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ПРОГРАММИРОВАНИЕ
MSC 35K70, 35B45, 35R02, 35A02, 65M32 DOI: 10.14529/mmpl40309
HOFF’S MODEL ON A GEOMETRIC GRAPH. SIMULATIONS
A.A. Bayazitova, South Ural State University, Chelyabinsk, Russian Federation, [email protected]
This article studies numerically the solutions to the Showalter-Sidorov (Cauchy) initial value problem and inverse problems for the generalized Hoff model. Basing on the phase space method and a modified Galerkin method, we develop numerical algorithms to solve initial-boundary value problems and inverse problems for this model and implement them as a software bundle in the symbolic computation package Maple 15.0. Hoff’s model describes the dynamics of H-beam construction. Hoff’s equation, set up on each edge of a graph, describes the buckling of the H-beam.
The inverse problem consists in finding the unknown coefficients using additional measurements, which account for the change of the rate in buckling dynamics at the initial and terminal points of the beam at the initial moment. This investigation rests on the results of the theory of semi-linear Sobolev-type equations, as the initial-boundary value problem for the corresponding system of partial differential equations reduces to the abstract Showalter-Sidorov (Cauchy) problem for the Sobolev-type equation. In each example we calculate the eigenvalues and eigenfunctions of the Sturm-Liouville operator on the graph and find the solution in the form of the Galerkin sum of a few first eigenfunctions. Software enables us to graph the numerical solution and visualize the phase space of the equations of the specified problems. The results may be useful for specialists in the field of mathematical physics and mathematical modelling.
Keywords: Sobolev-type equation; Hoff’s model.
Introduction
Take a finite connected oriented graph G = G(V; E) with vertex set V = [V] and edge set E = [Ej]. Associate to the each edge Ej two positive integers j, dj E R+, which have physical meaning in the context of our problem: the length and area of the cross-section of the edge.
Ej
set up the generalized Hoff equation
XjUjt + Ujxxt = ®\jUj + a2juj + ... + anju2n 1, n E N, (1)
and at each vertex we impose the boundary conditions
Uj (0j t) = Uk (0, t) = um(lmj t) = up(lpj t), Ej , Ek E E<a(Vi), Emj Ep E E (Vi) j (2)
^ ^ djujx(0j t) ^ ^ dmumx(lmjt') 0j (3)
j: Ej ^Ea{Vi) m: Em^E^ {Vi)
where E“(V,^d Ew (Vi) are the sets of edges with the source and target vertex Vi respectively. The parameters Xj E R+ correspond to the load on beam j, and the
parameters asj £ R for s = 1, 2,..., n characterize the material of beam j. The unknown functions Uj (x, t) of x £ (0, lj) and t £ R specify the deflection of beam j from the vertical direction.
The direct problem consists in finding the vector function u = (u1,u2, ...,Uj,...), each component Uj(x,t) of which satisfies the continuity condition (2) and the flow balance condition (3). Besides, the components Uj(x,t) must satisfy the Cauchy initial conditions
Uj(x, 0) = Uj0(x), x £ (0,lj). (4)
or the Showalter-Sidorov initial conditions
(xj + (Uj(x’0) — Uj°(x)) = 0 x £ (0,lj). ^
The inverse problem (in the case n = 2) consists in finding the solutions Uj to (1) and
the unknown coefficients aj = a1j and [j = a2j using the additional measurements
aj U0j(0) + [j U0j(0) = Vj, aj U0j (lj) + [j U0j (lj) = ^j, (6)
where Vj and ipj reflect the change of the rate in buckling dynamics at the endpoints of
the beam at the initial moment.
The paper aims to generalize the results of [1,2] to the case of the inverse problem for Hoff’s model with the Showalter-Sidorov conditions and to develop numerical algorithms for solving the stated problems. Hoff’s equation belongs to the class of semilinear Sobolev-type equations; for this reason we use here the theory of degenerate semigroups and relatively p-bounded operators. For more details concerning the theory of semigroups and Sobolev-type equations, as well as Cauchy and Showalter-Sidorov problems, see [3-7].
1. Hoff’s Model
Following [1,2], reduce (l)-(4) to the Cauchy problem u(0) = u0, and (l)-(3), (5) to the Showalter-Sidorov problem L(u(0) — u0) = 0 for the semilinear Sobolev-type equation LU = Mu + N(u). Consider the Hilbert space L2(G) as well as two spaces U and F with the natural structure of a Banach rather than Hilbert space. Define the operators L, M, N : U ^ F as
(Lu,v) = dj / (\jUjVj — Ujxvjx)dx, (Mu,v) = aijdj / UjVjdx,
j Jo j Jo
(N(u),v) = d^a2j J ijvjdx + ... + anj J u2'n-1Vjdx^ ,
where (■, ■) is the inner product on L2(G). For all a2j,a3j,...,anj £ R we have N £ C^(U; F) The operators L and M lie in L(U; F), that is, are linear and continuous. Moreover, L is Fredholm (that is, ind L = 0), while M is compact and (L, 0)-bounded whenever ker L = {0} or ker L = {0} and all coefficients a1j are nonzero and of the same sign.
Let us state the main results as theorems on the solvability of direct and inverse problems in the case of the Cauchy problem and Showalter-Sidorov problem for Hoff’s equations on the graph (the first three theorems are proved in [1,2]).
Theorem 1.
(i) If ker L = {0} then the phase space of (1) is U.
(ii) If ker L = {0} al I a1j are nonzero and of the same sign as all nonzero coefficients
asj for s = 2,n then the phase space of (1) is the simple manifold
M = {u E U : (Mu + N(u),Xk) = 0,missing}
where ker L = span{xfc : k = 1, 2,l} is an L2(G)-orthonormal basis of ker L identified with a basis of cokerL.
Denote by D the set of admissible values of the vectors ф and ф for which the solutions to the inverse problem are the coefficients aj and ,5j of the same sign (aj > 0 fij > 0 or aj < 0 ftj < 0 for all j). The structure of the set D is described in [2].
Theorem 2.
(i) if ker L = {0} then for all u0 E Urnd ф^ фj E R satisfying u0j (0) = 0 u0j (lj) = 0,
u0j(0) = ±u0j(lj^ фju^j(lj) = фju^j(0), and фju0j(lj) = фju0j(0) there exists a unique
solution to the inverse problem (l)-(4), (6).
(ii) If ker L = {0} then for all ф,ф E D and u0 = (u0i,u02,...,u0j,...) E U satisfying uoj(0) = 0 u0j(lj) = 0 u0j(0) = ±u0j(lj); and
\ d J u0j (lj) _ фj u0j (0))u0j + (ф' u0j(0) “ фj u0j (lj ))u0j ) dx> Xkj = 0,
\ j j 0 '
there exists a unique solution u E U, aj, fij E R\{0} to the inverse problem (l)-(4), (6) with aj (3j E R+.
Theorem 3.
(i) If ker L = {0} then for all u0 E U there exists a unique solution to the Showalter-Sidorov problem (l)-(3), (5).
(ii) If ker L = {0} and all coefficients asj = 0 fo r s = 1, ...,n are of the same sign, then u0 E U
Theorem 4.
(i) If ker L = {0} then for all u0 E U, фj, фj E R satisfyi ng u0j (0) = 0 u0j (lj ) = 0, u0j(0) = ±u0j(lj), фju^j(lj) = фju^j(0), and фju0j(lj) = фju0j(0) there exists a unique solution to the inverse problem (l)-(3), (5), (6).
(ii) If ker L = {0} then for all vectors ф,ф E D and u0 E U satisfyi ng u0j (0) = 0, u0j(lj) = 0, and u0j(0) = ±u0j(lj) there exists a unique solution u E U, aj, fij E R\{0} to the inverse problem (l)-(3), (5), (6) with ajfij E R+.
Proof, (ii) Suppose that u0 E U and the conditions u0j (0) = 0 u0j (lj) = 0 u0j (0) = ±u0j (lj) are fulfilled. Then there exists a unique pair of coefficients aj and fij satisfying (6). The condition ф,ф E D ensures that ajfij > 0 for all j. Moreover, by the construction of D, all coefficie nts a j are nonzero and of the same sign as the nonz его coefficients j Hence, the hypotheses of claim (ii) of Theorem 3 hold, and so there exists a unique solution u E U, aj, f3j E R\{0} to the inverse problem (l)-(3), (5), (6) with ajfij E R+.
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86 Вестник ЮУрГУ. Серия «Математическое моделирование и программирование»
2. The Results of Simulations
Basing on the theoretical results for Hoff’s model on a graph, we wrote a software bundle in the symbolic algebra package Maple 15.0. It enables us to:
(1) find numerical solutions to direct and inverse problems with given coefficients Xj, aij, j and ipj as Galerkin sums of the first few eigenfunctions;
(2) draw the graphs of the solutions to Hoff’s model on a graph;
(3) draw the phase space of Hoff’s model on a graph.
Example 1. Take the graph G with three vertiees V2, and V3 and two adjacent edges Ei, of length ¡i = n and area di = 1 of the cross-section, and E2, of length l2 = n and area d2 = 1 of the cross-section.
Consider on G the equations of Hoff’s mo del with Xi = 1, X2 = 1, aii = —10, a2i = —0, 5 ai2 = —0, 5 a22 = —0,19:
{
Ult + u\xxt + 10ui + 0, 5«3 — 0 u2t + u2xxt + 0, 5U2 + 0,19u3 — 0.
3 _ n (7)
The continuity conditions are ui(n,t) = u2(0,t), and the flow balance conditions are uix(0,t) = 0 ulx(n,t) = u2x(0,t), and u2x(n,t) = 0. Seek the solution to (7) as the Galerkin sums
N
uN (x,t) — Y1
i=l
N
uN (x,t) — Y1 vi(t)(i(x)’
i=l
where xi(x) = (&(x),(i(x)) are the eigenfunctions of the Sturm-Liouville problem on the graph. For N = 3, the eigenvalues and eigenfunctions are
Vi = 1 Xi = (1,1),
M2 = 0, 75, X2 = (cos(X), — sin(X)),
fi3 = 0,x3 = (cos x, — cos x),
so we look for the solution in the form
x
ui(x, t) = vi(t) + v2(t) cosx + v3(t) cos(^),
x
U2(x,t) = Vi(t) — V2(t) cos x — v3(t) sin(^).
In the case Xi = X2 = 1 claim (u) of 1 applies because 0 G (r(L).
Multiplying (7) by the functions Xk for k = 1, 2, 3, we obtain the system of differential
Table 1
The numerical solution to (8) with initial data ^i(Q) = -0, 01,v2(0) = -0, 01,v3(0) = 0, 02______________
t v1(t) v2 (t) v3(t)
0 —0, 01 -0,00738 0,02
0,05 —0,010116 —0,006685 0,018122
0,10 —0,009997 —0,006173 0,016736
0,15 —0,009748 —0,005774 0,015652
0,20 —0,00943 —0,005443 0,014757
0,25 —0,00908 —0,005159 0,013985
0,30 —0,008718 —0,004905 0,013296
equations
0, 4786v|(t)v3(t) + 0, 248v3(t) + 6, 2v3(t) + 0, 62v2(t)v3(t) +
+ 1, 736v1(t)v2(t)v3(t) + 1, 626v2(t)v32(t) + 16, 808v2(t) +
+3, 252v2(t)v2(t) + 0, 813v3(t) + 1, 626v1(t)vf(t) = 0,
1, 24v1(t)v2(t)v3(t) + 18, 6v3(t) + 0, 413v3(t) + 0, 868v|(t)v3(t) +
< +1, 86v2(t)v3(t) + 33, 615v1(t) + 3, 252v1(t)vf (t) + 6, 283v;1(t)+ (8)
+2,168v3(t) + 1, 626v2(t)v2(t) + 3, 252v1(t)v2(t) = 0,
0, 868v1(t)v|(t) + 18, 6v1(t) + 1, 24v1(t)v2(t) + 0, 744v2(t)v2(t) + 0, 62v2(t)v2(t)
+0,159v3(t) + 6, 2v2(t) + 0, 62v3(t) + 1, 626v|(t)v3(t) + 3, 252v2(t)v3(t)
^ +16, 808v3(t) + 3, 252v1(t)v2(t)v3(t) + 0, 813v3(t) + 2, 356v3(t) = 0.
Figure l.a depicts its phase space.
Let us solve the Showalter-Sidorov problem for (8) with the data v1(0) = —0,01, v2(0) = —0,01, and v3(0) = 0, 02, which corresponds to the initial condition
u1(x, 0) = —0, 01 — 0, 01 cos x + 0,02 cos(^)
u2(x, 0) = —0, 01 — 0, 01 cos x — 0,02 sin(^)
for (7). Table 1 lists some values of the solution to (8) in a neighbourhood of the point t=0
Example 2. Take the graph G with three vertiees V1, V2, and V3 and two adjacent edges
E1} of length l1 = n and area d1 = 7 of cross-section, and E2, of length l2 = n and area
d2 = 1 of cross-section.
Consider on G the equations of Hoff’s mo del with a11 = —1, a21 = —0, 5 a12 = —0, 7, «22 = —0, 9, A1 = 1, A2 = 4:
f u1t + u1xxt + u1 + Q, 5u3 = 0 /Q\
\ 4u2t + u2xxt + 0, 7u2 + 0, 9u3 = 0.
The continuity condition is u1(n,t) = u2(0,t), and the flow balance conditions are
7u1x(0,t) = 0 7u1x(n,t) = u2x(0,t^^d u2x(n,t) = 0. Seek the solution as the Galerkin
+ u(x.t)
Fig. 1. A1 = A2 = 1
v1(0) = —0, 01,v2(0) = —0, 01,v3(0) = 0, 02
sums
N N
uN(x,t) = Y1 vi(t)&(x), uN(x,t) = J2 vi(t)(i(x),
i=1 i=1
where xi(x) = (&(x),(i(x)) are the eigenfunctions of the Sturm-Liouville problem on the graph. For N = 2 the eigenvalues and eigenfunctions are
M1 = 0, X1 = (cosx, — cos 2x),
15 ( x 7x . 7x x
a2 = — ,x2 = cos —, cos--------sin — ,
P2 1^A2 V 4 4 4;’
and so we look for the solution in the form
x
u1(t, x) = v1(t) cos x + v2(t) cos 4,
7x 7x
u2 (t,x) = — v1 (t) cos 2x + v2 (t)(cos —— sin —).
A1 = 1 A2 = 4
0 £ v(L).
Multiplying (11) by the functions xk for k =1, 2, we obtain the system of differential equations
f —7v^(t)v2(t) + 93v1(t)v|(t) + 33v3(t) + 963v1 (t) + 30v2(t) + 4v3(t) = 0, , ,
\ 142v2(t) + 136v2(t) — 0, 2v^(t) + 3v1(t) + 9v2(t)v2(t) + 1v1(t)v|(t) + 5v3(t) = 0.
Figure 2.a depicts the phase space of (10).
v1(0) = —0, 01
v2(0) = —0, 01, and v3(0) = 0, 02, which corresponds to the initial condition
u1(x, 0) = —0, 01 — 0, 01 cosx + 0,02 cos(^), u2(x, 0) = —0, 01 — 0, 01 cos x — 0, 02sin(^)
Table 2
The numerical solution to (10) with the initial condition Ui(0) = —0, 032, u2(0) = 1
t u1(t) u2(t)
0 —0,0326786 1
0,4 —0,020976 0,65291
0,8 —0,013661 0,428574
1,0 —0,011055 0,347559
1,2 —0,008956 0,281959
1,4 —0,00726 0,228795
1,6 —0,005889 0,185683
1,8 —0,004778 0,150710
2,0 —0,003877 0,122333
X
Fig. 2. (a) The phase space of (10) with X1 = 1 and X2 = 4; (b) The solution to (10) with w1(0) = —0, 032, u2(0) = 1, and X1 = 1, X2 = 4
for (11). Table 2 lists some values of the solution to (10) in a neighbourhood of the point t = 0, while Figure 2.b depicts its graph.
Example 3. Take the finite oriented graph G with three vertices V1t V2, and V3, and with adjacent edges E1, of length l1 = n and area d1 = 7 of cross-section, and E2, of length l2 = n and d2 = 1 area of cross-section.
G X1 = 1 X2 = 4
f U1t + u1xxt + o: 11u 1 + a21u1 = 0 , .
\ 4u2t + u2xxt + a12u2 + a22U2 = 0-
The continuity condition is u1(n,t) = u2(0,t), and the flow balance conditions are 7u1x(0,t) = 0 7u1x(n,t) = u2x(0,t^, and u2x(n,t) = 0. We have to find the numerical solution to the inverse problem with Showalter-Sidorov initial conditions with p1 = —0,06,
p2 = —0, 6, = —0,096, ^2 = —0,96,
x
u10(x) = —0,1 cos x + 0, 3 cos 4,
7x 7x
u20(x) = 0,1 cos 2x + 0, 3(cos —— sin —).
Theorem 2 implies that
u10(0) = 0, 2, u10(n) = 0, 05(2 + 3>/2) « 0, 312, u20(0) = 0, 4, u20(n) = 0,1(1 + 3>/2) ~ 0, 524,
p1 = —0, 06 p2 = —0, 6 ^1 = —0, 096, ^2 = —0,96. Therefore, ö1 = 0, 003579 = 0 and ö2 = 0,024 = 0, so the hypotheses of claim (ii) of Theorem 2 hold. The unknown coefficients a1 = —0, 295, a2 = —1, 03, ß1 = —0,134, and ß2 = —2, 9 are nonzero and of the same sign.
The author is grateful to Georgy Anatolevich Sviridyuk for his support and interest in this work.
References
1. Bayazitova A.A. [The Showalter - Sidorov Problem for the Hoff Model on a Geometric Graph]. The Bulletin of Irkutsk State University. Series "Mathematics", 2011, vol. 4, no. 1, pp. 2-8. (in Russian)
2. Sviridyuk G.A., Bayazitova A.A. On Direct and Inverse Problems for the Hoff Equations on Graph. Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya Fiz.-Mat. NauM, 2009, no. 1 (18), pp. 6-17. (in Russian)
3. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Köln, Tokyo, VSP, 2003. DOI: 10.1515/9783110915501
4. Sviridyuk G.A., Zagrebina S.A. [The Showalter - Sidorov Problem as a Phenomena of the Sobolev-type Equations]. The Bulletin of Irkutsk State University. Series "Mathematics", 2010, vol. 3, no. 1, pp. 104-125. (in Russian)
5. Kostin, V.A. Towards the Solomyak - Yosida Theorem on Analytic Semigroups. St. Petersburg Mathematical Journal, 2000, vol. 11, no. 1, pp. 91-106.
6. Sviridyuk G.A., Brychev S.V. Numerical Solution of Systems of Equations of Leont’ev Type. Russian Mathematics (Izvestiya VUZ. Matematika), 2003, vol. 47, no. 8, pp. 44-50.
7. Sviridyuk G.A. Phase Spaces of Sobolev Type Semilinear Equations with a Relatively Strongly Sectorial Operator. St. Petersburg Mathematical Journal, 1995, vol. 6, no. 5, pp. 1109-1126.
Received May 7, 2014
МОДЕЛЬ ХОФФА НА ГЕОМЕТРИЧЕСКОМ ГРАФЕ. ВЫЧИСЛИТЕЛЬНЫЙ ЭКСПЕРИМЕНТ
A.A. Баязитова
Целью статьи является численное исследование задачи Шоуолтера-Сидорова (Коши) и обратной задачи для обобщенной модели Хоффа. На основе метода фазового пространства и модифицированного метода Галеркина разработаны алгоритмы численного решения начально-краевой и обратной задач для указанной модели и реализована в виде комплекса программ в системе компьютерной математики Maple 15.0. Уравнение Хоффа, заданное на каждом ребре графа, описывает выпучивание двутавровой балки, а модель Хоффа описывает динамику конструкции из двутавровых балок. Обратная задача состоит в определении неизвестных коэффициентов по результатам дополнительных измерений, показывающих изменение скорости динамики выпучивания в начале и конце балки в начальный промежуток времени. Проведенное исследование основано на результатах теории полулинейных уравнений соболевского типа, поскольку начально-краевая задача для соответствующей системы дифференциальных уравнений в частных производных сводится к абстрактной задаче Шоуолтера - Сидорова (Коши) для уравнений соболевского типа. В приведенных примерах вычисляются собственные значения и собственные функции для оператора Штурма-Лиувилля на графе, находится решение в виде галеркинской суммы по нескольким первым собственным функциям.
Ключевые слова: уравнение соболевского типа; модель Хоффа.
Литература
1. Баязитова, A.A. Задача Шоуолтера - Сидорова для модели Хоффа на геометрическом графе / A.A. Баязитова // Известия Иркутского государственного университета. Серия: Математика. - 2011. - Т. 4, 3Tä 1. - С. 2-8.
2. Свиридюк, Г. А. О прямой и обратной задачах для уравнений Хоффа на графе / Г.А. Свиридюк, A.A. Баязитова // Вестник Самарского государственного технического университета. Серия: Физико-математические науки. - 2009. - 3Tä 1 (18). _ с. 6-17.
3. Sviridyuk, G.A. Linear Sobolev Type Equations and Degenerate Semigroups of Operators / G.A. Sviridyuk, V.E. Fedorov. - Utrecht, Boston, Köln, VSP, 2003.
4. Свиридюк, Г.А. Задача Шоуолтера - Сидорова как феномен уравнений соболевского типа / Г.А. Свиридюк // Известия Иркутского государственного универси-тетата. Серия: Математика. - 2010. - Т. 3. № 1. - С. 104-125.
5. Костин, В.А. К теореме Соломяка - Иоспды для аналитических полугрупп /
В.А. Костин // Алгебра и анализ. - 1999. - Т. 11, 3Tä 1. - С. 118-140.
6. Свиридюк, Г.А. Численное решение систем уравнений леонтьевского типа / Г.А. Свиридюк, С.В. Брычев // Известия высших учебных заведений. Математика. - 2003. - № 8. - С. 46-52.
7. Свиридюк, Г.А. Фазовые пространства полулинейных уравнений типа Соболева с относительно сильно секториальным оператором / Г.А. Свиридюк // Алгебра и анализ. - 1994. - Т. 6, № 5. - С. 252-272.
Альфия Адыгамовна Баязитова, кандидат физико-математических наук, кафедра «Математический и функциональный анализ>, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].
Поступила в редакцию 1 мая 20Ц г.
