On linear ODEs with a time singularity of the first kind and unsmooth inhomogeneity

Transcription

1 Rachůnková e al. RESEARCH On linear ODEs wih a ime singulariy of he firs kind and unsmooh inhomogeneiy Irena Rachůnková, Svaoslav Saněk, Jana Vampolová and Ewa B Weinmüller 2* * Correspondence: ewa.weinmueller@uwien.ac.a 2 Deparmen for Analysis and Scienific Compuing, Vienna Universiy of Technology, Wiedner Haupsraße 8, A-4 Wien, Ausria Full lis of auhor informaion is available a he end of he aricle Absrac In his paper we invesigae analyical properies of sysems of linear ordinary differenial equaions (ODEs) wih unsmooh noninegrable inhomogeneiies and a ime singulariy of he firs kind. We are especially ineresed in specifying he srucure of general linear wo-poin boundary condiions guaranying exisence and uniqueness of soluions which are coninuous on he closed inerval including he singular poin. Moreover, we sudy he convergence behaviour of collocaion schemes applied o solve he problem numerically. Our heoreical resuls are suppored by numerical experimens. Keywords: linear sysems of ODEs; singular boundary value problem; ime singulariy of he firs kind; unsmooh inhomogeneiy; exisence and uniqueness; collocaion mehod; convergence AMS Subjec Classificaion: 34A2; 34A3; 34B5 Inroducion Singular boundary value problems (BVPs) arise in numerous applicaions in naural sciences and engineering and herefore, since many years, hey are in focus of exensive invesigaions. An imporan class of linear singular problems akes he form of he following BVP: y () = M α y() + f(), (, ], B y() + B y() = β, () where α, y is an n-dimensional real funcion, M is n n marix and f is an n-dimensional funcion which is a leas coninuous f C[, ]. We are mainly ineresed o find ou under which circumsances he above problem has a soluion y C[, ]. B and B are consan marices and i urns ou ha hey are subjec o cerain resricions for a well-posed problem wih a unique coninuous soluion. If α =, hen we say ha BVP () has a ime singulariy of he firs kind, while for α > i has a ime singulariy of he second kind, commonly referred o as essenial singulariy. Problems of ype (), where f may depend in addiion on he space variable y and may have space singulariy a y =, have been sudied in [2, 2, 23, 24]. The analyical properies of () have been discussed in [9, ], where he aenion was focused on he exisence and uniqueness of soluions and heir smoohness. Especially, he srucure of he boundary condiions which are necessary and sufficien for () o be well-posed was of special ineres. Our aim is o generalize hese analyical resuls

2 Rachůnková e al. Page 2 of 4 o he problem y () = M f() y() +, (, ], B y() + B y() = β, (2) where f C[, ] bu f()/ may no be inegrable on [, ]. While for he BVP () and is applicaions, linear and nonlinear, comprehensive lieraure is available, his is no he case for problem (2). Le us add ha he equaion in (2) can be obained from he regular equaion u (x) = Mu(x) + g(x) considered on [, ) by he ransformaion x = ln, and problems of ype (2) arise in he modelling of he avalanche run up [2]. Furher, we refer o papers [3, 4, 2, 6, 22], where he solvabiliy of close linear singular problems is discussed. Ineresing resuls abou well-posedness for linear boundary value problems wih ime singulariies in weighspaces have been proved in [, 3, 4, 5]. Alhough his framework is close o wha we are aiming a here, i is no quie complee. So, in a way our resuls are closing he exising gaps. Noe ha analyical resuls which are focused on he unique solvabiliy of equaion in (2) can be found in [25, 26] where, in linear case, he main aim is o describe smooh paricular soluion of he equaion and boundary condiions are no invesigaed here. The collocaion scheme proposed o solve he problem numerically, is based on an approximaion of he associaed inegral equaion and is less accurae han he scheme proposed here. To compue he numerical soluion of () polynomial collocaion was proposed in [8]. This was moivaed by is advanageous convergence properies for (), while in he presence of a singulariy oher high order mehods show order reducions and become inefficien []. Consequenly, we have implemened wo Malab collocaion based codes for singular BVPs [5, 7]. The code sbvp solves explici firs order ODEs [5], while bvpsuie can be applied o arbirary order problems also in implici formulaion and o differenial algebraic equaions [7]. Over recen years, boh codes were applied o simulae singular BVPs imporan for applicaions and proved o work dependably and efficienly. This was our moivaion o also propose and analyse polynomial collocaion for he approximaion of he iniial value problems (IVPs) (2). The paper is organized as follows: In Secion 2, we collec he preliminary resuls and inroduce he necessary noaion. In Secions 3, 4, and 5, hree case sudies are carried ou, he case of only negaive real pars of he eigenvalues of M, posiive real pars of he eigenvalues of M, and zero eigenvalues of M, respecively. These resuls are summarized and compared wih he case of smooh inhomogeneiy in Secion 6. Finally, he hree case sudies are used o formulae he respecive resuls for he general IVPs and erminal value problems (TVPs) and BVPs in Secion 7. We show convergence orders in conex of general IVPs in Secion 8 and illusrae he heoreical findings by experimens carried ou using bvpsuie in Secion 9. In Secion, we recapiulae he mos imporan resuls of he sudy.

3 Rachůnková e al. Page 3 of 4 2 Preliminaries We are ineresed in analyzing he BVP y () = M f() y() +, (, ], y C[, ], B y() + B y() = β, (3) where M R n n, B, B R m n, β R m and f C[, ]. Noe ha in general m n because he requiremen y C[, ] resuls in addiional n m condiions soluion y has o saisfy [9]. Throughou he paper, he following noaion is used. We denoe by R n and C n he n-dimensional vecor space of real-valued and complex-valued vecors, respecively, and denoe he maximum vecor norm by x := (x,..., x n ) = max i n x i. We denoe by C n [, ] he space of coninuous real vecor-valued funcions on [, ]. In his space, we use he maximum norm, y := max [,] y(), and he norm resriced o he inerval [, δ], δ >, is denoed by y δ := max [,δ] y(). Space C p n[, ] is he space of p imes coninuously differeniable real vecor-valued funcions on [, ]. If here is no confusion, we omi he subscrips n and wrie C[, ] = C n [, ], C p [, ] = C p n[, ]. For a marix A C m n we use he maximum norm, A = max i m j= n a ij. Before discussing he BVP (3), we firs consider he easier problem consising of he ODE sysem y () = M f() y() +, (, ], (4) subjec o iniial/erminal condiions, i.e. we deal wih he iniial value problem (IVP), y () = M f() y() +, (, ], y C[, ], B y() = β, (5) where B R m n, β R m, and m n, or wih he erminal value problem (TVP), y () = M f() y() +, (, ], y C[, ], B y() = β, (6)

4 Rachůnková e al. Page 4 of 4 where B R n n, β R n, respecively. Paricular aenion is paid o he srucure of iniial/erminal and boundary condiions which are necessary and sufficien for he exisence of a unique coninuous soluion on he closed inerval [, ]. I urns ou ha he form of such condiions depends on he specral properies of he coefficien marix M. Therefore, we disinguish beween hree cases, where all eigenvalues of M have negaive real pars, posiive real pars, or are zero. In he firs sep, we consruc he general soluion of (4). Le us denoe by J C n n he Jordan canonical form of M and le E C n n be he associaed marix of he generalized eigenvecors of M. Thus, M = EJE. (7) Moreover, le us inroduce new variables, v() := E y() and g() := E f(), hen we can decouple he sysem (4) and obain v () = J g() v() +. (8) By he variaion of consan, any general soluion of linear equaion (8) is a complexvalued funcion of he form v() = Φ()d + Φ() Φ (s) g(s) s where d C n is an arbirary vecor and ds = J d + J Φ() = J := exp(j ln()) = is he fundamenal soluion marix saisfying s J I g(s) ds, (, ], (9) J j (ln ) j, j! j= Φ () = J Φ(), Φ() = I, (, ], see [7, Chaper IV]. In case ha J consiss of m Jordan boxes, J, J 2,..., J m, he fundamenal soluion marix has he form of he following block diagonal marix, J = diag( J, J2,..., Jm ), where J k = λ k λ k, k =,..., m, ()

5 Rachůnková e al. Page 5 of 4 and J k = λ k (ln ) 2 (ln ) n k (n k )! (ln ) n k 2 (n k 2)! ln 2... ln ln......, (, ]. () Here, λ k = σ k +iρ k C is an eigenvalue of M and dimj +dimj 2 + +dimj m = n. The general soluion of equaion (4) is hen given by y() = M c + M s M I f(s) ds, (, ], (2) where c = Ed C n and M = E J E C n n. Also, ( M ) = M M I, (, ] (3) and M = ( ) M ( M ) = M M I, (, ]. (4) From he srucure of marix (), i is obvious ha he soluion conribuion relaed o he k-h Jordan box may become unbounded for =. Apparenly, he asympoic behavior of he soluion depends on he sign of he real par σ k of he associaed eigenvalue λ k. Therefore, we have o disinguish beween hree cases, σ k <, λ k =, and σ k >. We assume ha M has no purely imaginary eigenvalues o exclude soluions of he form iρ = cos(ρ ln ) + i sin(ρ ln ). We complee he preliminaries by wo echnical remarks, which will be frequenly used in he following analysis. Remark The main focus of our invesigaions is on correcly posed iniial/erminal condiions which guaranee he exisence of coninuously differeniable soluions of (4), y C [, ]. Since logarihm erms occur in marix (), he relaion lim + α (ln ) k =, α R +, k N, (5) is essenial when discussing he smoohness of y. Remark 2 By inegraing (4) we obain M s M I ds = s M = M I, (, ]. (6)

6 Rachůnková e al. Page 6 of 4 Moreover, if M has only eigenvalues wih negaive real pars, hen lim s + s M = due o Remark, and herefore s M I ds = ( M). (7) 3 Eigenvalues of M wih negaive real pars In his secion, we consider sysem (4), such ha all eigenvalues of M have negaive real pars. I urns ou ha in his case, i is necessary o prescribe iniial condiions of a cerain srucure o guaranee ha he soluion is coninuous on [, ]. Moreover, his coninuous soluion of he associaed IVP (5) is shown o be unique and is form is provided in Theorem 5. In he proof of his heorem, we require he following lemmas. Lemma 3 J = Le γ and le he n n marix J be of he form λ λ, λ = σ + iρ, (8) where σ. For σ =, we assume λ = and γ >. Then, for (, ], s J s γ ds = n j j= k= γ σ ( ln ) k, (9) k!(γ σ) j+ k and in paricular, s J s γ ds = n j=. (2) (γ σ) j+ Proof: Due o he form of J, he norm of s J for s (, ] is n s J = s λ j= ln s j j! n = s σ j= ( ln s) j. j! By repeaed inegraion by pars, we obain ( ln s) j j! s γ σ ds = sγ σ γ σ ( ln s) j j! s γ σ ( ln s) j + ds = γ σ (j )! j k= s γ σ ( ln s) k k!(γ σ) j+ k.

7 Rachůnková e al. Page 7 of 4 Therefore, due o (5), s J s γ ds = = n n j= j j= k= ( ln s) j n s γ σ ds = j! γ σ ( ln ) k k!(γ σ) j+ k. j= k= j s γ σ ( ln s) k k!(γ σ) j+ k Clearly, for = s J s γ ds = n j= (γ σ) j+ which complees he proof. Lemma 4 Then Assume ha all eigenvalues of he marix M have negaive real pars. lim + s M I ds =. (2) Proof: Le λ k = σ k + iρ k, k =,..., l, be eigenvalues of marix M and J k, k =,..., l, he associaed Jordan boxes of M. Then s M = Es J E, where s J = diag ( s J, s J2,..., s J l). Therefore, lim + s M I ds E E lim + s J s ds. The resul follows from (9) wih γ = and (5). Theorem 5 Le us assume ha all eigenvalues of M have negaive real pars. Then for every f C[, ] sysem (4) has a unique soluion y C[, ]. This soluion has he form y() = and saisfies he iniial condiion s M I f(s) ds, [, ] My() = f(). This condiion is necessary and sufficien for y o be coninuous on [, ]. Moreover, if f C r [, ], r, hen y C r [, ], and he following esimaes hold for all [, ]: y (k) () cons. f (k), k =,..., r.

8 Rachůnková e al. Page 8 of 4 Proof: The general soluion of sysem (4) can be spli ino wo pars y() = M c + M s M I f(s) ds ) = (c M s M I f(s) ds + M s M I f(s) ds =: y h () + y p (), (, ]. (22) Firs, we show ha y p C[, ]. Change of variable, u = s/, yields y p () = u M I f(u) du, (, ]. Le us now inroduce he funcions, z m () := z () := m s M I f(s) ds, m N, (23) s M I f(s) ds. (24) Then, by (2), lim z () z m () = lim m m m s M I f(s) ds m f lim s M I ds =. m Clearly z m () C[, ], for m N, and hence z is coninuous as he uniform limi of coninuous funcions. Consequenly, y p () C[, ]. Since all real pars of eigenvalues are negaive, y h is no coninuous a = and i is obvious ha y C[, ] if and only if c := c s M I f(s) ds =. Thus he unique coninuous soluion saisfying (4) has he form y() = and he esimae s M I f(s) ds, [, ], (25) y() cons. f, [, ], holds due o Lemma 4. This soluion is uniquely deerminaed by c = and here are no addiional condiions o be imposed. Noe ha c = is equivalen o My() = f()

9 Rachůnková e al. Page 9 of 4 which follows from y() = s M I f() ds = ( M) ( M M ) f() = M f(), according o Remark 2 and (7). We now examine he smoohness of y. Le f C [, ]. For he firs derivaive y, we have from (25) y () = s M f (s) ds, y () cons. f, [, ], due o Lemma 3. Similarly, if f C 2 [, ], i follows for he second derivaive y () = Clearly, if f C r [, ], hen s I M f (s) ds, y () cons. f, [, ]. y (r) () = s (r )I M f (r) (s) ds, y (r) () cons. f (r), [, ] and he resul follows. Theorem 5 shows ha if all eigenvalues of M have negaive real pars, hen here exiss a unique coninuous soluion y of IVP (5) if and only if B = M, β = f(), and m = n. Clearly, B has o be nonsingular. Noe ha for his specrum of M a erminal problem (6) canno be se up in a reasonable way. Remark 6 Ineresingly, in he above case, he ODE sysem in (5) has a limi for, which yields a represenaion for y () consisen wih (26). Le us consider he soluion y of (5) specified in (25). For f C [, ], y C[, ] and he Taylor expansion a = yields y () = M (y() + y (ξ)) + f() = ( My() + f() ) + My (ξ), where ξ (, ). Leing, resuls in Since My() = f(), and y () = lim (My() + f()) + My (). y () = lim (f() f()) + My () y () = (I M) f () which coincides wih wha we obain by evaluaing (26) a =.

10 Rachůnková e al. Page of 4 4 Eigenvalues of M wih posiive real pars In his secion we deal wih sysem (4) whose marix M has eigenvalues wih posiive real pars. I urns ou ha in his case here exiss a unique coninuous soluion of problem (6). Is smoohness depends no only on he smoohness of f bu also on he size of real pars of he eigenvalues of M. Before saing he main resul of his secion in Theorem 9, we show he following wo lemmas. Lemma 7 Le γ and le he n n marix J be of he form (8), J = λ λ, λ = σ + iρ, where σ >. Then for [, ] he funcion u() = ( ) J s sγ ds, saisfies he following inequaliies: (i) u() cons. γ for γ < σ, (26) n (ii) u() cons. σ j= n (iii) u() cons. σ j= ( ln ) j+ j! ( ln ) j j! for γ = σ, (27) for γ > σ. (28) Proof: We discuss separaely he cases γ < σ, γ = σ, and γ > σ. (i) Firs, le γ < σ. Then here exiss a consan ε > such ha σ = γ + 2ε. The erm ( ) ε n ( ( ln j s)) s j! j= is bounded on [, ] due o (5) and hence ( ) J s sγ ds = ( ) σ n ( ( ln j s)) s γ ds s j! j= cons. γ+ε s ε ds = cons. γ.

11 Rachůnková e al. Page of 4 (ii) For γ = σ funcion u can be esimaed by ( ) J ( ) σ n ( ( ln j s sγ ds = s)) s γ ds s j! j= n ( ( ln j = σ s s)) σ+γ ds j! n σ (iii) Finally, for γ > σ, we have ( ) J s sγ ds = n σ j= ( ln ) j j! j= n s ds cons. σ ( ) σ n ( ( ln j s)) s γ ds s j! j= j= ( ln ) j j! j= n s σ+γ ds cons. σ ( ln ) j+. j! j= ( ln ) j. j! Lemma 8 he funcion Le γ and le all eigenvalues of M have posiive real pars. Then u() = ( ) M s sγ ds, [, ], is bounded on [, ] and lim u() = for γ >. (29) + Proof: Le all eigenvalues of M have posiive real pars. Then u() = ( ) M s sγ ds E E ( ) J s sγ ds. Esimaes (26) o (28) and propery (5) imply u() cons. σ for [, ], where σ = min { } γ, σ 2. This means ha u is bounded in [, ]. If γ >, hen σ > and (29) follows. Theorem 9 Le us assume ha all eigenvalues of M have posiive real pars. Then for every f C [, ] and every consan vecor c, here exiss a soluion y C[, ] of (4). This soluion has he form M c + M y() = s M I f(s) ds for (, ], M f() for =. If he marix B R n n in (6) is nonsingular, hen for any β R n here exiss a unique soluion of TVP (6). This soluion is given by (3) wih c = B β. (3)

12 Rachůnková e al. Page 2 of 4 Le f C r+2 [, ]. Then he following saemens hold: (i) y C r [, ] C r+3 (, ] for r < σ + r +. (ii) y C r+ [, ] C r+3 (, ] for σ + > r +. Moreover, higher derivaives of y saisfy for [, ] (i) y (k) () cons. ( σ+ k ( + ln() nmax ) + f (k) ) for k =,,..., r, (ii) y (k) () cons. ( σ+ k ( + ln() nmax ) + f (k) ) for k =,,..., r +, where σ + is he smalles posiive real par of he eigenvalues of M and n max is he dimension of he larges Jordan box in J. Proof: The general soluion of equaion (4) can be wrien in he following form: y() = M c+ M s M I f(s) ds = M c+ ( ) M f(s) ds =: y h ()+y p (). (3) s s Since all eigenvalues have posiive real pars, i follows from (5) ha y h () = M c is coninuous on [, ]. Now, we show ha lim y p () exiss and herefore y C[, ]. Using he inegraion formula (6) we obain ( ) M f() ds = M ( M I)f(), s s and hence Therefore M f() = ( ) M f() ds M M f(). s s ( ) M f(s) ds ( M) f() = s s Since f C [, ], here exiss M (, ) such ha ( ) M f(s) f() ds+m M f(). (32) s s f(s) f() s M, s [, ]. (33) Relaion (32) ogeher wih (33) yield ( ) M f(s) s s ds ( M) f() M Since all eigenvalues of M have posiive real pars, (5) implies ( ) M s ds + ( M) M f(). lim + ( M) M f() =. Moreover, by Lemma 8 wih γ = propery (29) holds and herefore, lim + ( ) M f(s) s s ds ( M) f() =.

13 Rachůnková e al. Page 3 of 4 Thus, lim + y p () = ( M) f() and y C[, ]. I is clear from (3) ha he soluion y of (4) becomes unique if we specify he consan vecor c R n. Noe ha a =, y() saisfies n linearly independen condiions My() = f() for any c R n. Therefore, we have o specify c via he erminal condiions given in (6). Le β R n and le B R n n be nonsingular, hen i follows from B y() = B c = β ha he unique soluion of TVP (6) is given by (3), where c = B β. We now provide he esimae for y. To his aim, we uilize Lemma 8 wih γ = and he inequaliy M = E J E cons. J cons. σ+ ( + ln() nmax ). Hence, y() M c + M s M I f(s) ds ( ) M M B β + s f(s) ds s ( ) M M B β + f s s ds cons. σ+ ( + ln() nmax ) B β + cons. f. In order o discuss he smoohness of y, we firs sudy he general soluion of he homogeneous problem y h. Assume ha r < σ + r +. Then, we have y h() = ( M c ) = M M I c, y h() = ( M c ) = M(M I) M 2I c, y (k) h () = ( M c ) (k) = M(M I) (M (k )I) M ki c, k =,..., r, and i is easily seen ha y h C r [, ] C (, ]. We now urn o he smoohness of he paricular soluion of he inhomogeneous problem y p. Firs, we inegrae by pars y p () = M s M I f(s) ds ( = M ( M) M f() ( M) If() ( M) ( = (M) M f() f() + M ) s M f (s) ds. ) s M f (s) ds Noe ha M and M are commuaive if M and M are commuaive, since M M = (M M ). The laer propery will be shown in Lemma 9. Le us now assume ha f C 2 [, ]. Then, we can differeniae he above equaion and obain

14 Rachůnková e al. Page 4 of 4 ( y p() = (M) M M I f() f () + M M I = M I f() + M I s M f (s) ds. ) s M f (s) ds + M M f () If σ + >, hen we argue as a he beginning of he proof (in conex of y and σ + > ) and conclude ha y p C [, ]. Moreover, he following esimae holds: y p() f() cons. σ+ ( + ln() nmax ) + f f() cons. σ+ ( + ln() nmax ) + f cons. cons. ( σ+ ( + ln() nmax ) + f ), [, ]. ( ) M ds s This procedure can be ieraed: Le f C 3 [, ], hen we inegrae y p by pars and have ( y p() = (M I) M I f () + (M I) M I f() f () + M I Differeniaing he above represenaion of y p yields ) s I M f (s) ds. y y p () = M 2I f () + (M I) M 2I f() + M 2I s I M f (s) ds. If σ + > 2, hen y p C 2 [, ] and he following esimae holds: p () ( f() + f () )cons. σ+ 2 ( + ln() nmax ) + cons. ( σ+ 2 ( + ln() nmax ) + f ), [, ]. ( ) M I f (s) s Similarly, if f C r+2 [, ] and σ + > r, hen y p C r [, ] and he following esimae holds: ( ) y p (r) () cons. σ+ r ( + ln() nmax ) + f (r), [, ]. For σ + > r +, hen y p C r+ [, ] and ( ) y p (r+) () cons. σ+ r ( + ln() nmax ) + f (r+), [, ]. I follows from (4) ha if f C r+2 [, ] hen y p C r+3 (, ]. Consequenly, we have y p C r [, ] C r+3 (, ] for r < σ + r + and y p C r+ [, ] C r+3 (, ] for σ + > r +. We recall: if r < σ + r + hen he soluion y h C r [, ] C (, ] saisfies y (r) h () cons. σ+ r ( + ln() nmax ), [, ].

15 Rachůnková e al. Page 5 of 4 For σ + > r +, we have y h C r+ [, ] C (, ] and y (r+) h () cons. σ+ r ( + ln() nmax ), [, ]. The above smoohness resuls for y p and y h complee he proof. We recapiulae he case when all eigenvalues of M have posiive real pars: For any f C [, ] and any vecor β R n here exiss a unique coninuous soluion y of TVP (6) if and only if he marix B R n n is nonsingular. For each coninuous soluion y of (4), My() = f() holds independenly on c R n from (3). Consequenly, in his case a well-posed iniial problem (5) does no exis. Remark A coninuous soluion o (4) exiss also in he case when f is no coninuously differeniable in [, ]. However, in his case, we need some more srucure in f close o he singulariy. Le us assume ha f() = O ( α h()) as, (34) for some consan α > and a funcion h C[, δ ], δ >. Then, he soluion of (4) is sill coninuous on [, ]. To see his noe, ha due o (34) here exiss a δ 2 > such ha f() < cons. α h() for (, δ 2 ). Define δ := min{δ, δ 2 }. Then y p () = Moreover by (29), lim + δ ( ) M f(s) ds = s s ( ) M f(s) s s Then, according o (5), we deduce δ ( ) M f(s) ds + s s ds < lim cons. h δ + lim y p () = lim + + δ ( ) M f(s) ds =. s s δ ( ) M f(s) ds. s s ( ) M s sα ds =. Therefore, y C[, ]. Noe ha he funcion f() = ( α,..., αn ), where α i >, i =,..., n, also saisfies condiion (34). 5 Eigenvalues λ = Finally, we consider he case when all eigenvalues of he marix M are zero. We begin wih a scalar equaion (4) which for M = λ = immediaely reduces o y () = f(), (35) and show ha addiional srucure in he funcion f is necessary o guaranee ha he soluion y is coninuous on [, ]. To see his, assume ha f is a consan

16 Rachůnková e al. Page 6 of 4 funcion, f(). Then, any soluion y of equaion (35) has he following form: y() = y() + ds = y() + ln, (, ] s and, clearly, y is no coninuous a =. Moivaed by he scalar case, we require he inhomogeneiy f o saisfy addiional condiions providing he coninuiy of he associaed soluion. Le us denoe he eigenspace of M associaed wih he eigenvalues zero by X, he orhogonal projecion ono X by R, and he marix, which consiss of linearly independen columns of R by R. We also define he projecion H as H := I R. Before formulaing he main resul of his secion we show he following lemma. Lemma for α > Le us assume ha all eigenvalues of he marix M are zero. Then lim + s M s α ds =. (36) Proof: Le J k, k =,..., l, be he Jordan boxes of M. Then we can wrie s M = Es J E, s J = diag ( s J,..., s l) J and hus lim + s M s α ds E E s J s α ds. Applying (9) and (5) we obain (36). Theorem 2 Le all eigenvalues of he marix M be zero. Moreover, le us assume ha here exis a consan α > and a funcion h C[, δ], δ > such ha f() = O( α h()) for. (37) Then for any f C[, ], β R m, and a nonsingular m m marix B R, here exiss a unique soluion y C[, ] of IVP (5), where m = dim X. This soluion has he form y() = R(B R) β + and saisfies he iniial condiion s M s f(s) ds, (, ], My() =, (38) which is necessary and sufficien for y C[, ]. Moreover, y() R(B R) β + cons. ( f + α h δ ), [, ],

17 Rachůnková e al. Page 7 of 4 and if α r+, f C r [, ], and h C r [, δ], hen y C r+ [, ] and he following esimaes hold k y (k) ( cons. α ) (k j) h (j) δ, [, δ), j= k ( ( y (k) α cons. ) (k j) h (j) δ + ( ) (k j) f ) (j), [δ, ], where k =,..., r +. j= Proof: We spli he general soluion of (4) ino wo pars y() = y h () + y p () as defined in (22). To prove ha y p C[, ], we again use he funcions z m wih m N and z specified in (23) and (24). Due o (9), (36), and (37), we obain lim z m () z m () = lim s M s f(s) ds m m h δ α m lim s M s α ds =. (39) m Therefore, y p = z C[, ] and y p () = since f() = due o (37). We now examine he coninuiy of y h () = M (c + ) s M s f(s) ds =: M η, cf. (22). The fundamenal soluion marix is given by M = E J E, where J has he form J = diag( J,..., J l ) and E = ( v, h (), h(2),..., h(n ), v 2, h () 2,..., h(n2 ) 2,..., v l, h () l ),..., h (n l ) l, where for k =,..., l, v k are he eigenvecors of M, h () k,..., h(n k ) k are he associaed principal eigenvecors, and n k are he dimensions of he Jordan boxes J k. Clearly, because of he logarihmic erms occurring in J, see (), y h is no coninuous a = in general. Only when he conribuions including he logarihmic erms vanish, y h becomes coninuous on [, ]. I is clear from () ha he only bounded conribuions o y h are linear combinaions of he eigenvecors of M. Consequenly, any linear combinaion of principal vecors has o vanish. This is he case when η i =, i, n +, n + n 2 +,..., l k= n k + and arbirary η i for all i =, n +, n +n 2 +,..., l k= n k +. Thus, y h is coninuous on [, ] if and only if i is a consan linear combinaion of he eigenvecors of M. Wih oher words, by seing y h () := η, we have y() C[, ] My() = Mη = η Ker M. Consequenly, M y() = is necessary and sufficien for he soluion y() = η + s M I f(s) ds, [, ] (4)

18 Rachůnková e al. Page 8 of 4 o be coninuous on [, ]. The smoohness resuls for y follow from he smoohness of f and from Lemma. Noe ha he regulariy requiremen My() = conains n l linearly independen condiions and can be equivalenly expressed by Hy() =, y() = Ry() or y() Ker M. The remaining l free consans have o be uniquely specified by appropriaely prescribed iniial condiions. Le us consider he iniial condiions specified in (5), where B R m n and β R m. Since y p () = and y h () = η, he iniial condiion B y() = β is equivalen o B η = β. Due o he fac ha η Im R, here exiss a unique l-dimensional vecor d, l = dim X, such ha η = Rd, where R is he n l marix conaining he linearly independen columns of R. Clearly, he problem is uniquely solvable if and only if m = l = dim X and he m m marix B R is nonsingular. Hence, B η = β B Rd = β d = (B R) β η = R(B R) β, and he soluion y has he form y() = R(B R) β + This soluion is bounded by s M s f(s) ds, [, ]. y() R(B R) β + cons. ( α h δ + f ), [, ], due o y p () = = δ s M s f(s) ds s M s f(s) ds + δ s M s f(s) ds δ cons. f + cons. α h δ s M s α ds, [, ] cons. ( f + α h δ ), [, ], and (9). In order o derive an esimae for he firs derivaive, we subsiue he soluion given by (4) ino equaion (4) and use he propery Mη = and Lemma. If α hen he firs derivaive is bounded by y () cons. M y () cons. M α s M s α h(s) ds + cons. α h() cons. α h δ, [, δ), δ α s M s α h(s) ds + M cons. α h δ + cons. f, [δ, ]. δ s M s f(s) ds + f()

19 Rachůnková e al. Page 9 of 4 For f C [, ], h C [, δ], and α 2, we now derive a bound of he second derivaive, y () = M 2 s M s f(s) ds + M y () cons. ( α 2 h δ + α h ), [, δ), s M f (s) ds f() 2 y () cons. ( α 2 h δ + α h + 2 f + f ), [δ, ]. + f (), Analogously, for f C r [, ], h C r [, δ], α r +, we have y C r+ [, ] and y (r+) cons. y (r+) cons. r ( α ) (r k) h (k) δ, [, δ), k= r k= ( ( α ) (r k) h (k) δ + ( ) (r k) f (k) ), [δ, ]. Remark 3 Here, we deal wih a purely polynomial inhomogeneiy of he form f() = ( α,..., αn ), where, α i N, for i =,..., n. In his case, y C [, ]. To see his, we consider he componens of y, given by y() = γ + y k () = γ k + s M s f(s) ds, n ( s M ) kj sαj αj ds = γ k + j= n w j (), j= where w j () = αj obain ( s M ) kj sαj ds. We now differeniae w j, j =..., n, and w j() = α j αj s M s αj ds, w j () = α j (α j ) αj 2 s M s αj ds, w (αj) j () = α j! w (αj+) j () =. Therefore y() C [, ]. s M s αj ds,

20 Rachůnková e al. Page 2 of 4 In Theorem 2, we described he unique solvabiliy of IVP (5) in case when all eigenvalues of M are zero. The dimension of he corresponding eigenspace X was m < n and i urned ou ha a regulariy requiremen My() = has o be saisfied. If m = n, hen M = and he regulariy condiion holds. In his case we can also invesigae if a well-posed TVP (6) exiss. This quesion is deal wih in he nex lemma. Lemma 4 Consider sysem (4) wih he marix M =. Le f C[, ] and assume ha (37) is saisfied. Then, for any vecor β R n and a nonsingular marix B R n n here exiss a unique soluion of (6), bounded by y() = B β + f(s) s ds, y() B β + cons. ( f + α h δ ). Moreover, if f C r [, ], h C r [, δ], and α r +, hen y C r+ [, ] and he following esimaes hold: k y (k) ( () cons. α ) (k j) h (j) δ, [, δ), j= k y (k) ( () cons. ) (k j) f (j), [δ, ], where k =,..., r +. j= Proof: For M = he sysem (4) reduces o and is soluion is y () = f(), f(s) y() = y() + ds. s To show ha y C[, ], we follow he argumens given in he proof of Theorem 2. The erminal condiion B y() = β yields y() = B β. Moreover, y() B β + δ f(s) s f(s) ds + ds δ s B β + f ln(δ) + cons. h δ B β + cons. ( f + h δ). δ s α ds Esimaes for he higher derivaives of y follow in an analogous manner.

21 Rachůnková e al. Page 2 of 4 6 Differences beween linear sysems wih smooh and unsmooh inhomogeneiy Before discussing he case of an arbirary specrum of M which enables more general well-posed IVPs, TVPs, and BVPs, we summarize here he resuls from he previous secions and poin ou he differences when compared o he framework given in [9, 9], where linear sysems wih smooh inhomogeneiy, y () = M y() + f(), f C[, ], (4) were sudied. 6. Eigenvalues wih negaive real pars Le us consider he ODE sysem (4) and assume ha all eigenvalues of M have negaive real pars. Then, according o [9, 9], y C[, ] if and only if y() =. Therefore, he following IVP has a unique soluion: y () = M y() + f(), y() =. Moreover, y C r+ [, ] if f C r [, ], r. According o Theorem 5, ODE sysem (4) has a soluion y C[, ] if and only if My() = f(). Consequenly, he IVP specified below has a unique soluion, y () = M f() y() +, My() = f(). Moreover, y C r [, ] if f C r [, ], r. The condiions y() = and My() = f() are necessary and sufficien for he soluion y C[, ] in case of he sysem (4) and (4), respecively. 6.2 Eigenvalues wih posiive real pars For his specrum of M neiher for sysem (4) nor for (4) here exiss a well-posed IVP. In boh cases we need o specify he boundary condiions a = and solve he TVP. In paricular, he TVP y () = M y() + f(), B y() = β, where B R n n is nonsingular and β R n, has a unique soluion y C[, ]. This soluion saisfies y() =. If f C r [, ] and σ + > r + hen y C r+ [, ], cf. [9]. In conras o sysem (4), we need exra smoohness of he funcion f o obain a unique coninuous soluion of he TVP y () = M f() y() +, B y() = β, where B R n n is nonsingular and β R n. Theorem 9 saes ha y C[, ] if f C [, ]. Addiionally, if f C r+2 [, ] and σ + > r+ hen y C r+ [, ], r.

22 Rachůnková e al. Page 22 of Eigenvalues λ = If all eigenvalues of M are zero, hen he well-posed IVP associaed wih (4) akes he form y () = M y() + f(), My() =, B y() = β, where he m m marix B R is nonsingular, β R m, and m = dim X. The iniial condiion My() = is necessary and sufficien for he soluion o by coninuous. The remaining m condiions necessary for is uniqueness are specified by B y() = β. For f C r [, ], r, y C r+ [, ], see [9, 9]. In case of he unsmooh inhomogeneiy in (4), f has o saisfy an addiional requiremen, f() = O( α h()) as, α >, h C[, δ], δ >, o enable a coninuous soluion of he following IVP: y () = M f() y() +, My() =, B y() = β, where he m m marix B R is nonsingular, β R m, and m = dim X. Finally, if f C r [, ], h C r [, δ], and α r +, hen y C r+ [, ]. 7 General IVPs, TVPs and BVPs In his secion we sudy general IVPs, TVPs and BVPs. For he subsequen discussion, we have o inroduce he following noaion. X + is he invarian subspace associaed wih he eigenvalues wih posiive real pars; X (e) is he space of eigenvecors associaed wih eigenvalues λ = ; X is he invarian subspace associaed wih he eigenvalues wih negaive real pars; X (h) is he space of generalized eigenvecors associaed wih he eigenvalue λ = ; S is he orhogonal projecion ono X + ; R is he orhogonal projecion ono X (e) ; P := R + S is he projecion ono X + X (e) ; Q := I P is he projecion ono X X (h) ; Z is he orhogonal projecion ono X (e) X (h) ; N is he orhogonal projecion ono X ; H is he orhogonal projecion ono X (h). All projecions are consruced using he eigenbasis of M. Firsly, we discuss general IVPs (5) and TVPs (6), where all condiions which are necessary and sufficien o specify a unique soluions y C[, ] are posed a only one poin, eiher a = or a =. According o he resuls derived above, resricions on he specrum of M need o be made. A. For IVP (5) we assume ha he marix M has only eigenvalues wih nonposiive real pars and if σ = hen λ =.

23 Rachůnková e al. Page 23 of 4 A.2 For TVP (6) we assume ha he marix M has only eigenvalues wih nonnegaive real pars and if σ = hen λ =. Addiionally, if zero is an eigenvalue of M, hen he associaed invarian subspace is assumed o be he eigenspace of M. Resuls formulaed below wihou proofs are simple consequences of Theorems 5, 9, 2, and Lemma 4. Lemma 5 Le us assume ha f C[, ], Sf C [, ] and Zf saisfies condiion (37). (i) Assume A. o hold. Le y be a coninuous soluion of IVP (5). Then MNy() = Nf(), Hy() =. (ii) Assume A.2 o hold. Le y be a coninuous soluion of TVP (6). Then MSy() = Sf(). In boh cases My() = M(S + N)y() = f(). The saemen of Lemma 5 means ha he condiions which are necessary for he soluion of IVP (5) o be coninuous are equivalen o rank M = rank H + rank N = rank Q = n rank R iniial condiions, which he soluion y has o saisfy. In case of TVP (6), where A.2 holds, any soluion of (4) is coninuous on [, ] and no regulariy condiions have o be prescribed. From Theorems 5 and 2 we obain he following resul for a general IVP (5). Theorem 6 Le us assume ha A. holds, he m m marix B R is nonsingular, where he marix R consiss of he linear independen columns of he projecion marix R, and β R m. Then, for every f C[, ] such ha Zf saisfies (37), here exiss a unique soluion y C[, ] of IVP (5), y() = R(B R) β + This soluion is bounded by s M s f(s) ds. y() cons.( α h δ + f ) + R(B R) β. Le Nf C r+ [, ] and Zf C r [, ] saisfy condiion (37) wih α r +. Then y C r+ [, ]. The analogous resul for a general TVP (6) follows from Theorems 9 and 2.

24 Rachůnková e al. Page 24 of 4 Theorem 7 Le us assume ha A.2 holds, B R n n is nonsingular, and β R n. Then, for every f C[, ] such ha Rf saisfies (37) and Sf C [, ], here exiss a unique soluion y C[, ] of TVP (6), y() = M B β + M s M s f(s) ds, (, ]. This soluion saisfies My() = f() and is bounded by y() cons. ( + σ+ ( + ln() nmax )) B β + cons. ( f + α h δ ). Le r < σ + r +, Sf C r+2 [, ], and Zf C r [, ] saisfy condiion (37) wih α r, hen y C r [, ]. For σ + > r +, Sf C r+2 [, ], and Zf C r [, ] saisfying condiion (37) wih α r +, we have y C r+ [, ]. Here, σ + denoes he smalles posiive real par of he eigenvalues of M and n max is he dimension of he larges Jordan box of M. Furher, we sudy he linear BVPs of he form y () = M f() y() +, (, ], y C[, ], B y() + B y() = β, (42) where he marix M may have an arbirary specrum, B, B R m n, m n, β R m, and f C[, ]. I is clear from he previous consideraions ha he form of he boundary condiions which guaranee he exisence of a unique coninuous soluion of (42) will depend on he specral properies of he coefficien marix M. Before proceeding wih he analysis, we show he following auxiliary resuls. Lemma 8 λ =. Then Le R be a projecion ono he eigenspace associaed wih eigenvalues M R = R, [, ]. Proof: Le M and R be represened using he eigenbasis of M, his means M = EJE and R = E ˆRE, where ˆR is a diagonal marix wih ones a he posiions corresponding o he eigenvalues λ = and zero enries elsewhere. I is sufficien o show J ˆR = ˆR since his implies M R = E J E E ˆRE = E ˆRE = R. The muliplicaion J ˆR means an operaion on columns of J and due o he srucure of ˆR, he resul of he marix muliplicaion J ˆR is a marix whose columns corresponding o he eigenvecors associaed wih λ = remain unchanged and all oher columns vanish. Clearly, he nonrivial columns of J ˆR are uni vecors, since λ = for λ =, and he resul, J ˆR = ˆR, follows. Lemma 9 The projecion marices S, Z, and N and he marix M commuae.

25 Rachůnková e al. Page 25 of 4 Proof: We show he resul for he projecion S, for he oher projecions he proof is analogous. Firs, we prove ha for he Jordan canonical form J Ŝ = ŜJ holds for [, ], where M = EJE and S = EŜE. The marix Ŝ is a diagonal marix wih ones a he posiions corresponding o he eigenvalues wih posiive real pars and zero enries elsewhere. The muliplicaion ŜJ means operaions on rows of J, while he muliplicaion J Ŝ represens operaions on columns of J. Being aware of he block srucure of marix J we see ha he resul of ŜJ or J Ŝ is a marix conaining Jordan boxes corresponding o eigenvalues wih posiive real pars and J Ŝ = ŜJ for [, ]. This implies M S = E J E EŜE = EŜE E J E = S M and he saemen follows. Noe ha he above projecions S, Z, and N commuae wih M. To specify he boundary condiions which guaranee he well-posedness of BVP (42) he following lemma is required. Lemma 2 Consider he following BVP: y () = M f() y() +, (, ], Hy() =, MNy() = Nf(), Sy() = Sγ, Ry() = Rγ. Then, for every f C[, ], such ha Zf saisfies (37) and Sf C [, ], and for any consan vecor γ, here exis a unique coninuous soluion of he form y() = M P γ + M S s M I f(s) ds + (Q + R) s M I f(s) ds. Proof: According o he previous resuls, he conribuions o he soluion y depend on he signs of he eigenvalues of M. For he eigenvalues wih negaive real pars he conribuion has he form y () = N s M I f(s) ds, [, ]. For he eigenvalues wih posiive real pars he conribuion is given by y + () = M Sγ + M S s M I f(s) ds, (, ], and can be coninuously exended o =. Finally, for he eigenvalues λ =, we have y () = Rγ + Z s M I f(s) ds, [, ].

26 Rachůnková e al. Page 26 of 4 The soluion y is he sum of all conribuions, y() = (N + S + Z)y(). Therefore, we obain y() = Rγ + M Sγ + (N + R + H) = M P γ + M S s M I f(s) ds + (Q + R) s M I f(s) ds + M S s M I f(s) ds. s M I f(s) ds We now evaluae y a he boundaries o show ha he above boundary condiions are saisfied. According o (37), Zf() = holds. This yields (R + H)y() = Zy() = Rγ + Z s M I f() ds = Rγ + Therefore Hy() = and Ry() = Rγ. Moreover, Sy() = M Sγ + M S s M I f(s) ds Sy() = Sγ. Finally, we show ha MNy() = Nf(). Firs noe ha MNy() = NM According o (6), s M I dsf(). M s M I ds = M I for (, ]. Taking ino accoun (7) and leing +, we obain NM s M I ds = lim + N M N = N, s M I dszf() = Rγ. since he marix N M consis only of Jordan boxes corresponding o eigenvalues wih negaive real pars. Therefore MNy() = Nf(). We now urn o he general boundary condiions specified in (42). For he invesigaion of hese condiion, we have o rewrie he represenaion of he soluion y, especially he erm y, y () = Rγ + Z = Rγ + M R = R γ + M R s M I f(s) ds = Rγ + Z M s M I f(s) ds s M I f(s) ds + M H s M I f(s) ds + M H s M I f(s) ds s M I f(s) ds, where γ := γ + s M I f(s) ds.

27 Rachůnková e al. Page 27 of 4 Remark 2 Noe ha he funcion M H s M I f(s) ds = M H M s M I f(s) ds, is coninuous on [, ]. In order o see his, we again use funcions z m and z given by (23) and (24). Due o (5), (36), (37), and (39), we have lim (ln m )k z () (ln ) k z m () h δ α (ln ) k m lim s M s α ds = m for k N {}. Since each enry of he marix M H M cons. (ln ) k, k N {}, he funcion is a sum of erms is coninuous on [, ]. M H M s M I f(s) ds = M H M z Consequenly, he general coninuous soluion of he ODE sysem given in (42) can be represened as y() = M P γ + M P s M I f(s) ds + M Q and saisfies he following boundary condiions: s M I f(s) ds, (43) Hy() =, MNy() = Nf(), P y() = P γ. In he following lemma, we use he superposiion principle o rewrie he soluion (43) of (42) in a way convenien o discus he boundary condiions specified in (42). Lemma 22 Le us assume ha he inhomogeneiy f C[, ] is given in such a way ha Zf saisfies (37) and Sf C [, ]. Le he n m marix P be a marix consising of he linearly independen columns of P. Then he general coninuous soluion of (42) has he form y() = ỹ() + Y ()α, [, ], (44) where α is a consan m-dimensional vecor and ỹ is he unique soluion of ỹ (y) = M f() ỹ() +, [, ], Hỹ() =, MNỹ() = Nf(), P ỹ() =, and Y () is he unique coninuous fundamenal soluion marix saisfying Y () = M Y (), [, ], Y () = P.

28 Rachůnková e al. Page 28 of 4 The case of he general boundary condiions (42) is covered by he following lemma. Lemma 23 Le f C[, ] be given in such a way ha Zf saisfies (37) and Sf C [, ]. Then, here exiss a unique soluion y C[, ] of BVP (42) if and only if he m m marix B R + B P is nonsingular. Here, B, B R m n, β R m, and m = rank P. Proof: We use (43) and (44) o calculae y() and y(). Since Hy() = and lim M S =, we firs deduce y() = (H + P + N)y() = (P + N)(ỹ() + Y ()α) = (P + N)ỹ() + P Y ()α = (P + N)ỹ() + (R + S)Y ()α = (P + N)ỹ() + Rα. Moreover, from P y() = and M S = S, we have y() = (Q + P )y() = (Q + P )(ỹ() + Y ()α) = Qỹ() + (QY () + P )α = Qỹ() + P α. Finally, we subsiue y() and y() ino he boundary condiion and obain B y() + B y() = B ((P + N)ỹ() + Rα ) ( + B Qỹ() + P ) α = β. Thus, ( ) B R + B P α = β B (P ỹ() + Nỹ()) B Qỹ(), and he unknown vecor α can be uniquely deermined if he m m marix B R + B P is nonsingular. This complees he proof. The following heorem saed wihou proof is a consequence of he above resuls. Theorem 24 Consider BVP (42), where he inhomogeneiy f is given in such a way such ha f C[, ], Zf saisfies (37), and Sf C [, ]. Moreover, le B, B R m n, β R m, and m = rank P. Le us assume ha he m m marix B R + B P is nonsingular. Then, BVP (42) has a unique coninuous soluion y C[, ]. This soluion saisfies wo iniial condiions, Hy() =, MNy() = Nf() which are necessary and sufficien for y C[, ].

29 Rachůnková e al. Page 29 of 4 8 Collocaion mehod In his secion we propose and analyze he polynomial collocaion, cf. [8], for he numerical reamen of IVP (5) which we assume o be well-posed, y () = M f() y() +, B y() = β, where he marix M has only eigenvalues wih nonposiive real pars, and if σ = hen λ =. Moreover, B R m n, β R m, where rank R = m n. For he numerical reamen, we have o augmen he m iniial condiions specified by B y() = β by he n m linearly independen iniial condiions singled ou from he se Hy() =, MNy() = Nf(). Consequenly, we have o solve he iniial value problem, y () = M f() y() +, B y() = β, Hy() =, MNy() = Nf(). (45) We firs discreize he analyical problem (45). The inerval of inegraion [, ] is pariioned by an equidisan mesh, := { = < <... < I < I =, j = jh, j =,..., I = /h}, and in each subinerval [ j, j+ ], we inroduce k collocaion nodes jl := j +u l h, j =,..., I, l =,..., k, where < u < u 2 <... < u k. The compuaional grid including he mesh poins and he collocaion poins is shown in Figure.... j... jl... Figure The compuaional grid } {{ } j+... I h By P k,h we denoe he class of piecewise polynomial funcion of degree less or equal o k on each subinerval [ j, j+ ]. We approximae he analyical soluion y by a piecewise polynomial funcion p P k,h C[, ], p() := p j (), [ j, j+ ], j =,..., I such ha p saisfies ODE sysem (4) a he collocaion poins, p ( jl ) M jl p( jl ) = f( jl) jl, l =,..., k, j =,..., I, (46) ogeher wih he coninuiy relaions, p j ( j ) = p j ( j ), j =,..., I, (47)

30 Rachůnková e al. Page 3 of 4 and p saisfies he iniial condiions B p() = γ, Hp() =, MNp() = Nf(). (48) Noe, ha rank B + rank H + rank N = n. Since in each subinerval [ j, j+ ] p() = p j () is a polynomial of degree smaller or equal o k, he oal number of unknowns, he coefficiens in he ansaz funcion p, is (k + )In. On he oher hand, he sysem (46) consiss of kin equaions, (47) provides (I )n, and (48) n condiions, which ogeher add up o (k + )In. This means ha he collocaion scheme (46), (47), and (48) is closed. The collocaion applied o solve (4) was sudied in [8], where in paricular, unique solvabiliy of he collocaion scheme and he convergence properies have been shown. For reader s convenience, we recapiulae in he nex heorem an imporan auxiliary resul from [8] required in he subsequen invesigaions. Noe ha since analyical problem (45) has a unique soluion, is value y() is known. Therefore, in Theorem 4. [8], a slighly simpler problem is considered, where insead of he iniial condiions he correc value of y() := δ is prescribed. Theorem 25 (Theorem 4. in [8]) Le us consider he collocaion scheme, p ( jl ) M p( jl ) = M µ c jl jl β, l =,..., k, j =,... I, p() = δ, (49) jl where µ, β =,, and p P k,h C[, ]. Then problem (49) has a unique soluion provided ha h is sufficienly small. This soluion saisfies ( ) p() cons. δ + ln(h) d Mδ + ln(h) (β(d µ))+ C, [, ], where C = max j I max l k c jl, d is he dimension of he larges Jordan box of M associaed o he eigenvalue λ = and (x) + = { x x, x <. We are now in he posiion o formulae he convergence resul for he collocaion mehod. Theorem 26 Le us consider he iniial value problem y () M f() y() =, y() = δ, where Hδ = and MNδ = Nf(). Le us assume ha he funcion f saisfies Nf C k+ [, ], Zf = O( α z()), wih α k +, Zf C k [, ] and z C k [, ]. Le he funcion p P k,h C[, ] saisfy he collocaion scheme p ( jl ) M jl p( jl ) = f( jl) jl, l =,... k, j =,..., I, p() = δ.

31 Rachůnková e al. Page 3 of 4 Then p() y() cons. h k, [, ]. Proof: The idea of he proof is o inroduce an error funcion e P k,h C[, ] and invesigae how i is relaed o he global error p y of he scheme. Le e be defined as follows: e ( jl ) := y ( jl ) p ( jl ), l =,..., k, j =,..., I, e() :=. Since on each subinerval [ j, j+ ] he funcion e () is a polynomial of degree less or equal o k i is uniquely deermined by is values a k disinc poins in his inerval, k ( ) e j () = l i y ( ji ) p (), [ j, j+ ], h where i= l i () = w()/ (( u i )w (u i )), i =,..., k, w() = ( u )( u 2 ) ( u k ). Since y C k+ [, ] he inerpolaion error is O(h k ) and hence, e () = y () p () + O(h k ). By inegraion in [, ], we obain e() = y() p() + O(h k ) which means ha e differs from y p by O(h k ) erms. Moreover, we see ha e saisfies he following collocaion scheme: e ( jl ) M e( jl ) = y ( jl ) M ( y( jl ) p ( jl ) M ) p( jl ) M O( jl h k ) jl jl jl jl = f( jl) jl f( jl) jl M jl O( jl h k ) = O(h k ), e() =. According o Theorem 25, we obain e() = O(h k ). This ogeher wih e() = y() p() + O(h k ) yields p() y() cons. h k and he resul follows. The especially aracive propery of he collocaion is he so called superconvergence. For regular ODEs and cerain choices of he collocaion poins (Gaussian, Lobao, Radau), he convergence order in he mesh poins can be considerably higher han k, provided ha he soluion y is sufficienly smooh. For he Gaussian poins he superconvergence order is O(h 2k ). Since already for he problem

32 Rachůnková e al. Page 32 of 4 (4) counerexamples show ha he superconvergence does no hold [8], we do no expec i for he problem a hand eiher. However, he so-called small superconvergence uniform in can be shown, see nex heorem. The main prerequisie for he proof is he propery w(s) ds = (5) which holds for an appropriae choice of he collocaion poins. Theorem 27 Le Nf C k+2 [, ], Zf C k+ [, ] and Zf = O( α z()), where α k + 2 and z C k+ [, ]. If (5) holds, hen he esimae for he global error given in Theorem 26 can be replaced by p() y() cons. h k+ ln(h) (d )+. Proof: Consider again he error funcion e defined in Theorem 26. Due o he smoohness assumpions made for he problem daa y C k+2 [, ] follows. Therefore, e () = k i= ( ) j l i y ( ji ) p () h ( j = y () p () + hk k! w We inegrae e on [, ] and use (5) o obain j e() = y() p() + This implies + hk k! y(k+) ( j ) i= j w h ) y (k+) ( j ) + O(h k+ ), [ j, j+ ]. h k i+ ( ) s i k! y(k+) ( i ) w ds i h ( ) s j ds + O(h k+ ) = y() p() + O(h k+ ). h e ( jl ) M e( jl ) = y ( jl ) M ( y( jl ) p ( jl ) M ) p( jl ) M O(h k+ ) jl jl jl jl = M jl O(h k+ ), e() =. According o Theorem 25 we have e() cons. ( ln(h) (d )+ h k+), and finally p() y() cons. ( ln(h) (d )+ h k+). This complees he proof.

33 Rachůnková e al. Page 33 of 4 9 Numerical experimens In order o illusrae he heoreical resuls derived in he previous secion, we have consruced model problems and run he collocaion code bvpsuie on coherenly refined meshes o compare he empirically esimaed convergence orders of he scheme wih he heoreically prediced ones. 9. General IVP wih smooh soluion We firs deal wih a linear sysem of ODEs, y () = y() + f(), (, ], (5) subjec o iniial condiions ( B y() = 3 2 ) y() =, ( ) y() = ( ). (52) Here, f() = exp() + 2 exp() + sin() cos() + cos 2 () sin 2 () 2 exp() + 4 exp() + 2 sin() cos() + 2 cos 2 () 2 sin 2 () exp() + 4 exp() + cos 2 () sin 2 () + 4 2, where f() = (2 4 4) T and he exac soluion y C [, ] of IVP (5), (52) reads: y() = exp() + sin() cos() 2 exp() + 2 sin() cos() exp() + sin() cos() + 2 2, y() = 2 2. Firs of all, noe ha y() saisfies (52). The marix M has a double eigenvalue λ = λ 2 = 2, a single eigenvalue λ 3 =, and he Jordan canonical form is J = 2 2. Therefore, he second se of wo linearly independen iniial condiions in (52) are regulariy condiions, necessary and sufficien for y C[, ]. The remaining free consan has o be calculaed from he firs iniial condiion in (52). Wih he projecion marices N = 2 2, R = 2 2,

34 Rachůnková e al. Page 34 of 4 i is easily seen ha he condiion MNy() = Nf() is saisfied. Moreover, he marix B R = (3 2 ) = is nonsingular, and herefore IVP (5), (52) is well-posed. In Tables o 4, we illusrae he convergence behaviour for he collocaion execued wih equidisan and Gaussian collocaion poins. The number of he collocaion poins k was chosen o vary from o 8. However, in he simulaions shown here, we repor only on he values o 4 since he resuls for 5 o 8 are very similar. The maximal global error is compued eiher in he mesh poins, or uniformly in, Y h Y := Y h Y := max j I p( j) y( j ), max p(τ i) y(τ i ), τ i = ih, h = 3. i. The esimaed order of convergence p and he error consan c are esimaed using wo consecuive meshes wih he sep sizes h and h/2. Since Y h Y ch p for h, we have Y h Y = ch p, Y h/2 Y = c ( ) p ( ) h Yh Y p = ln 2 Y h/2 Y ln(2). Having p, we calculae he error consan from c = Y h/2 Y / ( h 2 ) p. According o he experimens, he empirical convergence orders very well reflec he heoreical findings. For Gaussian poins, we observe he small superconvergence order k + uniformly in. The superconvergence order 2k in he mesh poins does no hold in general, see case k = 4. For uniformly spaced equidisan collocaion poins we again observe he order k + which for his model is slighly beer han we can show heoreically. 9.2 General IVP wih unsmooh soluion Nex, we discuss an IVP whose soluion is less smooh han in he previous model. The problem reads: y () = M where M = ( f() y()+, ( )y() = 4, , f() = ) y() = exp() exp() sin() cos() (. ), (53)