Using Gröbner Bases for Solving
Linear 2Pt Boundary Value Problems

Special Semester on Gröbner Bases
Workshop on Symbolic Analysis
Castle of Hagenberg / Austria
11 May 2006

Markus.Rosenkranz@oeaw.ac.at

Radon Institute for Computational and Applied Mathematics
  Austrian Academy of Sciences
  A-4040 Linz, Austria

Outline of the Talk

Cooperation Prof. Buchberger—Prof. Engl within SFB F013.

Importance of boundary conditions for practical usage of differential equations:

Simple starting point: Linear two–point boundary value problems (briefly BVPs).

Parts (1) and (2):  Doctoral thesis -> JSC.
Parts (3) and (4):  Ongoing joint research with Georg Regensburger.

Two–Point Boundary Value Problems

Given f ∈ (C^∞)[a, b], find u ∈ (C^∞)[a, b] such that:

[a, b]	..................... Finite interval in ℝ
	(C^∞)[a, b]	..................... Smooth functions [a, b]  ->  ℂ
	T		..................... Linear differential operator of order n
	(Subscript[B, 1]),…,(Subscript[B, n])	..................... Two–point boundary operators

For simplicity, we assume constant coefficients in T:

Two–point boundary operators:

Traditional Solution by the Green's Function

We assume regularity:

Then every linear two–point BVP admits to write the solution as:

g	...................... Green's function (([a, b])^2) -> ℂ
	f	...................... Given forcing function [a, b] -> ℂ
	u	...................... Desired solution function [a, b] -> ℂ

Traditional solution method (see e.g. Kamke): Matrix inversion based on a fundamental system for T.

Importance of Green's function: Response to "point sources".

An Example: Beam Deflection

Beam deflected under a loading σ:

Fixed at both ends, but allowed to bend:

Green's function:

The Operator Perspective of BVPs

Everything seems to be on the functional level:

But actually, everything happens on the operator level:

Integro–Differential Algebras

A complex algebra ℱ with basis \(TraditionalForm\`ℱ\^#\), plus 5 linear operations fulfilling these axioms:

Boundary operators as abbreviations: \(TraditionalForm\`f\^← ≡ 
        f - \(∫\^*\)f'\) and \(TraditionalForm\`\(f\^ -> \) ≡ \(∫\_*\)f' - 
          f\).

Standard example: ℱ = (C^∞)[0, 1] with:

Common subalgebra: Exponential polynomials

Other example: \(TraditionalForm\`ℱ = \(C\^∞\)(ℝ\^2)\) with:

Integro–Differential Polynomials

Analogous to differential algebra -> differential polynomials:

Introduce abbreviations for basic operators:

Hence we construct the noncommutative algebra of integro–differential polynomials:

Here 𝒢𝓇𝓃 is the ideal generated by a set of polynomial identities ("Green's system").
Somehow mirroring integro–differential axioms.

The Green's System 1, 2

System["1. Equalities for Isolating Differential Operators",any[f],
({{D A = 1, "DA"}, {D B = -1, "DB"}, {D ⌈f⌉ = ⌈f⌉ D + ⌈f'⌉, "DM"}, {D L = 0, "DL"}, {D R = 0, "DR"}})
]

System["2. Equalities for Isolating Boundary Operators",any[f],
({{L A = 0, "LA"}, {R A = A + B, "RA"}, {L B = A + B, "LB"}, {R B = 0, "RB"}, {L ⌈f⌉ = (f^←) L, "LM"}, {R ⌈f⌉ = (f^→) R, "RM"}, {L L = L, "LL"}, {L R = R, "LR"}, {R L = L, "RL"}, {R R = R, "RR"}})
]

The Green's System 0, 3

System["Equalities for Algebraic Simplication",any[f, g],
({{⌈f⌉ ⌈g⌉ = ⌈f g⌉, "MM"}})
]

System["3. Equalities for Contracting Integration Operators",any[f],
({{\(A\ ⌈f⌉\ A = ⌈\(∫\^*\)f⌉ A - 
                  A\ ⌈\(∫\^*\)f⌉\), "AMA"}, {\(A\ ⌈f⌉\ B = ⌈\(∫\^*\)f⌉ B + 
                  A\ ⌈\(∫\^*\)f⌉\), "AMB"}, {\(B\ ⌈f⌉\ A = ⌈\(∫\_*\)f⌉ A + 
                  B\ ⌈\(∫\_*\)f⌉\), "BMA"}, {\(B\ ⌈f⌉\ B = ⌈\(∫\_*\)f⌉ B - 
                  B\ ⌈\(∫\_*\)f⌉\), "BMB"}, {\(A\ A = ⌈\(∫\^*\)1⌉\ A - 
                  A\ ⌈\(∫\^*\)1⌉\), "AA"}, {\(A\ B = ⌈\(∫\^*\)1⌉\ B + 
                  A\ ⌈\(∫\^*\)1⌉\), "AB"}, {\(B\ A = ⌈\(∫\_*\)1⌉ A + 
                  B\ ⌈\(∫\_*\)1⌉\), "BA"}, {\(B\ B = ⌈\(∫\_*\)1⌉ B - 
                  B\ ⌈\(∫\_*\)1⌉\), "BB"}})
]

The Green's System 4

System["4. Equalities for Absorbing Integration Operators",any[f],
({{A ⌈f⌉ D = -(f^←) 
                    L + ⌈f⌉ - 
                  A ⌈f'⌉, "AMD"}, {B ⌈f⌉ D = (f^→) 
                    R - ⌈f⌉ - 
                  B ⌈f'⌉, "BMD"}, {A D = -L + 1, "AD"}, {B D = R - 1, "BD"}, {\(A ⌈f⌉ 
                  L = ⌈\(∫\^*\)f⌉ L\), "AML"}, {\(B ⌈f⌉ 
                  L = ⌈\(∫\_*\)f⌉ L\), "BML"}, {\(A ⌈f⌉ 
                  R = ⌈\(∫\^*\)f⌉ R\), "AMR"}, {\(B ⌈f⌉ 
                  R = ⌈\(∫\_*\)f⌉ R\), "BMR"}, {\(A\ L = ⌈\(∫\^*\)1⌉\ L\), "AL"}, {\(B\ L = ⌈\(∫\_*\)1⌉\ L\), "BL"}, {\(A\ R = ⌈\(∫\^*\)1⌉\ R\), "AR"}, {\(B\ R = ⌈\(∫\_*\)1⌉\ R\), "BR"}})
]

New Formulation of BVPs

Remember: Given f ∈ (C^∞)[a, b], find u ∈ (C^∞)[a, b] such that T u = f and (Subscript[B, 1]) 
          u = … = (Subscript[B, n]) u = 0.

Hence, for every f ∈ (C^∞)[a, b] we must have:

Or purely on the level of operators:

Beam Deflection Again

Functional formulation from before:

New operator formulation:

Characteristic data: T = (D^4) and \(TraditionalForm\`⟨B\_1, B\_2, B\_3, 
          B\_4⟩ = ⟨LG, 
          RG, 
          LD\^2G, 
          RD\^2G⟩\).

Computation within the Theorema system:

Compute[
      Green[(D^4), ⟨L, R, L (D^2), 
          R (D^2)⟩], by -> GreenEvaluator]

Solving BVPs—A Primitive Approach

Heuristic strategy:

Problematic points in the above strategy:

Observe:

Solving BVPs—A Refined Approach

P = 
      Compute[Proj[
          (D^4), ⟨L, R, L (D^2), 
            R (D^2)⟩], 
        by -> GreenEvaluator]

T◆ = 
      Compute[((((D^4)))^◆), 
        by -> GreenEvaluator]

Crucial Properties of the Green's System:

But there are 233 such S–polynomials! -> Proof automated in Theorema (approximately 2000 lines).

Proof is relative to the axioms of integro–differential algebras.

The Problem Monoid—Ideas

Additive Structure on Boundary Conditions:

Encoding of Boundary Value Problems:

Identifying Boundary Value Problems

Noncommutative Multiplicative Structure on Boundary Value Problems:

Example of Multiplication / Factorization

Decomposing Boundary Value Problems:

In the Problem Monoid:

Abbreviation:

The Problem Monoid—Exact Definition

Formulation in the Language of Green's Algebra:

Problem Factorization versus Operator Factorization

Easy to check:

Notation:

Example from before:

The Factorization Lemma

Every factorization of the differential operator can be lifted to the level of problems:

Iteration yields:

First–Order Green's Operators

Solving Inhomogeneous Initial Value Problems à la Mikusi(Overscript[n, '])ski

Note:

Example:

Solution:

Returning to Green's Operators—on the Functional Level à la Mikusi(Overscript[n, '])ski

Introduce Multiplication on Collection 𝔊 of all Green's Functions of Regular Problems in 𝔅:

Note:

Localization in Noncommutative Rings

Even if R has no zero divisors, it may not have a ring of fractions, i.e. a quotient (skew) field!

The Ore condition turns out to be very tough!

Successful choice: \(TraditionalForm\`R = \(𝔊\^+\)\) and S = 𝔊:

The Ore Condition in the Problem Monoid

Proof of the Ore Condition:

Lots of Fundamental Formulae

The Fundamental Formulae à la Mikusi(Overscript[n, '])ski:

Recall: C = (([D, F])^(-1)) = 
          A - AF

Solving Inhomogeneous Boundary Value Problems à la Mikusi(Overscript[n, '])ski

Solution:

Conclusion

Computer algebra tools (noncommutative polynomials) for handling

Two new recent achievements:

Setting still very simple: variable coefficients, partial differential equations, systems, …?