ [_l,l]ziy||p°+ [-l,l]z||,i= ],...,«; 2 — graph of width of interval solving systems of linear equations
r<t)
Fig. 3. Graphs of interval solving systems of linear equations (1} and its width (2)
The norms of the matrices D h 1 / 2Dw" b the current variant is equal to 2.31851 and 0.03! 81 respectively. For rice.4 the grading graphs normalized per unit are also presented g(/) densities £(/), obtained by the formula (14), and the values of their width.
These estimates are several times more accurate than those obtained by the solution of SLAE (9)-(10), This fact is explained by the fact that in (14) the interval extension from the kernel function is only on the second argument, and the value of the first is set accurately.
The division of the integration interval was chosen for the reasons of uniform error distribution. At the center ofthe interval, the step value was set to 0.08571 and at the edges to 0.14120. The error of the approximate solution of the SIU is within 4%. The calculation time is about 1.5 seconds.
One of the principal benefits of interval and two-sided methods for solving various problems, in particular, the SMI is a General property of [1,2]:
Property (proof of interval and two-sided methods) The strict proof of existence of uniformly bounded solution (in particular, integral equation) can be carried out by means of one computational experiment by means of finding of numerical interval or bilateral solution.
2.2. Bilateral solution of the BIE with the DLP in the case of a non-smooth boundary
The application of known two-sided methods to the solution ofthe type (6) BIE may be difficult if there are corner points on the boundary or (and) the operator norm is too large (greater than or equal to one). As the study shows, in the neighborhood of angular points, the density of DLP retains continuity, but its derivative is unlimited. Although the DLP kernel is completely integrable, it is not limited. For this reason, the use of the method of claim 1, as well as the similar method [2] is not possible.
The study of other known approaches made it possible to establish that the solution ofthe problem with guaranteed accuracy allows us to calculate the interval quadrature method with the residual term evaluation and the allocation of a special point, as well as a two-sided method with a posteriori error estimation.
2.2.1. Interval method of quadratures with the evaluation of the remainder term, and highlights a singular point Consider BIE with integral type DLP:
9(M)= \K{M,N)£>(N)dr,+f(M), MsY, 0?)
r
where F - piecewise smooth bounded curve, r =U{Ar,;y6 a), Ar.~ the elements of the partition curve r,
a - the corresponding set of partition node numbers Mj £ F, f(M) - a function belonging to the Lipschitz class.
Taking into account (15), the density values at the nodes of the partition satisfy the relation:
&(Mt)= £ $(Mj} J K(M„N)drii +
jew
■j
+ \ K(Mi,N)(9{N)-9{M¡))drs+f{M), i em.
дг,.
Fig. 4. Graphs of interval estimates ofthe density function 9(f) and the corresponding absolute error values
Suppose that 0 = (ot+&2, где и{ДГ,;./е<щ} - the regular portion of the curve Г, а e«,} contains corner points.
On a regular section ofthe boundary density S(N) continuously
differentiable and belongs to the class of Lipschitz with constant . in the neighborhood of corner points, the density belongs to
the Gelder class C", therefore F\)- F\) ¡< La \ , where
0 <a< 1,0 <La«x>tPvP2ev{bTj\jea)1) ■ Degree a depends
on the size of the angle between the normal vectors to the boundary in the vicinity of the corner point. Suppose that M,) t£(-,ieci), then
WeV^ j K{Mt,NWN + V f
yew Af jwxd,
+ Z }/f(A/1,A')L-l,l]iu|yVM.prv+/(M), tear,
ATy
therefore,
j&m
J&r,
JEU72 ATy
(18)
+Kif"*)La | ATy | [-1.1] +/(A/), iem, where
Kj= \ K(M„N)dT„, K9m=K(M„AT,),
Af
K(M ,N) = K,{M,N) + K{reg\M ,N),
K(rt*\MtN) ~ regular part of the kernel, K,(M,N) ~ part of
the
kerne!
containing
the
singularity,
K^"g\M,N) = K^iM^ATj) ■
Component values
J K,(MltNyi^|Ar,.|° [-l,l]c/rv, i e m2,
AT,
calculated or estimated analytically.
To calculate the Lipschitz and holder constants, it is advisable to use the values of the density function and its derivatives obtained on the basis of the original equation (15).
Uniform interval density estimates can be calculated using ratios:
¿»(¿r^cHj + L, | AT, | [-1,
¿(AT,) s S; + La | AT, |" [-1,l],i e ©2.
Assume that the norm of the matrix {jy less than one:
max J enr ^ < I,
Uenr J
then on the basis of the simple iteration method it is possible to prove the existence of interval estimates of the solution E,so, are solutions ofa system of linear algebraic equations with interval free terms and a real matrix:
= S k9Ejiem, (19>
j (Em
where
If the norm of the matrix {¿..j not small enough, you can first resolve the source system with respect to the vector of nodal density values 19f ,i S CO :
{¿(K)}*...... = {s9 - Kj iV....., k+f,}'\. . <20>
>}=£ | K(M„N)(S{N)-sogyr,, + f(M\ ism. m aTj
The desired interval estimates E can be calculated based on
1
the ratio (18) in which instead of the values r appropriate interval estimates should be used Rjsoi- This method has advantages in comparison with the method described in paragraph 1, since the matrix b (17) is real and therefore has better properties, in particular, its norm is less than that of the corresponding interval matrix. The disadvantage of the described interval method is its limited order of accuracy: w + =0(h"), where h -the
option to split the boundary.
2.2.2. Two-sided method with a posteriori error estimation Higher order accuracy estimates can be obtained from a two-
way method with a posteriori error estimate [2]. Suppose that ggJ^Af,*)^ <a<l.
We calculate in some way the approximate density values at the nodes of the partition ,9tisco • Using the values S„,,ie&
built spline s(M), approximating exact solution 9{M)-
Denote Z(M) = &{M)-s{ A/), then taking into account (17) z(M) = \K(M,N)z(N)dYK —Tf{M), M e f, (21)
where //(A/) - the residual of the equation (17) after the substitution in it of the spline s(M) instead ofan exact solution &{M)'-
ti(A0 = s(M)- JAT(MtN)s(N)drv - f(M), M sT. (22) i
Because of (19) follows || Z(M) |[<|| tj(M) || /(1 - a), that's fair Statement 3. Under the above certain conditions, uniform two-sided estimates of the B1E (17) solution can be found by the formula:
9(M)es(M) + |-l,l] ||rj(M)\\/(1 -a) . (23)
2.2.3. The case when the norm of the core B1E (¡5) is greater than or equal to one
In the considered case it is possible to use described in item 2.2.1 interval method, or convert the BIE (17) in order to reduce the norm of the kernel. The corresponding algorithm can be formulated as follows. We transform the curvilinear integral of the first kind into the B1L (17) to a certain integral based on the
parametrization procedure:
b
m = JK(/, T)9(T)dT + At), t e (a, b), <24>
a
We transform the equation (24) can be written as:
+K™LJArJ+"[-U], ¿€£7.
where
ь
7(0 = \K,(t,r)$(T)dT+ f(t\ te(a,b),
a
/e7;= ['м.'(].'б1}= 1.....».
Т. - split interval intervals [я./>], А|(/,г)=Д/,г)-Л{/,г).
(25)
7=1
V-1
Denote
Then from (25) can be obtained SLAE
.....
J= i
relative to quantities ¡9 , i = i,...,where , h
K9=~ ¡Kj^di, i,j = I,...,«,
«
1 '' ~
hi=ti-ii-u i=l.....
'm
where
^ n ^
Taking into account (26) G = {8S- - .....„.
(26)
= .....
Therefore,
•?(0=J
"} tr I
J=1
>9(г-Ыг +
(27)
gin at the corner point, a special part of the kernel looks like K.U,r) = comi-I /(t2 + r2) and by /->0+ and re/?! tends to a Delta function S(t) [7]. Therefore by / = 0+ H a< 0 spline approximation error does not tend to zero:
||/:.(f,T)-i:.(f,T)|| = sup|j| > lim |б(т)-К-(0,т):/т = limf jf6(iV/r + o( 1)
с
J
= 1/2,
where it is considered that at e-> 0
К-(0,т)Л = о(1).
These causes make it reasonable in some cases the regulariza-tion of the BIE with the DLP. The study showed that the known regularization method based on the use of iterated kernels [5-7] is not effective in the problem being solved. The necessary result can be obtained by means of analytical reference of the corresponding integral operator. The regulator operator is constructed using the Mellin integral transformation.
Example 2. Consider the problem of calculating the electrostatic field, similar to the example 1, in the case where the conductor cross-section is a square with a side of a unit length (Fig.5). Because of the symmetry = there-
fore, it is enough to find a solution when y > 0 in form;
<p(M) = \&(N)^HM,N)dr!, + ^(M), MeS. • dv
where r = r, +T, +Ti (see. Fig.5),
¥{x,y) = q / (An) i/«[(x - A'0): + (v + ya f ] - /«[(-v - ^ )2 + (y - j.^)2]}, = %(xs - XM, >'iV - V(, ) - % (x.v - , VlV + V„ ).
У Л
+£4(0/,+/(0, t€(a,b).
j-'
When a sufficiently accurate approximation of the kernel K(t, r), core standards A",(/,r) and, therefore, the kernels of the
equation (27) may be less than one. In this case, the two-sided method described above can be used to find a solution (27).
2.2.4. The case of irregular cores
The results of computational experiments have shown that when implementing the above described method of reducing the norm of the kernel, the main difficulties are related to the presence of the kernel function K(t,r) features, as a result of which the approximation error increases significantly. For this reason, the use of even more accurate spline approximation may not be sufficient to reduce the norm of the BIE kernel. As an example, consider the case when the angle between tangents ai the corner point is 7T! 2. Then, in the local coordinate system with the ori-
-y
j л
->
X
-q
Fig.5
Function <p0(Xff,yN) определена в (3). Unknown density DLP &(хя,ун) is the solution of BIE:
3{M) = -2 [3{N)^HM,N)drN - 2<j/( M), M e Г, J dv
Denote
3(0 (') =p(x,fty„),
Mer,,i =1,2,3; fs(0.l/2) by i = l,3 and i e(-l/2,I/2) by / = 2.
The solved BIE can be written as a system of three integral equations with densities &.(f)j = 1,2,3 -
a = t9<i,< ...=
The integrals in (29) and in the expression of the residual
„ h
n(0 = ^(0-S \K(t,x)Sj{x)dx-f(t)t i=l.....n.
1
3(0+- J ÄW
TT J
t-1/2
f+ 1/2
(/-I /2)2 +(T+1/2)1 (/ +1 /2)2 +(i + l /2)"
</t +
* iij. tt J
!+(t-/) l + (r + f}~
4co+-tat*)
n i
1
- f 90
7T *
/+1/2
i+1/2
(r+l/2)'+(t-l/2) (f + l/2)"+(T+l/2y -dt+
(28a)
ch+
-VI Ml
- im
l+(r-f)
t-1/2
/-1/2
(r-1/2) +(t-l/2) (r-1/2) +(i+l/2)
/€[-1/2,1/2]. I
3(0+-
0
l
+- f A(t)
77- J
ch--2i//2(r), (29b)
1 + (t-/)2 1 +(T + /)2 /-1/2
</t +
/ + 1/2
(/-1/2)2 + (t- 1 / 2)" {/ +1 / 2)2 + (t-1 / 2)2
xi/i = -Vj(0./e[0,l/2].
(29c)
e, +e
(29)
/=i f
calculated accurately using analytical integration. According to the obtained estimates, the norm of the integral operator does not exceed 0.8. The results of calculations are presented in Fig.6-11.
The study shows that the density ¿¡.(/),/ = 1,2,3 belong Gelder
space C:\SXtd-m^KA^-hW where
4, < JltLal + /?,,La2 <(Jt,(0t + £) + &)/( 1 - 2J0S),< JaLa2 + A, /,= 0,5/c'oj(?ra/2),a e(0,l/2),
3//r || [|c- + I / JT || <921|(. + 3 / 7T || 5-j ||c +2Z,(^,),
¿(<// )— constant Lipschitz free terms of integral equations,
/ = 1,2,3.
The above mentioned relations can he used in the interval method to estimate the variation of the solution in the neighborhood of angular points. Estimates of derivatives of densities &,(t),i = 1,2,3 on the regular part of the boundary can be found
by differentiating the relations (28) by parameter t.
In solving this problem, the most effective of the above methods is a two-sided method (22), (23). To compute an approximate real solution Qk= {0.(/)J. , n, used the method of
quadratures based on the formula:
f<s> 0,5 ■•
0.4 --
0.5 l.O 1.5 2.0
Fig. 6. Graph of the approximate solution of the boundary integral equation
f(s) O.OlS
D.5 l.O
1.5 2.0 ^
Fig. 7. Graph of the residual module of the integral equation for n = 10
f(s> o. oio -O. 008 --
O, 002 --
jVwUffl^
Fig, 8. Graph of the residual module of the integral equation for n = 20
W
As follows from these results, the error of the approximate solution in the area of the right corner point is an order of magnitude higher than the error on the rest of the border. An effective way to reduce the error of the approximate solution in the neighborhood of angular points is the use of an uneven grid, as illustrated in Fig.10.11.
2.2.5. Two-sided method using regularization As noted, the core DLP bad approaches spline-functions with the existence at the boundary corner points. This and other circumstances (in particular, a relatively large norm of the operator DLP) determine the feasibility of the regularization procedure BIE. To do this, we present a system (28) in the form of two subsystems of integral equations of the 2nd kind corresponding to the left and right corner points:
+ - U(t)-(f^--r/r = /!(/), /£[0,1/2].
(/-1/2V +(t + 1/2V
A(0--f$(t)
* <>
1 112
£(/> + - [5,(1) if i
1 "t
TT J
(31a)
jdr = f2(t), /€[-1/2,0].
(31b)
{/-1/2)* + (i
/+1/2 (/ + 1 / 2)" +(t-l / 2)
----dx = f2(t), /e [0,1/2],
(f-1/2) +(x-l/2)
(31c)
Tdz=f}(t), / e[0,1 / 2].
(31d)
/-1/2
(/ -1 / 2)" +(t-1 / 2)
where
/¡(0=— J bWr
I "2
-- U(t)
rr J
/ + 1/2
+1/2)2 +{r+ 1 /2)r I 1
T<h-
1 + (t -/)2 l + (t + /y /-1/2
d\-
m '*m
(f + !/2):+(i + l/2}:
Tch~
xJn i+(t-/;
4 J**>
/-1/2
/-1/2
_(/-l/2)r + (r-l/2f (/-1/2}:+(t+1/2)"
di-2i//2(t),
I
x?
1
-1 U(r)
*-V2 1 "2
+xjm(t-V2f+(T+i/2y
/ + 1/2
f+ 1/2
(/ + l/2}3+(t-l/2}: {t+Mif +{x+M2)\
di-
-ch+
/-1/2
^i/r-2v/,(f),
/,(/)—- U(t)
7T »
1
I
1+(T-/)2 I + (T + /)" _ /-1/2
dx-
--f &(rb-—Tj—-—^dr-
ff l_ ■ (f-1/2)" + (t-1/2)"
-- í ¿«2(T)7----wch-2<fát),
i J * 14.\ ">Y 4. T.\nY
The regularization algorithm is based on the allocation of a special part of the kernel and the subsequent accurate treatment of the corresponding integral operators.
Functions fl(t),f2(t),f2(t)i£3(t) contain a regular part of the operator kernel and free members.
Carry out the replacement &,(/) = «(\/2-t),J\(t) = g ,(1/2-/), / e (0,1/2),
93(0 = v( 1 /2+4/2(0 = giC 1 /2+1), t e (-1 /2,0), then the first pair of equations is reduced to a symmetric system of equations w ith the same kernel k(t,i) = t/\n(rW)\.
Ml a
v(t)~ ]A(f,T)w(T)A = g2(i), t e(0,1/2), i)
Applying the Mellin integral transform [7] t> 0,
2m •
F(s)= jtsl/(f)dt, Res>0,
(1
we partially reverse the operators of the considered system of equations:
3(0 = />,(')'• f191(i)/íil(/,t)í/r+ } S2(t)Xa(t,r)ch+
0 -\n
1/3
+ ¡&i(z)K,1{t,T)di, i —1,2,
(32)
Where by ,* = ¡ t e (0,1 / 2), by | = 2 / e (-1 / 2,0) - Expressions of cores K,:(t,r) and free members p4(t) are not given as
rather bulky and have purely technical character. The second pair of equations (31c), (31 d) is given to the analogous equations.
The free terms and kernels of the system of integral equations obtained from the BIE as a resLdt of regularization are uniformly bounded and continuous functions substantially belter approaching splines. The value of the new system operator norm can be estimated from above by 0.8.
The main technical problem is the calculation of the interval values of the cores K;i{t,T) ,i - l,...,4,y = 1,...,3- This problem is
effectively solved by using standard methods of interval integration using quadrature formulas of high order accuracy [1,2],
For the solution of the system of BIE after regularization, a bilateral method with a posteriori error estimation is preferred [2-4]. Due to the improved properties of the nuclei, the problem of reducing the norm of the system operator can be solved on the basis of the corresponding algorithm much more effectively.
When calculating the first order of accuracy of the norm of the residual equations and the error of the approximate solution have the following values:
in the step of partitioning the circuit V h = 0.05: ||>/1|*0.04,||z|[*0.25;
when the step is half the size h = 0.025: ||/7|[s:0.027,||z||«<U6.
The number of partitions of the interval integration in calculating the values of kernels /£..(/,*■),/= 1.....4,y=l,...,3 was
taken equal to 60 .To improve the accuracy of the approximate solution, it is advisable to use an uneven grid, which thickens in the neighborhood of angular points. Thus, a 4-fold decrease in the grid step directly near the corner point leads to a decrease in the error of the solution by about 2 times: ||z|[« 0.08 ■
f (s> 0.3 +
r\
O.S l.O 1.5 2.0
Fig, 9. Graph of the module of the free term of the integral equation
F<sl
o. oio
Juik
o. oio --O,008 - -
J^mmy».- ,
Fig. 10. Graph of the residual module of the integral equation in the case n = 10 of nonuniform partition
Fig. 11. Graph of the residual module of the integral equation for o = 20 in case of uneven splitting
References
1. Kalmykov S.A., Shokin Y.I., Yuldasliev Z.H. (1986). Methods of interval analysis. Novosibirsk: Science. 224 p.
2. Dobronets B.S., Shaidurov V.V. (1990). Bilateral methods. Novosibirsk: Science, 208 p.
3. Dobronets B.S, (2004). Interval mathematics. Krasnoyarsk: KGU. 216 p.
4. Moore R.IL, Kearfott R.B., Cloud M.J.(2009). Introduction to interval analysis. Philadelphia: S1AM9.
5. Mikhlin S.G. (1977). Linear equations in partial derivatives. Moscow: Higher, school. 431 p.
6. Mikhlin S.G. (1962). Multidimensional singular integral equations. Moscow: Fizmatgiz. 256 p.
7. Kom G., Kom T. (1978), Handbook of mathematics. Moscow; Science. 832 p.
8. Nekrasov S.A. (1986). Bilateral method of solving Cauchy problems. USSR Computational Mathematics and Mathematical Physics. Vol. 26. No. 3, pp. 83-86.
9. Nekrasov S.A, (1988). The construction of two-sided approximations to the solution of a Caucasus problem. USSR Computational Mathematics and Mathematical Physics. Vol. 28. No. 3, pp. 23-29.
10. Nekrasov S.A. (1995). Bilateral methods for the numerical integration of initial - and boundary-value problems. Computational Mathematics and Mathematical Physics. Vol. 35. No. 10, pp. II89-1202.
I!. Nekrasov S.A. (2003). Efficient two-sided methods for the Caucasus problem in the case of large integration intervals. Differential Equations. Vol. 39. No. 7, pp. 1023-1027.
12. Nekrasov S.A. (2016). Numerical method for solving dynamic systems with lumped parameters which accounts for an input data error. Journal of Applied and Industrial Mathematics. Vol. 10, Issue 4, pp. 528-537.
T-Comm Tom 12. #5-2018
ИНТЕРВАЛЬНЫЕ И ДВУСТОРОННИЕ МЕТОДЫ РЕШЕНИЯ КРАЕВЫХ ЗАДАЧ И ИНТЕГРАЛЬНЫХ УРАВНЕНИЙ ТЕОРИИ ПОТЕНЦИАЛА
Арутюнян Тигран Робертович,
Московский Технический Университет связи и информатики (МТУСИ), Москва, Россия, rob57@mail.ru
Некрасов Сергей Александрович,
Южно-Российский государственный политехнический университет (НПИ), Новочеркасск, Россия, nekrasoff_novoch@mail.ru
Дннотация
Статья посвящена поиску решений начальных и краевых задач с гарантированной точностью. Описан и обоснован метод двухсторонних оценок для расчета переменного электрического поля в слое диэлектрика. Рассматривается соответствующая двухточечная краевая задача. Преимуществом предложенного метода оценок по сравнению с другими методами интегрирования краевой задачи является несколько меньшая степень зависимости от постоянной Липшица правых частей системы ОДУ. Рассмотрен пример применения двухстороннего метода для расчета переменного электрического поля в диэлектрическом слое. Во второй части статьи исследуются методы решения граничных интегральных уравнений теории потенциала с гарантированной точностью. При реализации этого подхода используется преобразование краевой задачи в интегральные уравнения с ядрами, имеющими регулярные свойства. Двусторонний метод решения краевых задач для уравнения Пуассона на основе МГИУ в случае гладкой границы обоснован. Предложенный метод не требует задания начальной аппроксимации решения, а также может быть применен даже в случаях, когда плотность и свободный член имеют интегрируемую особенность. Рассматривается эффективный двусторонний метод решения соответствующей системы, основанный на апостериорной оценке погрешности. Описанный метод имеет первый порядок точности. Если ядро интегрального уравнения дифференцируемо, то интервальные оценки могут быть вычислены для производных соответствующих порядков от функции плотности. Используя аналогичные соотношения, рассчитываются интервальные оценки для значений потенциала и его частных производных по координатам. Решен пример расчета с гарантированной точностью электростатического поля, создаваемого двумя заряженными нитями в присутствии электронейтрального цилиндрического проводника эллиптического сечения. Двустороннее решение ГИУ найдено в случае негладкой границы. Обоснован интервальный метод квадратур с оценкой оставшегося члена и выделением особой точки. Рассмотрены случаи, когда норма ядра ГИУ больше или равна единице и случай нерегулярного ядра. Результаты вычислительных экспериментов показали, что при реализации описанного выше способа снижения нормы, основные трудности связаны с наличием функции ядра, в результате чего погрешность аппроксимации значительно возрастает. По этой причине использование еще более точного сплайнового приближения может оказаться недостаточным для уменьшения нормы ядра ГИУ. В качестве примера рассмотрим случай, когда угол между касательными в угловой точке равен 90 градусам. Оператор регуляризатора построен с использованием интегрального преобразования Меллина. Рассмотрен пример расчета электростатического поля решен для случая, когда контур сечения проводника представляет собой квадрат со стороной единичной длины. Обоснован двусторонний метод с использованием регуляризации.
Ключевые слова: краевая задача, граничные интегральные уравнения, потенциал двойного слоя, интервальный метод, двухсторонний метод
Литература
1. Калмыков С.А., Шокин Ю.И., Юлдашев З.Х. Методы интервального анализа. Новосибирск: Наука, 1986.
2. Добронец Б.С., Шайдуров В.В. Двусторонние методы. Новосибирск: Наука, 1990.
3. Добронец Б.С. Интервальная математика. Красноярск: КГУ. 2004.
4. Moore R.E., Kearfott R.B., Cloud M.J. Introduction to interval analysis. Philadelphia: SIAM, 2009.
5. Михлин С.Г. Линейные уравнения в частных производных. М.: Высш. школа, 1977.
6. Михлин С.Г. Многомерные сингулярные интегральные уравнения. М.: Физматгиз. 1962.
7. Корн Г., Корн Т. Справочник по математике. М.: Наука, 1978.
8. Nekrasov S.A. A bilateral method of solving Cauchy problems. USSR Computational Mathematics and Mathematical Physics. 1986. Vol. 26. No. 3.
9. Nekrasov S.A. The construction of two-sided approximations to the solution of a cauchy problem. USSR Computational Mathematics and Mathematical Physics. 1988. Vol. 28. No. 3.
10. Nekrasov S.A. Bilateral methods for the numerical integration of initial- and boundary-value problems. Computational Mathematics and Mathematical Physics. 1995. Vol. 35. No. 10.
11. Nekrasov S.A. Efficient two-sided methods for the Cauchy problem in the case of large integration intervals. Differential Equations. 2003. Vol. 39. No. 7.
12. Nekrasov S.A. A numerical method for solving dynamical systems with lumped parameters which accounts for an input data error. Journal of Applied and Industrial Mathematics. 2016. Vol. 10, Issue 4.
Информация об авторах:
Арутюнян Тигран Робертович, магистрант, Московский Технический Университет связи и информатики (МТУСИ), Москва, Россия, Некрасов Сергей Александрович, д.т.н., профессор прикладной математики,кафедра прикладной математики Южно-Российского государственного политехнического университета (НПИ), Новочеркасск, Россия
