A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential

Transcription

1 Avances in Applie Mathematics an Mechanics Av. Appl. Math. Mech. Vol. 1 No. 4 pp DOI: /aamm.09-m0946 August 2009 A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential Wen-Xiu Ma 1 Xiang Gu 2 an Liang Gao 3 1 Department of Mathematics an Statistics University of South Floria Tampa FL USA 2 Department of Physics University of South Floria Tampa FL USA 3 Department of Applie Mathematics Northwestern Polytechnical University Xi an Shaanxi P.R. China Receive 27 April 2009; Accepte (in revise version) 27 May 2009 Available online 18 June 2009 Abstract. It is known that the solution to a Cauchy problem of linear ifferential equations: x (t) = A(t)x(t) with x( ) = x 0 can be presente by the matrix exponential as exp( A(s) s)x 0 if the commutativity conition for the coefficient matrix A(t) hols: A(s) s A(t) = 0. A natural question is whether this is true without the commutativity conition. To give a efinite answer to this question we present two classes of illustrative examples of coefficient matrices which satisfy the chain rule t t exp( A(s) s) = A(t) exp( A(s) s) but o not possess the commutativity conition. The presente matrices consist of finite-times continuously ifferentiable entries or smooth entries. AMS subject classifications: 15A99 34A30 15A24 34A12 Key wors: Cauchy problem chain rule commutativity conition funamental matrix solution. Corresponing author. URL: mawx/ mawx@math.usf.eu (W.X. Ma) xgu@physics.cas.usf.eu (X. Gu) gaoliang box@126.com (L. Gao) c 2009 Global Science Press

2 574 W.X. Ma X. Gu L. Gao / Av. Appl. Math. Mech. 4 (2009) pp Introuction It is known that one of the important problems in the theory of ifferential equations is how to solve Cauchy problems of linear ifferential equations 1. If the funamental matrix solution is foun unique solutions to the Cauchy problems of linear ifferential equations can be automatically presente. However on one han it is not always possible to compute the funamental matrix solution explicitly. On the other han linear ifferential equations are also use in solving nonlinear integrable equations in both continuous an iscrete cases 2 3. Therefore the explicit representation of solutions to the Cauchy problems of linear ifferential equations is a crucial issue in the theory of both linear an nonlinear ifferential equations. Let us specify a system of linear ifferential equations on an interval I=(a b) R as follows: x (t) = A(t)x(t) + f (t) (1.1) where f (t) R n is continuous on I an A(t) is an n n matrix of real continuous functions on I. Any higher-orer scalar linear ifferential equation of Kovalevskaia type can be transforme into the above linear system. Of significant importance in the theory of ifferential equations is how to solve the Cauchy problem on I: x (t) = A(t)x(t) + f (t) x( ) = x 0 where I an x 0 R n are given. Various examples of fining solutions to Cauchy problems of ifferential equations both linear an nonlinear can be foun in 1 4. If the coefficient matrix A(t) commutes with its integral A(s) s: A(t) A(s) s = 0 t I (1.2) then the funamental matrix solution U(t ) of the homogeneous system x (t)= A(t)x(t) is etermine by the matrix exponential (see say 5): U(t ) = exp A(s) s t I. (1.3) That is to say if we have the commutativity conition (1.2) the following chain rule hols: t t exp A(s) s = A(t) exp A(s) s t I (1.4) an so the unique solution to the Cauchy problem (1) is given by the variation of parameters formula: x(t) = U(t )x 0 + U(t s) f (s) s t I. (1.5)

3 W.X. Ma X. Gu L. Gao / Av. Appl. Math. Mech. 4 (2009) pp However if the commutativity conition (1.2) oes not hol then we cannot expect to have the proceeing chain rule (1.4). A counterexample on the real line given by Liu 6 is t 1 A(t) =. (1.6) It is compute in 6 that e t 2 /2 2 t exp A(s) s = (et2 /2 1) an thus the chain rule (1.4) oes not hol for the matrix A(t) etermine by (1.6). It is natural that we woner if the commutativity conition (1.2) is necessary to guarantee the chain rule (1.4) at a given point I. Namely is there any matrix A(t) of real continuous functions which satisfies the chain rule (1.4) but violates the commutativity conition (1.2) at a point I? This is about the solution representation of the Cauchy problem at a point I an thus will help us answer the question raise by Ma an Shekhtman in 7: Can we have the matrix exponential representation (1.3) for the funamental matrix solution without the commutativity conition (1.2)? This is one of the most funamental questions an shoul not be overlooke in the theory of ifferential equations. In this paper we woul like to present two classes of illustrative examples which satisfy the chain rule (1.4) but o not satisfy the commutativity conition (1.2) at a fixe point I. The paper is organize as follows. In Section 2 two classes of concrete examples of continuous matrices are constructe in which matrices consist of finite-times continuously ifferentiable entries or smooth entries. Other matrix forms are also analyze uner two transformations to present ifferent classes of the require matrices. In Section 3 a few further remarks are finally given. 2 Presenting illustrative examples Let I=(a b) R. We will use the following result by Ma an Shekhtman 7 in our construction of illustrative examples. Assume that B(t) = c F(t) t I (2.1) where F(t) is a non-constant ifferentiable function an c is a nonzero complex number satisfying e c =1 + c (see 8 for more etails on such complex numbers c). Then on the interval I we have t eb(t) = B (t) e B(t) but B (t) B(t) 0. (2.2)

4 576 W.X. Ma X. Gu L. Gao / Av. Appl. Math. Mech. 4 (2009) pp Let us now fix I. We are going to construct two classes of continuous matrices A(t) on I which satisfy our requirements: t eb(t) = A(t) e B(t) but A(t) B(t) 0 on I (2.3) where B(t)= A(s) s. That is to say the matrix exponential e B(t) prouces the unique solution e B(t) x 0 to the Cauchy problem of the corresponing homogeneous linear ifferential equations with x( )=x 0 but the commutativity conition (1.2) is not satisfie. Example 2.1. (Matrices with Finite-times Continuously Differentiable Entries) First let m N an we select <t 1 <b an a complex number =0 so that h := 1 sin m (s t 1 ) s = 0 an c :=h satisfies e c =1 + c. Define a continuous matrix A(t) on I as follows: { A1 if t < t A(t) = 1 A 2 if t t 1 sin A 1 = m (t t 1 ) sin m (t t 1 ) 0 g(t) sin A 2 = m (t t 1 ) (2.4a) (2.4b) (2.4c) where g is a nonzero smooth function on t 1 b). Obviously any matrix of this class is (m 1)-times continuously ifferentiable on I but its m-th orer erivative oesn t exist at t=t 1. Actually at t=t 1 its (1 1)th entry has no m-th orer erivative but its (1 2)th entry may have the m-th orer erivative which epens on what kin of functions g(t) the matrix A(t) involves. A irect computation yiels { B1 if t < t B(t) := A(s)s = 1 B 2 if t t 1 B 1 = sin m (s t 1 ) s sin m (s t 1 ) s B 2 = c c + g(s) sin m (s t 1 ) s t 1. (2.5a) (2.5b) (2.5c) Example 2.2. (Matrices with Smooth Entries) Secon we select <t 1 <b an a complex number =0 so that the number c = e 1 t 1

5 W.X. Ma X. Gu L. Gao / Av. Appl. Math. Mech. 4 (2009) pp satisfies e c =1 + c. Define a continuous matrix A(t) on I as follows: A(t) = A 1 = A 2 = A 1 if t < t 1 A 2 if t = t 1 A 3 if t > t 1 (t t 1 ) 2 e 1 t t 1 (t t 1 ) 2 e 1 A 3 = t t 1 0 g(t) e 1 t 1 t (2.6a) (2.6b) (2.6c) where g(t) is a nonzero Laurent polynomial (possibly polynomial) in t t 1. Obviously any matrix of this class is infinitely ifferentiable on I. Actually any orer left-sie an right-sie erivatives of the (1 1)th an (1 2)th entries at t=t 1 are equal to zero. A irect computation gives rise to B(t) := B 1 = B 2 = B 1 if t < t 1 A(s) s = B 2 if t = t 1 B 3 if t > t 1 t t c e 1 1 t t c e 1 1 c c B 3 = (2.7a) (2.7b) c c + g(s) e 1 t 1 t 1 s s. (2.7c) Proof: Now let us show that the above two classes of matrices provie the require examples. Namely we nee to verify that the two classes of matrices efine above satisfy the requirements in (2.3). When t<t 1 we have the commutativity conition: an thus the chain rule hols on the sub-interval (a t 1 ). When t>t 1 the ifferentiable functions A(t) B(t) = 0 (2.8) t eb(t) = A(t) e B(t) (2.9) F(t) = c + g(s) sin m t (s t 1 ) s F(t) = c + g(s) e 1 1 s s t > t 1 t 1 t 1 have nonzero erivatives on the sub-interval (t 1 b) an so they are not constant functions on (t 1 b). It then follows from the aforementione result in (2.1) an (2.2) that

6 578 W.X. Ma X. Gu L. Gao / Av. Appl. Math. Mech. 4 (2009) pp the chain rule (2.9) hols but the commutativity conition (2.8) is not satisfie on the sub-interval (t 1 b). When t=t 1 the chain rule (2.9) follows from the continuity of t eb(t) an A(t)e B(t). To conclue for the above two classes of matrices the chain rule (2.9) hols for all t I but the commutativity conition (2.8) is not satisfie on the whole interval I. Note that two transformations B(t) PB(t)P 1 B(t) B(t) + α(t)i 2 (2.10) where P = P 1 = α(t) is an arbitrary ifferentiable function an I 2 is the ientity matrix of size 2 o not change the properties in (2.2). Therefore base on these two transformations we can also begin with the following forms of basic complex matrices: B(t) = F(t) c 0 F(t) 0 c c 0 F(t) 0 which satisfy (2.2). Similarly other classes of matrices satisfying our requirements in (2.3) can be presente. However all examples of matrices constructe above have complex entries. To construct examples of purely real matrices we use the real matrix representation of complex numbers as in 7: ā ij (t) = Re(aij (t)) Im(a ij (t)) Im(a ij (t)) Re(a ij (t)) (2.11) an introuce real square matrices ā11 (t) ā 12 (t) Ā(t) := ā 21 (t) ā 22 (t) 4 4 (2.12) associate with A(t)=(a ij (t)) 2 2 efine by either (2.4) or (2.6). These matrices give us the require matrices of real continuous functions for which the chain rule (2.9) hols without the commutativity conition (2.8).

7 W.X. Ma X. Gu L. Gao / Av. Appl. Math. Mech. 4 (2009) pp Conclusions an remarks We have analyze a few sets of possibilities to represent solutions to Cauchy problems of linear ifferential equations by the matrix exponential. Two classes of illustrative examples of continuous matrices were presente whose entries are finite-times ifferentiable an smooth respectively. The corresponing systems of linear ifferential equations have the matrix exponential form for solutions to their Cauchy problems but their coefficient matrices o not possess the commutativity conition. We remark that as i in 7 we can construct much bigger size matrices to satisfy our requirements in (2.3) by inserting arbitrary sub-matrices. Two of the key points in our construction above are to use the complex number c satisfying e c =1 + c an to efine coefficient matrices piecewise. The presente examples are counterparts of the insightful examples by Horn an Johnson 9 about the relation between A B = 0 an e A+B = e A e B where A an B are two square matrices of the same size. Furthermore it is known 6 that the chain rule (2.9) may not hol if we o not require the commutativity conition (2.8). Therefore our results also provie a complete supplement to the chain rule without the commutativity conition. It is interesting to note that our illustrative examples using e c =1 + c oes not obey the conition 10: e λ 1(t) λ 2 (t) λ 1 (t) λ 2 (t) 1 = 0 (3.1) where λ 1 (t) an λ 2 (t) are istinct eigenvalues of B(t). It is evient that every matrix B(t) in Section 2 efine by either (2.1) with any t I or (2.5) with t t 1 or (2.7) with t t 1 has two eigenvalues c an 0. The above conition (3.1) with λ 1 =c an λ 2 =0 clearly becomes e c =1 + c which violates our selection criteria for the constant c. Actually the conition (3.1) at a point t I is sufficient to guarantee that the chain rule implies the commutativity t eb(t) = B (t)e B(t) B(t) B (t) = 0 at this point t I (see e.g for etails) but it is not necessary since the matrices B(t) efine by either (2.5) or (2.7) with t=t 1 generate counterexamples. Ziebur 11 gave more general examples of matrices than the class of matrices B(t) in (2.1) which satisfy (2.2) but violate (3.1). Discussions on chain rules for general functions of matrices can also be foun in We finally point out that the efinition of A(t) in our examples epens on the initial time I. It remains an open question whether there is a continuous matrix A(t) not epening on I such that the chair rule (2.9) hols without the commutativity conition (2.8) over the interval I. This is the exact question on the matrix funamental solution of linear ifferential equations presente in 7.

8 580 W.X. Ma X. Gu L. Gao / Av. Appl. Math. Mech. 4 (2009) pp Acknowlegments The work was supporte in part by the Establishe Researcher Grant an the CAS Faculty Development Grant of the University of South Floria Chunhui Plan of the Ministry of Eucation of China Wang Kuancheng Founation the National Natural Science Founation of China (Grant Nos an ) an the Doctorate Founation of Northwestern Polytechnical University (Grant No. CX200616). References 1 E. L. INCE Orinary Differential Equations Dover Publications Inc. New York W. X. MA Complexiton solutions to the Korteweg-e Vries equation Phys. Lett. A 301 (2002) pp W. X. MA AND Y. YOU Rational solutions of the Toa lattice equation in Casoratian form Chaos Solitons & Fractals 22 (2004) pp G. W. BLUMAN AND S. C. ANCO Symmetry an Integration Methos for Differential Equations Springer New York W. MAGNUS On the exponential solution of ifferential equations for a linear operator Communications on Pure an Applie Mathematics Comm. Pure Appl. Math. 7 (1954) pp J. H. LIU A remark on the chin rule for exponential matrix functions The College Mathematics Journal 34 (2003) pp W. X. MA AND B. SHEKHTMAN Do the chain rules for matrix functions hol without commutativity? To appear in Linear an Multilinear Algebra (2009) available online. 8 S. R. VALLURI D. J. JEFFREY AND R. M. CORLESS Some applications of the Lambert W function to physics Can. J. Phys. 78 (2000) pp R. A. HORN AND C. R. JOHNSON Topics in Matrix Analysis Cambrige University Press Cambrige J. F. P. MARTIN On the exponential representation of solutions of linear ifferential equations J. Differ. Equations 4 (1968) pp A.D. ZIEBUR The chain rule for functions with a matrix argument Linear Algebra Appl. 194 (1993) pp A. D. ZIEBUR Chain rules for functions of matrices Linear Algebra Appl. 283 (1998) pp