Applications of Mathematics

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1 Applications of Mathematics Emil Vitásek Approximate solution of an inhomogeneous abstract differential equation Applications of Mathematics, Vol 57 (2012), No 1, Persistent URL: Terms of use: Institute of Mathematics AS CR, 2012 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use Each copy of any part of this document must contain these Terms of use This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 57(2012) APPLICATIONS OF MATHEMATICS No 1, APPROXIMATE SOLUTION OF AN INHOMOGENEOUS ABSTRACT DIFFERENTIAL EQUATION* Emil Vitásek, Praha (Received November 19, 2009) Abstract Recently, we have developed the necessary and sufficient conditions under which a rational function F(hA) approximates the semigroup of operators exp(ta) generated by an infinitesimal operator A The present paper extends these results to an inhomogeneous equation u (t)=au(t)+f(t) Keywords: abstract differential equations, semigroups of operators, rational approximations, A-stability MSC 2010: 34K30, 34G10, 35K90, 47D03 1 Preliminaries Let X bea(complex)banachspaceandlet Abeaninfinitesimalgeneratorofa continuoussemigroupofoperators U(t), t [0, T](fortherelevantliteraturesee,for example,[1],[2],[4],[6]) Let Fbearationalfunctionwithpolesintherighthalf-planeofthecomplexplane andletitberegularatinfinity Further,letthecoefficientsofthepolynomialsin the numerator and denominator of F be real and let F approximate the exponential function with order p, ie, let (11) exp(z) = F(z) + O(z p+1 ) for z 0, where p is a positive integer Theapproximationofthegivensemigroup U(t)willbemeantinthefollowing sense:dividetheinterval [0, T]into Nsubintervals [t j, t j+1 ]ofthelength h = T/N * The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research plan No AV0Z

3 bymeshpoints 0 = t 0 < t 1 < < t N = T anddefinethesequence {u j, j = 0, 1,,N} Xbytherecurrence (12) u j+1 = F(hA)u j, j = 0,,N 1, u 0 = η Further, suppose that lim u j = U(t)η h 0 jh t holdsforany η Xandany t [0, T]Thenwesaythattherationalfunction Fapproximates the semigroup U(t) or, alternatively, that the method(12) is convergent on the class of abstract differential equations of the form (13) u (t) = Au(t), t [0, T], with the initial condition (14) u(0) = η X The following theorem was proved in[5] Theorem 11 A rational function F with its poles in the right half-plane, regular atinfinityandsatisfying(11)withsome p 1generatestheconvergentmethod(12) ifandonlyifthereexistsaconstant M = M(t)suchthat (15) F j (ha) M foranysufficiently small handforany j satisfying 0 jh t Moreover, if η D(A p+1 )thentheconvergenceisoforder h p Theaimofthispaperistogeneralizetheseresultstothecaseofanonhomogeneous equation 2 Main result Let us investigate the differential equation of the form (21) u (t) = Au(t) + f(t), t [0, T], with the initial condition (22) u(0) = η X 32

4 Here, A is an infinitesimal generator of a continuous semigroup of operators U(t) as insection1andthefunction f := [0, T] XiscontinuousItiswellknownthatif we suppose, moreover, that the initial value η lies in D(A) then the classical solution of the problem(21) (22) exists and is given by (23) u(t) = U(t)η + t 0 U(t τ)f(τ)dτ However,(23)hassenseforany η X eventhoughthefunction u(t)neednot be differentiable in the general case Nevertheless, we will suppose it to be the generalized solution of the problem(21) (22) In the nonhomogeneous case, we will not construct the approximations of the solution of(21) (22) directly from a rational function F approximating the exponential as was described in Section 1 but we will use the so-called selfstarting block methods as they were introduced in[3] For the readers convenience, the definition and basic properties of such methods will be summarized in Appendix Apply now the SB-method(35) to the problem(21) (22) We obtain (24) u jk+1 = u jk + hc Au jk+1 + hc f jk+1 u (j+1)k u jk Au (j+1)k d 1 Au jk d 1 f jk + h + h d k Au jk d k f jk f (j+1)k Let G Abethetensorproductofamatrix G(oforder k)andtheoperator A, ie G A := D(A) D(A) X Xdefinedby g 11 A g 1k A (25) G A = g k1 A g kk A This notation allows to rewrite(24) in the form (26) (I hc A) u jk+1 u (j+1)k = (I + hd A) u jk u jk + hc f jk+1 f (j+1)k d 1 f jk + h d k f jk 33

5 where Disthediagonalmatrixwiththecomponentsofthevector donthediagonal The operator I hc A is generally unbounded Thus, the question about the solvability of(24) should be answered first Before formulating the corresponding theorem, we recall that the resolvent R(λ, A) of A has to satisfy the inequality (27) R(λ, A) n M (R(λ) ω) n for n = 1, 2, andforany λforwhich R(λ) > ω,and M, ωarerealconstants Itissosince Aistheinfinitesimalgeneratorofastronglycontinuoussemigroupof operators, see, eg,[1] We also recall that the semigroup fulfills the inequality (28) U(t) M exp(ωt) Further,realizethatanymatrixoftheform I zcis(atleastforsufficiently small z)nonsingularsothatitispossibletowriteitsinverseintheform (29) (I zc) 1 = 1 Q(z) where p 11 (z) p 1k (z) p k1 (z) p kk (z) (210) Q(z) = det(i zc) and p ij (z)isthedeterminantofthematrixoforder k 1obtainedfromthematrix (I zc)byomittingthe jthrowand ithcolumnandmultiplyingby ( 1) i+j Note thatany p ij (z)isapolynomialin zofdegreeatmost k 1 Theorem21Let Ahaveitsspectruminthehalf-plane R(λ) ωandlet Chave itseigenvaluesinthehalf-plane R(λ) > 0Thentheoperator (I hc A)hasfor sufficiently small h a bounded inverse, and (211) (I hc A) 1 = M holds, where (212) m ij = p ij (ha)q 1 (ha) m 11 m 1k m k1 m kk andthepolynomials p ij (z)and Q(z)aregivenby(29)and(210),respectively 34,,

6 Proof Inthisproofweusesomeresultsfromthetheoryoffunctionsofunbounded operators, see again, eg,[1] The degree of the polynomial Q is exactly k, since CisregularMoreover,thereexistsaconstant λ C > 0suchthatallrootsof Q lieinthehalf-plane R(λ) λ C,sincetheeigenvaluesof Clieintheopenrighthalfplane Withoutlossofgeneralitywecansupposethat ω > 0andletuschoose h 0 insuchawaythat h 0 < λ C /ω Thenthespectrumoftheoperator haliesinthe half-plane R(λ) h 0 ω < λ C forany 0 < h h 0 Further,thedegreeofanyofthe polynomials p ij (z)isatmost k 1andthedegreeofthepolynomial Q(z)isexactly k aswehavealreadysaidabove Consequently,therationalfunction p ij (z)q 1 (z)is regularattheinfinity,anditisalsoregularinthehalf-plane R(λ) < λ C,asfollowsfromthepropertiesoftherootsofthepolynomial Q Thus,theoperators p ij (ha)q 1 (ha)arecorrectlydefinedandtheyareboundedoperatorsin X Thedefinitionofthefunctions p (j) i gives immediately that (213) (δ (s) i zc is )p sj (z)q 1 (z) = δ (j) i, where δ (s) i is the Kronecker symbol Note that formula(213) is nothing else than thecommonlyknowncramer srule Since δ (s) i zc is isapolynomialofdegree1 andsince p sj (z)q 1 (z)hasarootattheinfinity,itfollowsthat (214) (δ (s) i I c is ha)p sj (ha)q 1 (ha)x = δ (j) i x for any x X, and the operatorsbehind the summation sign are well-defined bounded operators But(214) gives immediately that (215) (I hc A)Mx = x forany x X XInthesimilarwayweprovethat (216) M(I hc A)x = x forany x D(A) D(A) Nowequations(215) (216)completetheproofof theorem Supposing that C satisfies the assumptions of Theorem 21, we can rewrite(26) intheform (217) u (j+1)k = F(hA)u jk + q j, j = 0, 1,, 35

7 wheretherationalfunction Fisgivenby (218) F(z) = P(z)Q 1 (z), the polynomial P by (219) P(z) = and (220) q j = h i=1 p ks (z)(1 + d s z) c is m ki f jk+s + h d i m ki f jk Hence, if the matrix C has its eigenvalues in the right half-plane the corresponding SB-method can be used even for the approximation of the generalized solution Naturally, the method must be understood in the form(217) The convergence is controlled as can be expected by the behaviour of the powers of the operator F(hA) Before formulating the corresponding convergence theorem we prove an auxiliary assertion Lemma21LetanSB-methodoforder p 1begivenandletthecorresponding matrix C have its eigenvalues in the right half-plane of the complex plane Further, let(15)besatisfiedfinally,let f(t)becontinuousin [0, T]anddefine (h)by (221) (h) = sup [F n (ha) U n (h)]f(τ) nh t 0 τ t Then (222) lim h 0 (h) = 0 Proof Supposethat(222)isnottrue Thenthereexist ε 0 > 0andsequences {h k }, {n k },and {τ k }satisfying i=1 (223) h k 0, n k h k t, τ k t such that (224) [F n k (h k A) U n k (h k )]f(τ k ) ε 0 36

8 holdsfor k = 1, 2, Passingifnecessarytosubsequences,wecanassumeherethat n k h k t 0, τ k t 1 for k Undertheseassumptionswehave (225) [F n k (h k A) U n k (h k )]f(t 1 ) 1 2 ε 0, k = 1, 2 This estimate follows from the definition of supremum, since f(t) is continuous and F n (ha) U n (h) isbounded(see(27)and(15))ontheotherhand, (226) [F n k (h k A) U n k (h k )]f(t 1 ) [F n k (h k A) U(t)]f(t 1 ) + [U(t) U(n k h k )]f(t 1 ) 0 invirtueoftheorem21andthecontinuityof U(t)Thus,wehaveacontradiction proving the lemma Now we have all ready to prove the convergence theorem for the approximation of problem(21) (22) Theorem 22LetanSB-methodoforder p 1begivenandletthecorresponding matrix C have its eigenvalues in the right half-plane of the complex plane Further,let(15)besatisfied Finally,let u jk betheapproximatesolutionofthe problem(21) (22),where f(t)iscontinuous, f(t) D(A)for t [0, T]and Af(t)is also continuous(cf(217) (220)) Then (227) lim u jk = u(t) h 0 jh t Proof Ifwetakeintoaccount(217)wecanwritetheapproximation u jk in the form (228) u jk = F j (ha) + F j 1 ν (ha)q ν, where q j aregivenby(220) FromTheorem11weknowthat F j (ha)η U(t)η for h 0and jh tsoitremainstoprovethat (229) F j 1 ν (ha)q ν t 0 U(t τ)f(τ)dτ Toachievethisletusinvestigatetheoperators q j Beginwiththeobviousidentity m kr (δ r (i) I c ri ha) = δ (i) k I(see(216))andrewriteitintheform r=1 (230) m ki = ha r=1 c ri m kr + δ (i) k I 37

9 After substituting(230) into(220), we obtain (231) q ν = h 2 A + h 2 A = h 2 A i=1 c is r=1 d i i=1 r=1 c ri m kr f νk+s + h c ri m kr f νk + h c ri m kr( i=1 r=1 ( + h c ks f νk+s + d k f νk ) i=1 i=1 d i δ (i) k f νk c is f νk+s + d i f νk ) c is δ (i) k f νk+s The continuity of f implies that (232) f νk+s = f νk + ϕ s, where (233) ϕ s = o(1) for h 0 Observingnowthatthenormsoftheoperators m kr areuniformlybounded(note that m kr = p kr (ha)q 1 (ha)andthedegreeof p kr islessthanthedegreeof Q(z)) and that the function Af(t) is continuous and, therefore, also bounded, we obtain from(231) (233) that ( (234) q ν = h c ks + d k )f νk + ψ ν, where (235) ψ = o(h) But c ks + d k = 1sincetheorderofthemethodusedisatleast1and,hence, (234) implies that (236) q ν = hf νk + ψ ν The substitution of(235) in the left-hand part of(229) gives (237) F j 1 ν (ha)q ν = h F j 1 ν (ha)f νk + o(1) 38

10 Obviously, (238) j 1 h tjk F j 1 ν (ha)f νk U(t jk τ)f(τ)dτ 0 = h F j 1 ν (ha)f νk t(ν+1)k t νk U(t jk τ)f(τ)dτ Theintegralinthelastsumoftheright-handtermof(238)canbeestimatedas (239) t(ν+1)k t νk U(t jk τ)f(τ)dτ = hu(t jk t νk )f νk + o(h), sincethefunction U(t jk τ)f(τ)iscontinuousthus,itremainstoinvestigatethe behaviouroftheexpression h j 1 (F j 1 ν (ha) U(t jk t νk ))f νk But (240) (F j 1 ν (ha) U(t jk t νk ))f νk j 1 h = h (F j 1 ν (ha) U j ν (h))f νk = h (F j 1 ν (ha) U j 1 ν (h))f νk + h U j 1 ν (h)(i U(h))f νk IfweusenowLemma21andtheestimate(28)theassertionofthetheoremfollows immediately Let an ordinary differential equation 3 Appendix (31) u (t) = f(t, u), t [0, T], with the initial condition (32) u(0) = η R be given The right-hand term of(31) is supposed to be defined, continuous, and Lipschitzianwithrespectto uinthestrip 0 t T, < y < sothatthe existence and uniqueness of(31),(32) is guaranteed in the whole interval [0, T] 39

11 Letaninteger k, k 1,realnumbers µ 1,,ν k 1,asquarematrix Coforder k, a k-dimensional vector d and a positive real number h be given Putting (33) t rk = rh, r = 0, 1, (these points will be called the basic points), (34) t rk+i = t rk + µ i h,, k 1 (theintermediatepoints),anddenotingtheapproximatesolutionatthepoint t j by u j,theselfstartingblockmethod(sb-methodbriefly)isdefinedbytheformula (35) u rk+1 = u rk i + hc f rk+1 + hf rk d, r = 0,, u (r+1)k f (r+1)k where i = (1,,1) and f j = f(t j, u j ) OnestepoftheSB-methodconsists therefore in computing k values of the approximate solution simultaneously from the generally nonlinear system of equations and the next step is started with the last of them Note that the Lipschitz property of f guarantees that the system(35) has at least for any sufficiently small h exactly one solution ThelocaltruncationerrorofanSB-methodisdefinedintheusualway,ieby u(t + µ 1 h) u (t + µ 1 h) (36) L(u(t); h) = u(t + µ k h) u(t)i hc u (t + µ k h) hu (t)d where µ k = 1 Usingthisdefinition,thegivenSB-methodwillbesaidtohavethe order p(p positive integer), if (37) L i (u(t); h) = O(h p+1 ) for i = 1,,k, where L i isthe ithcomponentofthevector L Theorderofthemethoddependsonlyontheparametersofthemethodanddoes not depend on the particular function u For example, the assertion that the order ofthemethodisatleast1isequivalentto kalgebraicequalities (38) c is + d i = µ i, i = 1,,k 40 The following two theorems can be proved very simply(see[3]) Theorem 31 The selfstarting method of order at least 1 is convergent

12 Theorem 32 Let the solution of(31),(32) have p + 1 continuous derivatives in [0, T]ThentheerrorofanSB-methodoforder pisoftheorder O(h p ) The subclass of overimplicit block methods formed by such SB-methods for which µ i = i/kandtheorderofwhichisatleast kisnotempty,aswasshownalsoin[3] We denote them as SBK-methods Note that these methods play an important role in the study of methods for solving stiff differential equations References [1] N Dunford, J T Schwartz: Linear Operators 1 General Theory Interscience Publishers, New York-London, 1958 [2] T Kato: Perturbation Theory for Linear Operators Springer, Berlin-Heidelberg-New York, 1966 [3] M Práger, J Taufer, E Vitásek: Overimplicit multistep methods Apl Math 18(1973), [4] A E Taylor: Introduction to Functional Analysis John Wiley& Sons, New York, 1958 [5] E Vitásek: Approximate solutions of abstract differential equations Appl Math 52 (2007), [6] K Yosida: Functional Analysis Springer, Berlin-Heidelberg-New York, 1971 Author s address: E Vitásek, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, CZ Prague 1, Czech Republic, vitas@mathcascz 41