CC BY
37
UDC 519.688
A PARALLEL ALGORITHM FOR SYMBOLIC SOLVING PARTIAL
DIFFERENTIAL EQUATIONS
© Natalia Aleksandrovna Malaschonok
Tambov State University named after G.E. Derzhavin, Internatsionalnava, 33, Tambov, 392000, Russia, Candidate of Physics and Mathematics, Associate Professor of Mathematical Analysis Department, e-mail: [email protected]
Key words: parallel algorithms; computer algebra; partial differential equations; Laplace-Carson transform; compatibility conditions.
A parallel algorithm for symbolic solving partial differential equations by means of Laplace-Carson transform is produced. The problem is reduced to solving linear algebraic systems with polynomial coefficients, for which efficient parallel algorithms exist. It permits to construct a fast parallel algorithm for systems of partial differential equations. An algorithm includes a procedure to obtain compatibility conditions for initial data.
1 Introduction
An application of Laplace and Laplace-Carson transform is useful in many problems of solving differential equations (for example [1, 2, 3, 4]) It reduces a system of partial differential equations to an algebraic linear system with polynomial coefficients. Parallel algorithms for solving such systems are being developed actively (for example, [5, 6]). It enables to construct parallel algorithms for solving linear partial differential equations with constant coefficients and systems of equations of various order, size and types. The application of Laplace-Carson transform permits to obtain compatibility conditions in symbolic way for many types of PDE equations and systems of PDE equations.
The steps, at which parallel calculations are possible and reasonable we denote by term Block. If indexes are contained, the ways of parallelization are pointed by them.
2 Input data
Denote m = (mi,..., mn), Consider a system
^ ^ __ Q m
E EE j Qmi ...d Xn U*(x) = fj(x), (1)
fc=i m=0 m
where j = 1,... ,K, uk (x), k = 1,..., K, — are unknown functions of x = (xi,..., xn) G R+ , fj G 5, amk are real numbers, m is the order of a derivative, and k -the number of an unknown function. Here and further summing by m = (mi,..., mn) is executed for mi + ... + mn = m. We consider all input functions reducible to the form; fj(t) = fj(x), xj < t < tj+i, i = 1,... ,Ij,xi = 0,+i = w,
where
S
fj(t) = E jt)«*1. i = 1-Ij, j =l,...,k, (2)
s=1
__ri
and j (x) = cSlxl-
Denote by A a class of functions which are reducible to the form (2),
We solve a problem with initial conditions for each variable. Introduce notations for them. Denote bv rv a set of veetors 7 = (y1, ..., Yn) such that yv = 1, 7» = 0 , if i < v, and 7» equals 0 or 1 in all possible combinations for i > v, The number of elements in rv equals 2v-i,
Denote P = (Pb ..., Pn), Pi = 0,..., m», a set of indexes such that the derivative of uk(x) of the order Pi with respect to the variables with numbers i equals Ug (x(y)) at the point x = xY with zeros at the positions ^ for which the coordinates 7^ of 7 equal 1. For example, if zeros stand only at the places with the numbers 1, 2, 3 , the n 7 = (1,1,1, 0,..., 0), Functions Ug7(x(y)) must also belong to A, To be short we shall not write down the expressions for
ulr(x(7))-
The algorithm component is the definition of compatible initial conditions. The system (1) is to be solved under such conditions,
fj
l = 1,...,K.
fj
the bounds of smoothness intervals,
3 Laplace-Carson transform
Consider the space S of functions f (x), x = (xi,...,xn) G R+, R+ = {x : x» ^ 0, i = 1,..., n} , for which M > 0, a = (ai,..., an) G Rn , a» > 0, i = 1,..., n , exist such that for
n
all x G R+ the following is true: |f (x)| ^ Me“^ ax = aixi,
i=1
S
LC : f (x) M F(p) = pW e-pxf (x)dx,
0
p = (pi,... ,pn), p1 = pi.. .pn, n
px = pixi, dx = dxi... dxn.
i=i
LC is performed symbolically at the class A,
4 Parallel LC algorithm
4.1 LC of a system
Let LC : uk M Uk,Ug7(x(y)) m UjkY(p(Y)), fj M Fj , the notation p(Y) is correspondent to the notation x(y) , Denote by ¡71| the “length’’ of 7 — the number of units in 7, pm = pm1 .. .pmn ■
Block 10
The LC of the left-hand side of the system (1) excluding images of initial conditions is written formally.
Block lr
uns trough the set of multiindexes of w^r(xr),
Then
d m
LC ; —-----------—--Uk (x) i——
d mi x1 . ..dm" xn K '
pmuk(p) + Y, £ £(-i)lb"pm1-Sl-71.. .p;;“"“fc“'>-Uksrl(p<Y>).
v=1 =0
Denote
n mv
j = £a^£ £ £(-i)Mp™-*^.. .p;;>-u‘,7(p<Y>).
m v=l =0 Y€rv
As a result of Laplace-Carson transform of the system (1) according to initial conditions we obtain an algebraic system relative to Uk
KM KM
£ ££4*pmuk(p) = f-y, £ <jk= i,...,k. (3)
k=1 m=0 m k=1 m=0
Block 2k
k runs from 1 to K.
These blocks performs LC of the right-hand parts of (1), A allows a further parallelization of calculations,
4.2 Solution of algebraic system Block 3
As a result of Laplace-Carson transform of the system (1) according to initial conditions we obtain the algebraic system (3) relative to Uk,
Efficient methods of parallel solving such systems are developed (for example [5, 6]),
At this stage the problem of definition of compatibility conditions arises (see blocks 4s,5), With respect to compatible conditions we use the inverse Laplace-Carson transform and obtain the correct solution of PDE system,
4.3 Compatibility conditions
Call a rational fraction "a proper fraction " if the degree of each variable (over C) in numerator is less then its degree in denominator.
Call a set of equations, defined by conditions
• the solutions of algebraic system may be represented as sums of proper fractions with exponential coefficients;
• the denominators of these proper fractions may be reduced to a product of linear functions.
(Note that the class B does not exhaust all cases that admit pure symbolic computations.)
Denote by D the determinant of the system (3), Dj the maximal order minors of the
extended matrix of (3), A case when there is a set Q of zeros of D with infinite limit point
at Repk > 0, k = 1,..., n, is of most interest. Solving the system (1) we obtain Uk as
fractions with D in the denominators. The inverse Laplace-Carson transform is possible if ak , k = 1,..., n, exist such that these functions are holomorphie in the domain Re pk > ak . So we make a demand: Dj has zeros at Q of multiplicity not less than multiplicity of corresponding D
and after LC -1 transform - to initial conditions. They turn to be dependent. We obtain the so-called compatibility conditions.
Block 4s
s depends upon the number of relations, from which the compatibility conditions arise.
The blocks calculate the values of numerators at zeros of denominators.
The block implements parallel solving of the system of equations, produced by relations for compatibility conditions.
-1 Uk
-1
5 Example
We take a simple example to demonstrate the method and the places where parallelization is possible.
It is convenient here to change notations for unknown functions, their Laplace transform, variables, initial conditions.
Example 1
Take a system of two equations with two unknown functions on R+ ,
/ = /(x,y); g = g(x,y)-
Initial conditions: /(0,y) = a(y); /(x, 0) = b(x); g(0,y) = c(y); g(x, 0) = d(x),
the class B,
Block 5
Block 6k
Block lr , r 1.2.
a(y) a(q), b(x) ß(p),
c(y) ^ ¿(q^ d(x) ^ Y(p).
Block 2k , k 1.2.
LC:
/(x,y) ^ ^q^ g(x,y) ^ v(p,q).
As a result of LC we obtain the algebraic system:
pu — pa(q) + qv — qY(p) = 1/p, qu — qß (p) + pv — p^(q) = 1/q.
Block 3
Then
—ap2 + ßq2 + (£ — Y)pq —p2 + q2 + (a — ß)p2q2 — (£p2 — Yq2)pq
u =------------------------------v =----------------------------------------------------
p2 — q2 , pq(p2 — q2) .
The denominator D: D(p, q) = pq(p2 — q2).
Block 4s , s=l.
The set of zeros of D with infinite limit points at the right half-plane is q = p. q = p u v
a — ß + y — ^ = 0.
Block 5
For example we may take ß = 0; y = p; ^ = 2; a = 0.
Then
2 p + 2p2 + q + 2q2 + 2pq
u —-----------, v — — , ,
p + q pq(p + q)
Block 6s , h 1.2.
LC-1 :
/ = _ f 2y y < X
f 1 2x, y ^ x,
_ . (2 + yK y < x
g ' y(2 + x), y ^ x.
References
1. Dahiya R.S., Jabar Saheri-Nadjafi. Theorems on n-dimensional Laplace transforms and their applications // 15th Annual Conf. of Applied Math., Univ. of Central Oklahoma, Electr. Journ. of Differential Equations, Conf,02, 1999. P. 61-74.
2. Dimovski I., Spiridonova M. Computational approach to nonlocal boundary value problems by multivariate operational calculus // Mathem. Sciences Research Journal. Dec. 2005. V. 9. N. 12. P. 315-329.
3, Malaschonok N. An algorithm for symbolic solving of differential equations and estimation of accuracy // Computer Algebra in Scientific Computing, CASC 2009, Springer-Verlag Berlin Heidelberg, 2009, P. 213-225,
4, Picone M. Nuovi metodi risolutivi per i problemi d’integrazione delle equazioni lineari a derivate parziali e nuova applicazionne della trasformata multipla di Laplace nel caso delle equazioni a coefficienti costanti // Atti Accad,Sci,Torino,75, 1940, P. 1-14,
5, Watt S.M. Pivot-Free Block Matrix Inversion, Proc 8th International Symposium on Symbolic and Numeric Algorithms in Symbolic Computation (SYNASC), IEEE Computer Society, 2006, P. 151-155, URL: http://www.csd.uwo.ca/ watt/pub/reprints/2006-svnase-bminv.pdf,
6, Malaschonok G.I. Parallel Algorithms of Computer Algebra // Materials of the conference dedicated for the 75 years of the Mathematical and Physical Dep, of Tambov State University, (November 22-24, 2005), Tambov: TSU, 2005, P. 44-56,
GRATITUDES: Supported by the Sci, Program Devel, Sci, Potent, High, School, RNP 2.1.1.1853.
Accepted for publication 7.06.2010.
ПАРАЛЛЕЛЬНЫЙ АЛГОРИТМ СИМВОЛЬНОГО РЕШЕНИЯ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ С ЧАСТНЫМИ
ПРОИЗВОДНЫМИ
© Наталия Александровна Малашонок
Тамбовский государственный университет им. Г.Р. Державина, Интернациональная, 33, Тамбов, 392000, Россия, кандидат физико-математических наук, доцент кафедры математического анализа, e-mail: [email protected]
Ключевые слова: параллельные алгоритмы, компьютерная алгебра, уравнения в частных производных, преобразование Лапласа-Карсона, условия согласованности.
Представлен параллельный алгоритм символьного решения системы уравнений с частными производными с помощью преобразования Лапласа-Карсона. Задача сводится к решению линейной алгебраической системы с полиномиальными коэффициентами, для которой существуют быстрые параллельные алгоритмы, это позволяет сконструировать быстрый параллельный алгоритм для систем дифференциальных уравнений с частными производными. Составной частью алгоритма является процедура получения условий согласованности для начальных условий.
