Solving systems of linear differential equations over integro ... - Ricam

Solving systems of linear differential equations over integro-differential algebras Srinivasarao Thota & Shiv Datt Kumar (joint work with: Markus Rosenkranz) Department of Mathematics Motilal Nehru National Institute of Technology Allahabad, India

International Conference on Applications of Computer Algebra

June 27, 2012 S.Thota & S.D.Kumar ()

Systems of LDEs over IntDiff algebras

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Outline

1

Classical formulation of boundary problems for systems of LDEs

2

Integro-differential algebra

3

Algebraic formulation of boundary problems for systems of LDEs

4

Example

5

Sample computations in Maple

6

References

S.Thota & S.D.Kumar ()


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Systems of linear differential equations

Systems of first order n linear differential equations have the matrix form u 0 (x) = M(x)u(x) + f (x), where u = (ui ), ui ∈ C 1 [a, b], M = (mij ) ∈ C [a, b]n×n is the system matrix and f = (fi ) ∈ C [a, b]n the forcing function. We want to find u in terms of symbolic f .



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Initial value problems An initial-value problem (IVP) is to find u(x) which satisfies u 0 (x) = M(x)u(x) + f (x) and which also satisfies an initial condition of the form u(x0 ) = y0 , where x0 ∈ [a, b] and y0 ∈ Rn . General existence-uniqueness theorem:

Theorem (Apostol [1]) Assume M(x) ∈ C [a, b]n×n is a matrix function continuous on an open interval (a, b). If x0 ∈ [a, b] and y0 ∈ Rn are given, then the system u 0 (x) = M(x)u(x), u(x0 ) = y0 6= 0 has a unique solution u(x).



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Solution of IVPs Theorem (Apostol [1]) Given M(x) ∈ C [a, b]n×n and f (x) ∈ C [a, b]n , both continuous on an open interval (a, b), the solution of the IVP u 0 (x) = M(x)u(x) + f (x), u(x0 ) = y0 on [a, b] is given by the formula u(x) = U(x)U(x0 )

−1

Z

x

y0 + U(x)

U(ξ)−1 f (ξ) dξ,

x0

where U(x) is the fundamental matrix.



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Duhamel’s formula

If M(x) is a constant matrix, then U(x)U(x0 )−1 ≡ e M(x−x0 ) . The precise formula for the solution of IVP is Z x M(x−x0 ) Mx u(x) = e y0 + e e −Mξ f (ξ) dξ, x0

which is called Duhamel’s formula.



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Two-point boundary problems A system of linear differential equations with two-point boundary conditions is: u 0 (x) = M(x)u(x) + f (x), x ∈ [a, b], Γa u(a) + Γb u(b) = 0,

(1) (2)

where Γa , Γb ∈ Rn×n . Note The boundary conditions (2) include initial conditions as a special case, where Γa = I and Γb = 0. Given the fundamental matrix U(x) of the homogeneous linear system and a pair of matrices Γa , Γb given by (2), the boundary condition matrix Γ is defined by Γ = Γa U(a) + Γb U(b).



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Condition for uniqueness of solutions to (1)(2).

Theorem (Starr-Rokhlin [3]) If the matrix Γ = Γa U(a) + Γb U(b) is regular, then there is a unique solution u(x) to the system (1)(2). Furthermore, the solution to the homogeneous system of equations with boundary conditions defined by (2) is u(x) ≡ 0.

Definition (Starr-Rokhlin [3]) A continuous function g : [a, b] × [a, b] → C [a, b]n×n is the Green’s matrix for a boundary problem (1)(2) if 1

∂ ∂x g (x, y )

2

g (x + 0, x) − g (x − 0, x) = I for all x ∈ [a, b],

3

∂ ∂x g (x, y )

4

Γa · g (a, y ) + Γb · g (b, y ) = 0 for all y ∈ [a, b].

is continuous except at x = y , + M(x)g (x, y ) = 0 for all x, y ∈ [a, b], x 6= y ,



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Solution of two-point BPs Theorem (Whyburn [4]) If the matrix Γ is regular, then there exists a unique Green’s matrix g for the system u 0 (x) = M(x)u(x) + f (x), Γa u(a) + Γb u(b) = 0 given by the formula ( g (x, y ) =

U(x)Γ−1 Γa U(a)U(y )−1 −1

−U(x)Γ

Γb U(b)U(y )

−1

if y ≤ x, if y ≥ x,

with U(x), the fundamental matrix for homogeneous system, and the solution is given by Z b u(x) = g (x, ξ)f (ξ) dξ. a



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BPs with Stieltjes boundary conditions More general boundary conditions are known as Stieltjes boundary conditions. Such conditions take the form Z b ∞ X Γi u(zi ) + F (ξ)u(ξ) dξ = 0, i=1

a

where Z = (z1 , z2 , . . . ) is a point sequence of the first species, Γ1 , Γ2 , . . . is an absolutely convergent sequence of constant matrices and F (x) is square matrix of order n and the elements of F (x) are of bounded variation on [a, b]. For two points Z Γa u(a) + Γb u(b) +

b

F (ξ)u(ξ) dξ = 0,

(3)

a

the boundary condition matrix given by (3) with fundamental matrix is Z b Γ = Γa U(a) + Γb U(b) + F (ξ)U(ξ) dξ. (4) a S.Thota & S.D.Kumar ()


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Theorem (Whyburn [4]) If the matrix Γ defined by (4) is regular, then there exists a unique Green’s matrix g (x, y ) for the system u 0 (x) = M(x)u(x) + f (x), Z b Γa u(a) + Γb u(b) + F (ξ)u(ξ) dξ = 0 a

given by the formula Ry ( U(x)Γ−1 Γa U(a) + a F (t)U(t)dt U(y )−1 i h Rb g (x, y ) = −U(x)Γ−1 Γb U(b) + y F (t)U(t)dt U(y )−1

if y ≤ x, if y ≥ x,

where U(x) is the fundamental matrix for the homogeneous system, and the solution of the system is given by Z u(x) =

b

g (x, ξ)f (ξ) dξ. a



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Integro-differential algebra Definition (Rosenkranz-Regensburger [2]) We call (F, ∂, ∫ ) an integro-differential algebra over K if F is a commutative K -algebra with K -linear operators ∂ and ∫ such that (∫ f )0 = f ,

(5)

(fg )0 = f 0 g + fg 0 ,

(6)

(∫ f 0 )(∫ g 0 ) + ∫ (fg )0 = (∫ f 0 )g + f (∫ g 0 )

(7)

are satisfied, where 0 is the usual shorthand notation for ∂. 1 2

3

∂ and ∫ respectively derivation and integral of F. Axiom (5) is called section axiom, Axiom (6) is called Leibnitz axiom, and Axiom (7) is called differential Baxter axiom. The projectors J = ∫ ◦∂ and E = 1 − ∫ ◦∂ respectively called the initialization and the evaluation of F. S.Thota & S.D.Kumar ()


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We apply the action of the operators ∂, ∫ and E componentwise to prove the following.

Proposition Let F be an integro-differential algebra over a field K . Then the matrix ring F n×n is again an integro-differential algebra over K .

Proof. Let F = (Fij ), G = (Gij ) ∈ F n×n . Then the section axiom (5) clearly satisfied, since (∫ P F )0 = F . For i, jP= 1, . . . , n we have P n n n 0 0 0 r =1 (Fir Grj ) = r =1 (Fir Grj ) + r =1 (Fir Grj ), hence (6) is satisfied and we Pn Pn Pn Pnhave also 0 0 0 0 0 r =1 Fir (∫ Grj ), r =1 (∫ Fir )(∫ Grj ) + r =1 ∫ (Fir Grj ) = r =1 (∫ Fir )Grj + hence (7) is also satisfied.



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IVPs over integro-differential algebras

Theorem (Rosenkranz-Regensburger [2]) Let (F, ∂, ∫ ) be an ordinary integro-differential algebra. Given a fundamental matrix U ∈ F n×n , the system u 0 = Mu + f , Eu = 0 has the unique solution u = U ∫ U −1 f , for every f ∈ F n .



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Proof. We have u 0 = (U ∫ U −1 f )0 = U 0 ∫ U −1 f + UU −1 f by Leibnitz axiom and section axiom. Since u1 , . . . , un are solutions of u 0 = Mu, we have MU = U 0 . For checking the initial condition, note that E ∫ U −1 f is already the zero vector, so E(u) = 0 for E is multiplicative. Hence u ∈ F n is a solution of the first order system. For uniqueness, If u is a solution of homogeneous system, then choose c = (c1 , . . . , cn )T ∈ K n such that u = c1 u1 + · · · + cn un and the initial conditions yield E(Uc) = 0. Since E is linear functional , E(Uc) = (EU)c and det EU = E(det U). Since U ∈ F n×n is regular, EU ∈ K n×n is regular, so c = (EU)−1 0 = 0 and u = 0.



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BPs over integro-differential algebras

Restate the system (3) as the following: (D − M)u ≡ u 0 − Mu = f ,

(8)

where ∂I = D, M ∈ F n×n and D − M ∈ F[∂, ∫ ]n×n . Note Stieltjes boundary conditions take a form of a matrix whose entries are (scalar) Stieltjes conditions and these conditions form right a ideal generated by vector characters in the matrix operator ring F[∂, ∫ ]n×n . We want to find an operator G that satisfies (D − M)G = 1 and ΓG = 0, where Γ is a boundary condition matrix. This G is known as the Green’s operator.



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A boundary problem for a systems is given by a pair (T , B), where T ≡ D − M and B ≤ (F ∗ )n is a space of boundary conditions.

Definition A boundary problem is called regular if for every forcing function there exists exactly one solution that satisfies the boundary conditions.

Definition u ∈ F n is a solution of (T , B) for a given f ∈ F n if Tu = f and u ∈ B ⊥ . Let β1 , . . . , βn be a basis for B. Then the boundary problem can be as follows: Given f ∈ F n , find u ∈ F n such that Tu = f , β1 (u) = · · · = βn (u) = 0. The boundary conditions can be any linear functionals. In particular, they can be point evaluations of derivatives or Stieltjes boundary conditions. S.Thota & S.D.Kumar ()


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Algorithmic criterion of regularity

The following Lemma is a generalization of scalar case (Rosenkranz-Regensburger [2]) to vector case.

Lemma A boundary problem (T , B) with dim(Ker (T )) = dim(B) is regular iff the matrix  Pn  Pn i=1 β1i (u1i ) · · · i=1 β1i (uni )   .. .. .. β(u) =   . . . Pn Pn i=1 βni (u1i ) · · · i=1 βni (uni ) is regular, where βi = (βij ) and ui = (uij ), i, j = 1, . . . , n are any basis of B and Ker(T ) respectively.



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Finding the Green’s operator The general algorithm for scalar is applicable to this case. The main steps to compute Green’s operator are: 1

Compute the fundamental right inverse T ∈ F[∂, ∫ ]n×n by T = U ∫ U −1 , with the help of variation-of-parameters formula.

2

Compute the projector P ∈ F[∂, ∫ ]n×n onto Ker(T ) along B ⊥ . This can be done as follows: Let Im(P) = [u1 , . . . , un ] and Ker(P) = [β1 , . . . , βn ]⊥ . Since (T , B) is regular, β(u) is invertible. Set (β 1 , . . . , β n )T = β(u)−1 (β1 , . . . , βn )T . Then the projector P is given by P(u) =

n X

β i (u)ui

i=1

Now the Green’s operator for (T , B) is G = (1 − P)T . S.Thota & S.D.Kumar ()


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Example Consider a system of equations u10 = u2 , u20 = f2 with u1 (0) = 0, u1 (1) = 0. Matrix notation: u 0 = Mu + f , where u = (u1 , u2 )T , f = (0, f2 )T and 0 1 M= 0 0 β1 = (L 0), β2 = (R 0), where L(u) = u1 (0), R(u) = u1 (1) Fundamental Matrix: x U= 1 S.Thota & S.D.Kumar ()

1 0

Fundamental Right Inverse: A xA − Ax T = 0 A Projector: P=

L + xR − xL 0 −L + R 0

Green’s operator: G=

A−x R1 − 0

R1 0

xA − Ax − x 2 R1 − 0 x +A

R1 ! 0

Solution: u(x) = Gf R Rx R1 R1 x −x 0 f2 − 0 xf2 dx − x 2 0 f2 dx, − 0 xf Systems of LDEs over IntDiff algebras

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Sample computation with Maple We used the Mapple package IntDiffOp implemented by Anja Korporal [4]. T:=MATRIXDIFFOP(M); T :=

D 0

−1 D

U:=FundamentalMatrix(T); U :=

x 1

1 0

FRIM:=FundamentalRightInverseMatrix(T); A x.A-A.x FRIM := 0 A



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References

Apostol, Tom M.: Calculus, volume II. John Wiley & Sons (Asia) Pte. Ltd., New York - Chichester - Brisbane - Toronto - Singapore (2002). Brown, R. C., Krall, A. M.: Ordinary differential operators under Stieltjes boundary conditions. Trans. Amer. Math. Soc. 198, 73-92 (1974). Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York - Toronto London (1955). Korporal, A., Regensburger, G., Rosenkranz, M.,: Regular and singular boundary problems in Maple, presented at ISSAC’ 10.



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Rosenkranz, M., Regensburger, G.: Solving and factoring boundary problems for linear ordinary differential equations in differential algebras. J. Symbolic Comput. 43(8), 515-544 (2008). Rosenkranz, M., Regensburger, G., Tec, L., Buchberger, B.: Symbolic analysis for boundary problems: From rewriting to parametrized Gr¨ o bner bases. In Langer, U., Paule, P., eds.: Numerical and Symbolic Scientific Computing: Progress and Prospects. Springer Wien New York, Vienna (2011). Starr, P., Rokhlin, V.: On the numerical solution of two-point boundary value problems II, Research Report YALEU/DCS/RR-802, (1990). William, M. Whyburn: Differential equations with general boundary conditions, Bull, Amer.Math.Soc. Volume 48, Number 10, 692-704 (1942).



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THANK YOU



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