Chapter D05 Integral Equations

Transcription

1 D05 Integrl Equtions Chpter D05 Integrl Equtions Contents 1 Scope of the Chpter 2 2 Bckground to the Problems Introduction Clssifiction of Integrl Equtions Structure of Kernel Singulr nd Wekly Singulr Equtions Fredholm Integrl Equtions Eigenvlue problem Equtions of the first kind Equtions of the second kind Volterr Integrl Equtions Equtions of the first kind Equtions of the second kind Recommendtions on Choice nd Use of Avilble Routines Fredholm Equtions of Second Kind Volterr Equtions of Second Kind Volterr Equtions of First Kind Utility Routines User-supplied Routines Index 6 5 Routines Withdrwn or Scheduled for Withdrwl 7 6 References 7 [NP3390/19/pdf] D05.1

2 Introduction D05 D05 Integrl Equtions 1 Scope of the Chpter This chpter is concerned with the numericl solution of integrl equtions. Provision will be mde for most of the stndrd types of eqution (see below). The following re, however, specificlly excluded: () Equtions rising in the solution of prtil differentil equtions by integrl eqution methods. In cses where the prime purpose of n lgorithm is the solution of prtil differentil eqution it will normlly be included in Chpter D03. (b) Clcultion of inverse integrl trnsforms. This problem flls within the mbit of Chpter C06. 2 Bckground to the Problems 2.1 Introduction Any functionl eqution in which the unknown function ppers under the sign of integrtion is clled n integrl eqution. Integrl equtions rise in gret mny brnches of science; for exmple, in potentil theory, coustics, elsticity, fluid mechnics, rditive trnsfer, theory of popultion, etc. In mny instnces the integrl eqution origintes from the conversion of boundry-vlue problem or n initil-vlue problem ssocited with prtil or n ordinry differentil eqution, but mny problems led directly to integrl equtions nd cnnot be formulted in terms of differentil equtions. Integrl equtions re of mny types; here we ttempt to indicte some of the min distinguishing fetures with prticulr regrd to the use nd construction of lgorithms. 2.2 Clssifiction of Integrl Equtions In the clssicl theory of integrl equtions one distinguishes between Volterr equtions nd Fredholm equtions. In Fredholm eqution the region of integrtion is fixed, wheres in Volterr eqution the region is vrible. Thus, the eqution cy(t) =f(t)+λ is n exmple of Fredholm eqution, nd the eqution cy(t) =f(t)+λ t K(t, s, y(s)) ds, t b (1) K(t, s, y(s)) ds, t (2) is n exmple of Volterr eqution. Here the forcing function f(t) ndthekernel function K(t, s, y(s)) re prescribed, while y(t) isthe unknown function to be determined. (More generlly the integrtion nd the domin of definition of the functions my extend to more thn one dimension.) The prmeter λ is often omitted; it is, however, of importnce in certin theoreticl investigtions (e.g., stbility) nd in the eigenvlue problem discussed below. If in (1), or (2), c = 0, the integrl eqution is sid to be of the first kind. If c = 1, the eqution is sid to be of the second kind. Equtions (1) nd (2) re liner if the kernel K(t, s, y(s)) = k(t, s)y(s), otherwise they re nonliner. Note. In liner integrl eqution, k(t, s) is usully referred to s the kernel. We dopt this convention throughout. These two types of equtions re brodly nlogous to problems of initil- nd boundry-vlue type for n ordinry differentil eqution (ODE); thus the Volterr eqution, chrcterised by vrible upper limit of integrtion, is menble to solution by methods of mrching type whilst most methods for treting Fredholm equtions led ultimtely to the solution of n pproximting system of simultneous lgebric equtions. For comprehensive discussion of numericl methods see Atkinson [1], Bker [2], Brunner nd vn der Houwen [3] nd Delves nd Wlsh [5]. In wht follows, the term integrl eqution is used in its generl sense, nd the type is distinguished when pproprite. D05.2 [NP3390/19/pdf]

3 D05 Integrl Equtions Introduction D Structure of Kernel When considering numericl methods for integrl equtions, prticulr ttention should be pid to the chrcter of the kernel, which is usully the min fctor governing the choice of n pproprite qudrture formul or system of pproximting functions. Vrious commonly occurring types of singulrity cll for individul tretment. Likewise provision cn be mde for cses of symmetry, periodicity or other specil structure, where the solution my hve specil properties nd/or economies my be effected in the solution process. We note in prticulr the following cses to which we shll often hve occsion to refer in the description of individul lgorithms () A liner integrl eqution with kernel k(t, s) = k(s, t) issidtobesymmetric. This property plys key role in the theory of Fredholm integrl equtions. (b) If k(t, s) =k( + b t, + b s) in liner integrl eqution, the kernel is clled centro-symmetric. (c) If in Equtions (1) or (2) the kernel hs the form K(t, s, y(s)) = k(t s)g(s, y(s)), the eqution is clled convolution integrl eqution; in the liner cse g(s, y(s)) = y(s). (d) If the kernel in (1) hs the form K(t, s, y(s)) = K 1 (t, s, y(s)), s t, K(t, s, y(s)) = K 2 (t, s, y(s)), t < s b, where the functions K 1 nd K 2 re well-behved, whilst K or its s-derivtive is possibly discontinuous, my be described s discontinuous or of split type; in the liner cse K(t, s, y(s)) = k(t, s)y(s) nd consequently K 1 = k 1 y nd K 2 = k 2 y. Exmples re the commonly occurring kernels of the type k( t s ) nd the Green s functions (influence functions) which rise in the conversion of ODE boundry-vlue problems to integrl equtions. It is lso of interest to note tht the Volterr eqution (2) my be conceived s Fredholm eqution with kernel of split type, with K 2 (t, s, y(s)) 0; consequently methods designed for the solution of Fredholm equtions with split kernels re lso pplicble to Volterr equtions. 2.4 Singulr nd Wekly Singulr Equtions An integrl eqution my be clled singulr if either () its kernel contins singulrity, or (b) the rnge of integrtion is infinite, nd it is sid to be wekly-singulr if the kernel becomes infinite t s = t. Sometimes solution cn be effected by simple dpttion of method pplicble to non-singulr eqution: for exmple, n infinite rnge my be truncted t suitbly chosen point. In other cses, however, theoreticl considertions will dictte the need for specil methods nd lgorithms. Exmples re: (i) Integrl equtions withsingulr kernels of Cuchy type; (ii) Equtions of Wiener Hopf type; (iii) Vrious dul integrl equtions rising in the solution of boundry-vlue problems of mthemticl physics; (iv) The well-known Abel integrl eqution, n eqution of Volterr type, whose kernel contins n inverse squre-root singulrity t s = t. Problems of inversion of integrl trnsforms lso fll under this heding but, s lredy remrked, they lie outside the scope of this chpter. 2.5 Fredholm Integrl Equtions Eigenvlue problem Closely connected with the liner Fredholm integrl eqution of the second kind is the eigenvlue problem represented by the homogeneous eqution y(t) λ k(t, s)y(s) ds =0, t b. (3) [NP3390/19/pdf] D05.3

4 Introduction D05 D05 Integrl Equtions If λ is chosen rbitrrily this eqution in generl possesses only the trivil solution y(t) =0. However,for certin criticl set of vlues of λ, th e chrcteristic vlues or eigenvlues (the ltter term is sometimes reserved for the reciprocls µ =1/λ), there exist non-trivil solutions y(t), termed chrcteristic functions or eigenfunctions, which re of fundmentl importnce in mny investigtions. The nlogy with the eigenproblem of liner lgebr is redily pprent, nd indeed most methods of solution of eqution (3) entil reduction to n pproximtely equivlent lgebric problem Equtions of the first kind The Fredholm integrl eqution of the first kind (K µi)y =0. (4) k(t, s)y(s) ds = f(t), t b, (5) belong to the clss of ill-posed problems; even supposing tht solution corresponding to the prescribed f(t) exists, slight perturbtion of f(t) my give rise to n rbitrrily lrge vrition in the solution y(t). Hence the eqution my be closely stisfied by function bering little resemblnce to the true solution. The difficulty ssocited with this instbility is ggrvted by the fct tht in prctice the specifiction of f(t) is usully inexct. Nevertheless gret mny physicl problems (e.g., in rdiogrphy, spectroscopy, stereology, chemicl nlysis) re ppropritely formulted in terms of integrl equtions of the first kind, nd useful nd meningful solutions cn be obtined with the id of suitble stbilising procedures. See Chpters 12 nd 13 of Delves nd Wlsh[5] for further discussion nd references Equtions of the second kind Consider the nonliner Fredholm eqution of the second kind y(t) =f(t)+ K(t, s, y(s)) ds, t b. (6) The numericl solution of eqution (6) is usully ccomplished either by simple itertion or by more sophisticted itertive scheme bsed on Newton s method; in the ltter cse it is necessry to solve sequence of liner integrl equtions. Convergence my be demonstrted subject to suitble conditions of Lipschitz continuity of the functions K withrespect to the rgument y. Exmples of Fredholm type (for which the provision of lgorithms is contemplted) re: () (b) the Uryson eqution the Hmmerstein eqution u(t) 1 0 F (t, s, u(s)) ds =0, 0 t 1, (7) u(t) where F nd g re rbitrry functions Volterr Integrl Equtions Equtions of the first kind Consider the Volterr integrl eqution of the first kind t 0 k(t, s)g(s, u(s)) ds =0, 0 t 1, (8) k(t, s)y(s) ds = f(t), t. (9) Clerly it is necessry tht f() = 0; otherwise no solution to (9) cn exist. D05.4 [NP3390/19/pdf]

5 D05 Integrl Equtions Introduction D05 The following types of Volterr integrl equtions of the first kind occur in rel life problems: equtions withunbounded kernel t s = t, equtions withsufficiently smoothkernel. These types belong lso to the clss of ill-posed problems. However, the instbility is pprecibly less severe in the equtions with unbounded kernel. In generl, non-singulr Volterr eqution of the first kind presents less computtionl difficulty thn the Fredholm eqution (5) with smooth kernel. A Volterr eqution of the first kind my, under suitble conditions, be converted by differentition to one of the second kind or by integrtion by prts to n eqution of the second kind for the integrl of the wnted function Equtions of the second kind A very generl Volterr eqution of the second kind is given by y(t) =f(t)+ t K(t, s, y(s)) ds, t. (10) The resemblnce of Volterr equtions to ODEs suggests tht the underlying methods for ODE problems cn be pplied to Volterr equtions. Indeed this turns out to be the cse. The min dvntges of implementing these methods re their well-developed theoreticl bckground, i.e., convergence nd stbility, see Brunner nd vn der Houwen [3], Wolkenfelt [6]. Mny Volterr integrl equtions rising in rel life problems hve convolution kernel (cf. Section 2.3(c)), see [3] for references. However, subclss of these equtions which hve kernels of the form k(t s) = M λ j (t s) j, (11) j=0 where {λ j } re rel, cn be converted into system of liner or nonliner ODEs, see [3]. For more informtion on the theoreticl nd the numericl tretment of integrl equtions we refer the user to Atkinson [1], Bker [2], Brunner nd vn der Houwen [3], Cochrn [4] nd Delves nd Wlsh [5]. 3Recommendtions on Choice nd Use of Avilble Routines Note. Refer to the Users Note for your implementtion to check tht routine is vilble. The choice of routine will depend first of ll upon the type of integrl eqution to be solved. 3.1 Fredholm Equtions of Second Kind () Liner equtions D05AAF is pplicble to n eqution with discontinuous or split kernel s defined in 2.3.(d). Here, however, both the functions k 1 nd k 2 re required to be defined (nd wellbehved) throughout the squre s, t b. D05ABF is pplicble to n eqution with smoothkernel. Note tht D05AAF my lso be pplied to this cse, by setting k 1 = k 2 = k, but D05ABF is more efficient. 3.2 Volterr Equtions of Second Kind () Liner equtions D05AAF my be used to solve Volterr eqution by defining k 2 (or k 1 ) to be identiclly zero. (See lso (b).) [NP3390/19/pdf] D05.5

6 Introduction D05 D05 Integrl Equtions (b) Nonliner equtions D05BAF is pplicble to nonliner convolution Volterr integrl eqution of the second kind. The kernel function hs the form D05BDF K(t, s, y(s)) = k(t s)g(s, y(s)). The underlying methods used in the routine re the reducible liner multi-step methods. The user hs choice of vriety of these methods. This routine cn lso be used for liner g. is pplicble to nonliner convolution eqution hving wekly-singulr kernel (Abel). The kernel function hs the form K(t, s, y(s)) = k(t s) t s g(s, y(s)). The underlying methods used in the routine re the frctionl liner multistep methods bsed on BDF methods. This routine cn lso be used for liner g. 3.3 Volterr Equtions of First Kind () (b) Liner equtions See (b). Nonliner qutions D05BEF is pplicble to nonliner eqution hving wekly-singulr kernel (Abel). The kernel function hs the form k(t s) K(t, s, y(s)) = g(s, y(s)). t s 3.4 Utility Routines The underlying methods used in the routine re the frctionl liner multistep methods bsed on BDF methods. This routine cn lso be used for liner g. D05BWF genertes the weights ssocited with Adms nd BDF liner multistep methods. These weights cn be used for the solution of non-singulr Volterr integrl nd integro-differentil equtions of generl type. D05BYF genertes the weights ssocited with BDF liner multistep methods. These weights cn be used for the solution of wekly-singulr Volterr (Abel) integrl equtions of generl type. 3.5 User-supplied Routines All the routines in this chpter require the user to supply functions or rel procedures defining the kernels nd other given functions in the equtions. It is importnt to test these independently before using them in conjunction withnag Librry routines. 4 Index Fredholm eqution of second kind, liner, non-singulr discontinuous or split kernel: liner, non-singulr smoothkernel: Volterr eqution of second kind, liner, non-singulr kernel: nonliner, non-singulr, convolution eqution: nonliner, wekly-singulr, convolution eqution (Abel): Volterr eqution of first kind, nonliner, wekly-singulr, convolution eqution (Abel): Weight generting routines, weights for generl solution of Volterr equtions: weights for generl solution of Volterr equtions with wekly-singulr kernel: D05AAF D05ABF D05AAF D05BAF D05BDF D05BEF D05BWF D05BYF D05.6 [NP3390/19/pdf]

7 D05 Integrl Equtions Introduction D05 5 Routines Withdrwn or Scheduled for Withdrwl None since Mrk References [1] Atkinson K E (1976) A Surveyof Numericl Methods for the Solution of Fredholm Integrl Equtions of the Second Kind SIAM, Phildelphi [2] Bker C T H (1977) The Numericl Tretment of Integrl Equtions Oxford University Press [3] Brunner H nd vn der Houwen P J (1986) The Numericl Solution of Volterr Equtions CWI Monogrphs, North Hollnd, Amsterdm [4] Cochrn J A (1972) The Anlysis of Liner Integrl Equtions McGrw Hill [5] Delves L M nd WlshJ (1974) Numericl Solution of Integrl Equtions Clrendon Press, Oxford [6] Wolkenfelt P H M (1982) The construction of reducible qudrture rules for Volterr integrl nd integro-differentil equtions IMA J. Numer. Anl [NP3390/19/pdf] D05.7 (lst)