Symbolic solving of differential equations with partial derivatives Текст научной статьи по специальности «Математика»

Вестник РУДН Серия Математика. Информатика. Физика. № 2 (2). 2010. С. 10-14

УДК 510.676, 519.7

Symbolic Solving of Differential Equations with Partial Derivatives

N. A. Malaschonok

Tambov State University Internatsionalnaya 33, 392622 Tambov, Russia

An algorithm for the symbolic solving of systems of linear partial differential equations by means of multivariate Laplace-Carson transform (LC) is produced. Considered is a system of K linear equations with M as the greatest order of partial derivatives and right hand parts of a special type, that permits a symbolic Laplace-Carson transform. Initial conditions are input. As a result of Laplace-Carson transform of the system according to the initial conditions, we obtain an algebraic system of equations. There exist efficient methods to solve large size systems of such types. It gives a possibility to implement the method for solving the large PDE systems. A method to obtain compatibility conditions is discussed. The application of LC allows one to execute it in a symbolic way.

Ключевые слова: systems of partial differential equations, Laplace-Carson transform, symbolic solving.

1. Introduction

The Laplace transform has been useful in various problems of differential equations theory, including problems of partial equations (for example [?, 1—5]). On the other hand, there are many ways to use computer algebra systems for numerical or symbolic solving of PDE systems, for example the well known usage of MAPLE for characteristics method that permits to simplify equations in many cases (for instance [6,7]).

We produce an algorithm for symbolic solving of systems of linear partial differential equations by means of multivariate Laplace-Carson transform. Considered are the systems of arbitrary number К of unknown functions and equations of arbitrary order M of derivatives in the cases, described in section 2, under conditions a)-b). The method allows one not to reduce (or to reduce to canonical form) the problem at initial stage, it reduces it to solving a linear algebraic system with polynomial coefficients where efficient methods were developed (for example [8-10]). So large systems of linear PDE may be solved in real time.

The application of Laplace-Carson transform permits to obtain compatibility conditions in a symbolic way for many types of PDE equations and systems of PDE equations.

2. Method

Consider the space S of functions f (x), x = (x\,... ,xn) £ R", R" = {x : Xi > 0, i = 1,..., n}, for which M > 0,a = (ai,..., an) £ R", en > 0, i = 1,... ,n,

n

exist such that for all X £ R™ the following is true: |/(^)| < M.eax, ax = aixi.

i=1

On the space S the Laplace-Carson transform (LC) is defined as follows:

LC : f (x) ^ F(p) = p1 J e-pxf (x)dx,

0

n

p =(pi,...,pn), p1 = Pi ...pn, px = ^2 piXi, dx = dxi ... dxn.

i=1

Supported by the Sci. Program "Devel. Sci. Potent. High. School", RNP.2.1.1.1853 and 1474.

Denote m = (m1,..., mn). Consider a system

dm

• Qmt Xl . . . Qmnx

km

J u

akk —^— uk w = fj w, (i)

k=1 m=0 m

where j = 1,... ,K, Uk(x), k = 1,... ,K, — are unknown functions of x = (x1,..., xn) G R+, fj G S, a^ are real numbers, m is the order of a derivative, and k — the number of an unknown function. Here and further summing by m = (m1,... ,mn) is executed for m1 + ... + mn = m.

We solve a problem with initial conditions for each variable. Introduce notations for them. Denote by r v a set of vectors 7 = (7^ ... ,^n) such that = 1, = 0, if i < v, and ■ji equals 0 or 1 in all possible combinations for i > v. The amount of elements in r v equals 2"-1.

Denote fa = (fa,..., fa), fa = 0,..., mi, a set of indexes such that the derivative of uk(x) of the order fai with respect to the variables with numbers i equals up^(x(l)) at the point x = x1 with zeros at the positions ^ for which the coordinates of 7 equal 1. For example, if zeros stand only at the places with the numbers 1, 2, 3, then 7 = (1,1,1, 0,..., 0).

Let LC : uk ^ Uk, up^ (x(l)) ^ Ujk^(p(l)), fj ^ Fj, the notation p(l) is correspondent to the notation x(l). Denote by H71| the "length" of 7 — the number of units in )m = % Then

ry rm — nmi

P = LJ1 . . . Pn

dm

LC : —---Uk (x) ^

dmix1 ...dm™Xn k '

n mu

^Pmuk(P) + j](-1)||7Vr-ki-71 ...prm™-k™-1™(P(1)).

v=1 kv=0 ierv

Denote

m v=1 =0 7er^

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

EEE akk Pm Uk (P) = F3 - £ £ ^ j = 1,...,K. (2)

k=1m=0 m k=lm=0

The algorithm component is the definition of compatible initial conditions. The system (1) should be solved under such conditions.

Denote by D the determinant of the system (2), Di — the maximal order minors of the extended matrix of (2). A case when there is a set Q of zeros of D with infinite limit point at Repk > 0, k = l,...,n is of most interest. Solving the system (2), 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 holomorphic in the domain Re pk > ak,. So we make a demand: Di = 0 at Q. This demand produces requirements to the LC images of initial conditions functions, and after LC-1 transform to initial conditions. They turns to be dependent. We obtain the so-called compatibility conditions.

The algorithm of solving the system (1) consists of four main steps: 1. Laplace-Carson transform of the system (1).

2. Solving of the algebraic system (2).

3. Establishing of compatibility conditions.

4. Inverse Laplace-Carson transform of the solutions of (2) — it is the solution of the system (1).

At the present stage of research we guaranteer symbolic computations if we may carry out the following:

a) For LC:

Represent input functions as sums of exponents with polynomial coefficients.

Call a rational fraction "a proper fraction" if the degree of each variable (over C) in nominator is less than its degree in denominator.

b) For LC-1:

- represent the solutions of algebraic system as sums of proper fractions with exponential coefficients;

- reduce the denominator of these proper fractions to a product of functions linear with respect to each variable.

Note that the class b) does not exhaust all cases that admit pure symbolic computations. We produce two very simple examples: for a case from b) and out of it. It is convenient in these cases to change notations for unknown functions, their Laplace transform, variables, initial conditions.

3. Examples

1. Take a system of two equations with two unknown functions on R +.

df da df dg

+ = + = y> f = f(x, y); 9 = g{x, y).

ax dy dy dx

Initial conditions:

f(0, y) = a(y); f(x, 0) = b(x); g(0, y) = c(y); g(x, 0) = d(x),

LC : f(x, y)^u(p, q), g(x, y) ^ v(p, q), a(y) ^ a(q), b(x) ^ P(p), c(y) ^ 5(q), d(x) ^ 7(p), pu — pa( q) + q v — q7 (p) = l/p, qu — q/3 (p) + pv — pS(q) = 1/q.

Then

_ — ap2 + / q2 + (5 — 7 )pq _ —p2 + q2 + (a — /) p2q2 — (5p2 — 7q2 )pq p2 — q2 ' pq(p2 — q2)

The denominator D: D(p, q) = pq(p2 — q2).

The set of zeros of D with infinite limit points at the right half-plane is q = p. Substituting q = p into the nominator of u and v, we obtain the compatibility condition: a — / + 77 — § = 0.

2 2

For example we may take / = 0; 7 = 5 = a = 0. Then

2 p + 2p2 + q + 2 q2 + 2pq U =--, V =---r-.

p + q pq(p + q)

LC-1 :

f = _(2y, y<x, =((2 + У)x, y<x,

I 2x, y^x, g = \y(2 + x), y^x.

d2f df

2. Consider the equation of parabolic type —^ — —- = xy. Initial conditions:

ox2 ay

fin \ Î \ °f(x' У) f{0, y) = a(y);

o x

К у); f(x, 0) = c(x);

x=0

LC : f(x, y) ^ u(p, q), a(y) ^ a(q), b(y) ^ |3(q), c(x) ^ J(p),

LC: p2u — p2a — pf — qu + q7 = —.

pq

Then

1 + p3q a + p2q f — pq 27 pq( p2 — q)

Substituting p = ^Jg into the nominator of u, we obtain the compatibility condition:

a = 7; 7 = —r-. For example a = 7 = 1, 7 = —r-. Correspondingly, we obtain

2 2

1 + p3q a + p2q f — pq 27 pq( p2 — q) '

2

x 2

and as a result of LC-1 f(x, y) = 1--— under the initial conditions a(y) =

22

c(x) = 1, b(y) = .

4. Conclusion

Let us adduce advantages of the algorithm presented in the paper.

1. The Laplace-Carson transform is a way for symbolic solving of differential equations as it reduces the solution process to algebraic manipulations.

2. Representation of righthand side functions by sums of exponents with polynomial coefficients (in a case when it is possible) makes the Laplace-Carson transform completely symbolic.

3. The algebraic system obtained after the Laplace transform may be solved by methods most convenient and efficient for each specific case.

4. Representation of the solutions of algebraic system as sums of proper fractions with exponential coefficients provides a symbolic character of the inverse Laplace transform.

5. The application of Laplace-Carson transform permits to obtain compatibility conditions in a symbolic way for many types of PDE equations and systems of PDE equations.

Литература

1. Dahiya R. S., Saberi-Nadjafi J. 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. — Pp. 61-74.

2. Dimovski I., Spiridonova M. Computational Approach to Nonlocal Boundary Value Problems by Multivariate Operational Calculus // Mathem. Sciences Research Journal. — 2005. — Vol. 9, No 12. — Pp. 315-329.

3. Malaschonok N. Parallel Laplace Method with Assured Accuracy for Solutions of Differential Equations by Symbolic Computations // Computer Algebra and Scientific Computing, CASC 2006. — LNCS 4196. — Springer, Berlin, 2006. — Pp. 251-261.

4. Burghelea D., Holler S. Laplace Transform, Dynamics and Spectral Geometry. — 2005. — arXiv:math.DG/0405037v2. ArXiv:math.DG/0405037v2.

5. Podlubny I. The Laplace Transform Method for Linear Differential Equations of the Fractional Order. — 1997. — arXiv:funct-an/9710005v1. ArXiv:funct-an/9710005v1.

6. Голоскоков Д. П. // Уравнения математической физики. Решение задач в системе Maple. — СПб: Питер, 2004.

7. Scott A. S. The Method of Characteristics and Conversation Laws // Journal of Online Mathematics and its Applications. — 2003. — http://mathdl.maa.org/ mathDL/4/?pa=content&sa=viewDocument&nodeId=389.

8. Strojohann A. Algorithms for Matrix Canonical Forms // Ph. D. Thesis. — Zurich: Swiss Federal Inst. of Technology., 2000.

9. Malaschonok G. I. Effective Matrix Methods in Commutative Domains // Formal Power Series and Algebraic Combinatorics. — Berlin: Springer, 2000. — Pp. 506517.

10. Malaschonok G. Solution of Systems of Linear Equations by the p-adic Method // Programming and Computer Software. — 2003. — Vol. 29, No 2. — Pp. 59-71.

UDC 510.676, 519.7

Символьное решение дифференциальных уравнений в частных производных

Н. А. Малашонок

Тамбовский государственный университет Интернациональная ул., д. 33, Тамбов, 392622, Россия

Предлагается алгоритм для символьного решения систем дифференциальных уравнений в частных производных посредством многомерного преобразования Лапласа— Карсона. Рассмотрена система К уравнений с M как наивысшим порядком частных производных и правой частью особого типа, который допускает символьное преобразование Лапласа—Карсона. Начальные условия являются входными. В результате Лаплас— Карсоновского преобразования системы по начальным условиям получаем алгебраическую систему уравнений. Существуют эффективные методы решения систем такого типа. Это дает возможность применять предлагаемый метод для решения больших систем уравнений в частных производных. Обсуждается метод получения условий совместности. Применение преобразования Лапласа—Карсона позволяет выполнить это в символьном виде.

Key words and phrases: системы дифференциальных уравнений в частных производных, преобразование Лапласа-Карсона, символьное решение.