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International Journal of Mathematics andComputer Applications Research (IJMCAR) ISSN 2249-6955 Vol. 3, Issue 1, Mar 2013, 77-82 © TJPRC Pvt. Ltd. ON SOLVING LINEAR BOUNDARY VALUE PROBLEMS SYMBOLICALLY IN MAPLE SRINIVASARAO THOTA & SHIV DATT KUMAR Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad, India ABSTRACT In this survey paper, we discuss solving linear boundary problems symbolically and new Maple package for treating boundary problems for linear ordinary differential equations, allowing two-/multipoint as well as Stieltjes boundary conditions. We employ the algebra of integro-differential operators for expressing differential operators, boundary conditions, and Green's operators. Implemented IntDiffOp Maple package for solving linear boundary problems include computing Green's operators as well as composing and factoring boundary problems. KEYWORDS: Maple, Linear Boundary Value Problem, Greens Operator, Integro-Differential Algebra, Integro- Differential Operator, Non-Commutative Groebner Bases INTRODUCTION A great deal of differential equations appears in the form of boundary value problems and there is no systematic hold up for solving boundary value problems even though the boundary problems play an important role in many applications and scientific computing. In this paper, we discuss regular boundary problems in Maple with algorithm and example. The first versions of this package with functions of regular boundary problems were introduced by Anja Korporal, Georg Regensburger, and Markus Rosenkranz in [2] [3] and [17]. The main aim of this paper is to distinguish between the algorithms and systems used to solve boundary value problems (BVPs). Early research of symbolic analysis on boundary problems used NCAlgebra but NCAlgebra is rather unreliable in many respects and using TH ORE is much better than NCAlgebra by using a generic preprocessing strategy that avoids the costly computation of a Groebner basis for each BVP. TH ORE is a general platform for proving, solving, computing in mathematics, invented by Bruno Buchberger. In Maple, there is another implementation [16] treating boundary problem for linear ordinary differential equations, allowing two/multipoint as well as Stieltjes boundary conditions. By the use of the algebra of integro-differential operators we can express differential operators, boundary conditions, and Green's operators in Maple. In Section 2 we recall the integro-differential algebras and in Section 2.1 we recall the algebra of integro- differential operators providing the algebraic structure for computing with boundary problems. We provide its implementation in Maple [16], where we use a normal form approach in contrast to [4]. In Section 3 we will present the method of solving linear boundary value problems in Maple and in Section 3 we will give outline of symbolic approach in Maple and in Section 3.2 we will show the computations with an example and finally in Section 4 we will give conclusion. INTEGRO-DIFFERENTIAL ALGEBRAS We first recall the definition of integro-differential algebra with examples, see [3] and [5] for further details. For simplicity, in algebraic settings corresponding to the special choice of ( ) C R F . Definition 2.1. We call ( , , ) F an integro-differential algebra over K if F is a commutative K-algebra with K-linear operators and such that 78 Srinivasarao Thota & Shiv Datt Kumar (2.1) ( ) ' ; (2.2) ( ) ' ' '; (2.3) ( ) ' ( ' ) ( ) ' ( ' ) ( ' ) f f f g f g f g f g f g f g f g are satisfied, where ' is the usual shorthand notation for . We refer to and respectively as the derivation and integral of F and to (2.1) as section axiom. In general, a K-linear operator is usually called a derivation if it satisfies (2.2). Moreover, we call a section of an integral for if it satisfies (2.3). Axiom (2.1) is called the section axiom since it says that 1 F , so is required to be a section of . In differential algebra, axiom (2.2) is commonly called the Leibniz axiom, obviously encoding the product rule of differentiation. In contrast, axiom (2.3) captures integration by parts and is new in this form [2.3]; called it the differential Baxter axiom. The projectors J and 1 E respectively called the initialization and the evaluation of F. We say that an integro-differential algebra over a field K is ordinary if Ker ( ) = K. For an ordinary integro-differential algebra, the evaluation can be interpreted as a multiplicative linear functional (character) : E F K . Example 2.2. Let us see the terminology for the projectors E and J in the standard example ( ) C R F with d dx and . x a Here E f = f (a) evaluates f at the initialization point a, and J f = f – f (a) enforces the initial condition. Example 2.3. The analytic functions on the real interval [a, b] form an integro-differential subalgebra [ , ] C a b of [ , ] C a b over R C K or K . It contains in turn the integro-differential algebra [ , ] Kx K x e of exponential polynomials, defined as the space of all K-linear combinations of , N n x x e n and K . Finally, the algebra of ordinary polynomials K[x] is an integro-differential subalgebra in all cases. Integro- Differential Operators We will continue from the Section 2 to define the algebra of integro-differential. We call ( , , ) F an integro- differential algebra if ( , ) F is a commutative differential algebra over a commutative ring K and s is a K – linear right inverse (section) of ' , meaning ( )' f f , such that the defferential Baxter axiom ( )' ( ' ) ( )' ( ' ) ( ' ) f g f g f g f g holds. We call 1 E the evaluation of F. We say that an integro-differential algebra over a field K is ordinary if Ker ( ) = 1. For an ordinary integro-differential algebra, the evaluation can be interpreted as a multiplicative linear functional (character) : E F K . This allows treating initial value problems, but for doing boundary problems we need additional characters : F K (for example, for ( ) R F C with the usual derivation and the integral operator On Solving Linear Boundary Value Problems Symbolically in Maple 79 : ( ) ÷ x a f f t dt for fixed point R a , evaluations : ( ) ÷ c E f f c at various points R c ). Let ( , , ) F be an ordinary integro-differential algebra over a field K and let * F be a set of characters : F K including E. The integro-differential operators , F are defined in [19] as the K-algebra generated by the symbols and , the "functions" f F and the "functionals" , modulo the Noetherian and confluent rewrite system of Table 2.1. Table 2.1: Rewrite Rules for Integro-Differential Operators ( ) f g f g f f - ' 0 1 f f f ( ) ( ) ' ( ) ( ) f f f f f f E f E f f The representation of integro-differential operators in Maple implementation is based on the fact that every integro-differential operator has a unique normal form as a sum of a differential, integral, and boundary operator. The normal forms of differential operators are as usual i i f , integral operators can be written uniquely (up to bilinearity) as sums of terms of the form f g , and the normal forms of boundary operators are given by , , , (2.4) , i i j j i f g h N with only finitely non-zero summands. Stieltjes boundary conditions are boundary operators where , , , 1 i i j f a K and g . They act on F as linear functional in the dual space F * . From Table 2.1 formulas can be derived for expressing the product of integro-differential operators directly in terms of normal forms. Implementing these formulas leads to faster computations since we need not reduce in each step. In this package, we use for the underlying "integro-differential algebra" all the smooth functions in one variable representable in Maple, together with the usual derivation and the integral operator 0 , x both computed by Maple . We take as characters { | } c E c R . SOLVING LINEAR BVPS IN MAPLE In this section, we discuss how to compute the boundary problems in Maple package. Outline of Symbolic Approach in Maple For an integro-differential algebra F, a boundary problem is given by a monic differential operator 1 1 1 0 n n n T c c c and boundary conditions 0 1 , , n B B . . Given a forcing function f F , we want to find u F such that (2.1) holds. A boundary problem is given by a pair (T, B), where T is a surjective linear map and * B F is an orthogonally closed subspace of the dual space. We call u F a solution of (T, B) for a given f F if Tu = f and u B . A 80 Srinivasarao Thota & Shiv Datt Kumar boundary problem is regular if for each f there exists a unique solution u. The Green's operator of a regular problem maps each f to its unique solution u. We also write 1 ( , ) T B for the Green's operator. A boundary problem is regular iff B is a complement of Ker (T) so that ker F T B as a direct sum. For 1 [ , ] F C a b , a monic differential operator T is always surjective and dim Ker (T) = n < 1. Moreover, variation of constants can be used to compute a distinguished right inverse: If T has order n and 1 , , n u u . is a fundamental system for it, the fundamental right inverse is given by * 1 1 (3.1) , n i i i T u d d where d is the determinant of the Wronskian matrix W for ( 1 , ..., n u u ) and d i the determinant of the matrix W i obtained from W by replacing the i-th column by the n-th unit vector. Equation (3.1) is valid in arbitrary integro-differential algebras provided the n-th order operator T has a fundamental system ( 1 , ..., n u u ) with invertible Wronskian matrix. Regularity of a boundary problem (T, B) can be tested algorithmically as follows. If ( 1 , ..., n u u ) is a basis for Ker (T) and 1 ( , ..., ) n B B for B, we have a regular problem iff the evaluation matrix 1 1 1 1 ( ) ... ( ) ( ) ( ) ... ( ) n m m n B u B u B u B u B u . . is regular and hence this implies m = n. The algorithm for computing the Green's operator is described in detail in [19]; see also [4]. The main steps consist of computing the fundamental right inverse * , T F from a given fundamental system as in B(u) and the projector , P F onto Ker (T) along B . Then the Green's operator is computed as * (1 ) G P T . We created data types for the different kinds of operator, representing integro-differential operators as triples intdiffop(a, b, c), where a is a differential operator, b an integral operator and c a boundary operator. Differential operators are represented as lists diffop( 1 2 f, f, ...) and integral operators as lists of pairs of the form intop(intterm( 1 1 f, g ), intterm( 2 2 f, g ),...). In this implementation, we split sums in the g i and move scalar factors to the coefficients f i . Due to (2.4), a boundary operator boundop contains a list of evaluations at different points. Each evaluation evop is a triple containing the evaluation point, the local part , i i f and the global part , , j j g h . Hence we use boundop(evop(c,evdiffop(f 0 ,…),evintop(evintterm(g 1 , h 1 ),...),...) for the representation of boundary operators. For displaying the operators, we use D for , A for and E[c] for the evaluation E c . For a boundary problem we need to enter a monic differential operator T and a list of boundary conditions (b 1 ,…,b m ) as described in Section (2.4) in the form bp(T, bc(b 1 ,...,b m )). We use the Maple function dsolve for computing a fundamental system of T. Sample Computation We start with the same example as in Section 2, i.e. the boundary problem u’’ = f, u(0) = u(1) = 0. The Green's operator can be compute as follows and from the Green's operator, we can extract the Green's function [3]. On Solving Linear Boundary Value Problems Symbolically in Maple 81 T:= DIFFOP(0,0,1): b1:= BOUNDOP(EVOP(0, EVDIFFOP(1), EVINTOP())): b2 := BOUNDOP(EVOP(1, EVDIFFOP(1), EVINTOP())): Bp := BP(T, BC(b1, b2)); Bp := BP ( D 2 , BC (E[0], E[1] ) ) IsRegular(Bp) true GreensOperator(Bp) ( x . A ) - ( A . x ) - ( ( x E[1] ) . A ) + ( (x E[1] ) . A . x ) GreensFunction(%) x 0 and x and x 1, x x 0 x and x and 1. CONCLUSIONS In this article, we have presented a method for solving linear boundary value problems by symbolic techniques in Maple. Unlike the usual methods transforming the given differential equation and its boundary conditions into a system of polynomial equations that can be solved for the desired Green's operator via noncommutative Grobner bases. REFERENCES 1. Stakgold, Greens functions and boundary value problems. John Wiley & Sons, New York (1979). 2. Rosenkranz, M., Buchberger, B., Engl, H.W.: Solving linear boundary value problems via non-commutative Grobner bases. Appl. 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Reidel Publishing Company, Dordrecht, 1986. 15. Rosenkranz, M., 2003a. The Greens algebra: a polynomial approach to boundary value problems. Ph.D. Thesis, June 2003, Johannes Kepler University, Research Institute of Symbolic Computation, A-4040 Linz, Austria. 16. A. Korporal, G. Regensburger, and M. Rosenkranz: Integro-differential operators via normal forms and boundary problems in Maple, ISSAC, 2010. 17. Korporal, A., Regensburger, G., Rosenkranz, M.: A Maple package for integro-differential operators and boundary problems. ACM Commun. Comput. Algebra 44(3) (2010) 120-122 Also presented as a poster at ISSAC 10. 18. Rosenkranz, M., Regensburger, G.: Solving and factoring boundary problems for linear ordinary differential equations in differential algebras. J. Symbolic Comput. 43(8) (2008) 515-544. 19. Regensburger, G., Rosenkranz, M., Middeke, J.: A skew polynomial approach to integro-differential operators. 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