Wavelets in the Transport Theory of Heterogeneous Reacting Solutes
Рассматривается дисперсионное уравнение одномерной конвекции (адвекции) теории транспорта гетерогенных реагирующих растворенных веществ в пористых средах. Вейвлетное решение получено в рамках мультирезольвентного анализа....
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Інститут гідромеханіки НАН України
2002
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| Цитувати: | Wavelets in the Transport Theory of Heterogeneous Reacting Solutes / C. Cattani, E. Laserra // Прикладна гідромеханіка. — 2002. — Т. 4, № 3. — С. 75-77. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-49472009-12-30T12:00:33Z Wavelets in the Transport Theory of Heterogeneous Reacting Solutes Cattani, C. Laserra, E. Рассматривается дисперсионное уравнение одномерной конвекции (адвекции) теории транспорта гетерогенных реагирующих растворенных веществ в пористых средах. Вейвлетное решение получено в рамках мультирезольвентного анализа. Розглядається дiсперсiйне рiвняння одновимiрної конвекцiї (адвекцiї) теорiї транспорту гетерогенних реагуючих розчинених речовин у пористих середовищах. Вейвлетний розв'язок одержано в рамках мультирезольвентного аналiзу. In this paper we consider the one-dimensional convection (advection) dispersion equation of the transport theory of heterogeneous reacting solutes in porous media. A wavelet solution is in the framework of multi-resolution analysis. 2002 Article Wavelets in the Transport Theory of Heterogeneous Reacting Solutes / C. Cattani, E. Laserra // Прикладна гідромеханіка. — 2002. — Т. 4, № 3. — С. 75-77. — Бібліогр.: 7 назв. — англ. 1561-9087 http://dspace.nbuv.gov.ua/handle/123456789/4947 35А35, 47А58, 65N13, 65F10, 76W05 en Інститут гідромеханіки НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Рассматривается дисперсионное уравнение одномерной конвекции (адвекции) теории транспорта гетерогенных реагирующих растворенных веществ в пористых средах. Вейвлетное решение получено в рамках мультирезольвентного анализа. |
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Article |
| author |
Cattani, C. Laserra, E. |
| spellingShingle |
Cattani, C. Laserra, E. Wavelets in the Transport Theory of Heterogeneous Reacting Solutes |
| author_facet |
Cattani, C. Laserra, E. |
| author_sort |
Cattani, C. |
| title |
Wavelets in the Transport Theory of Heterogeneous Reacting Solutes |
| title_short |
Wavelets in the Transport Theory of Heterogeneous Reacting Solutes |
| title_full |
Wavelets in the Transport Theory of Heterogeneous Reacting Solutes |
| title_fullStr |
Wavelets in the Transport Theory of Heterogeneous Reacting Solutes |
| title_full_unstemmed |
Wavelets in the Transport Theory of Heterogeneous Reacting Solutes |
| title_sort |
wavelets in the transport theory of heterogeneous reacting solutes |
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Інститут гідромеханіки НАН України |
| publishDate |
2002 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/4947 |
| citation_txt |
Wavelets in the Transport Theory of Heterogeneous Reacting Solutes / C. Cattani, E. Laserra // Прикладна гідромеханіка. — 2002. — Т. 4, № 3. — С. 75-77. — Бібліогр.: 7 назв. — англ. |
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AT cattanic waveletsinthetransporttheoryofheterogeneousreactingsolutes AT laserrae waveletsinthetransporttheoryofheterogeneousreactingsolutes |
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2025-07-02T08:05:46Z |
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2025-07-02T08:05:46Z |
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1836521668977098752 |
| fulltext |
ISSN 1561 -9087 �ਪ« ¤ £÷¤à®¬¥å ÷ª . 2002. �®¬ 4 (76), N 3. �. 75 { 77��� 35A35, 47A58, 65N13, 65F10, 76W05WAVELETS IN THE TRANSPORT THEORY OFHETEROGENEOUS REACTING SOLUTESC. CATTANI�, E. LA SERRA��� Dipartimento di Matematica \G.Castelnuovo", Univ. di Roma \La Sapienza", �� Dipartimento diMatematica e Informatica, Univ. di Salerno, Italia�®«ã祮 03.04.2002In this paper we consider the one-dimensional convection (advection) dispersion equation of the transport theory ofheterogeneous reacting solutes in porous media. A wavelet solution is in the framework of multi{resolution analysis.� áᬠâਢ ¥âáï ¤¨á¯¥àᨮ®¥ ãà ¢¥¨¥ ®¤®¬¥à®© ª®¢¥ªæ¨¨ ( ¤¢¥ªæ¨¨) ⥮ਨ âà ᯮàâ £¥â¥à®£¥ëå ॠ-£¨àãîé¨å à á⢮à¥ëå ¢¥é¥á⢠¢ ¯®à¨áâëå á। å. �¥©¢«¥â®¥ à¥è¥¨¥ ¯®«ã祮 ¢ à ¬ª å ¬ã«ìâ¨à¥§®«ì¢¥â®£® «¨§ .�®§£«ï¤ õâìáï ¤÷ᯥàá÷©¥ à÷¢ïï ®¤®¢¨¬÷à®ù ª®¢¥ªæ÷ù ( ¤¢¥ªæ÷ù) ⥮à÷ù âà ᯮàâã £¥â¥à®£¥¨å ॠ£ãîç¨å஧稥¨å à¥ç®¢¨ ã ¯®à¨áâ¨å á¥à¥¤®¢¨é å. �¥©¢«¥â¨© ஧¢'燐ª ®¤¥à¦ ® ¢ à ¬ª å ¬ã«ìâ¨à¥§®«ì¢¥â®£® «÷§ã.INTRODUCTIONThe transport theory of reacting solutes in a porousmedium is considered under the hypotheses that themotion of solute transport is unidirectional, isother-mal and devoid of instabilities; it takes place in aheterogeneous porous medium and the content, thedensity and the viscosity of the water in the medi-um are constant during the process. Since we aremainly interested in the chemical reaction equationswe assume that the physical parameters de�ning themedium are una�ected by the transport, leaving thesize of the pores, their distribution in the solid (andso on) unchanged. The chemical species de�ning thesolid are at rest (while the chemical species de�n-ing the solute are mobile) in the medium, so thatthe mathematical model of the transport solution un-der chemical reactions (source-free) is the followingalgebraic-di�erential system [6](�; D;Q constants)8>>>>>>>><>>>>>>>>: �@ci@t = Lci;L def= D @2@x2 �Q @@xfr �ci; @ci@t � = 0; (i = 1; : : : ;#P ; r = 1; : : :) ;(1)also called the convection-dispersion system. Sys-tem (1) is a di�erential and algebraic system in theunknown #P functions ci(x; t), #P being the num-ber of tenads [?] involved in the chemical reaction.The algebraic equations (1)3 express the relationships
among concentrations of reaction participants, to beful�lled irrespective of the contributions of the in-dividual processes (both chemical and/or physical)in
uencing the concentrations ci.1. ONE DIMENSIONALSOLUTETRANSPORTEQUATIONIn a one-dimensional domain
� R let x bethe coordinate, I a �nite interval of time t, (I def=ft : 0 < t < T; T <1g). We consider a su�cient-ly fast and reversible reaction of heterogeneous type[6],where the medium's original solution consists oftwo reacting cations M1;M3 in equilibrium with acation exchanger 1 Me. The displacing solution con-tains a reacting cation M1 and a non{reacting anionM2. The transport a�ecting reaction is represent-ed by the chemical reactions (for binary cation ex-change) M1 +M3Me *)M3 +M1Me:There are four tenads in the system, three reactingM1;M3;Me and one chemical non{reacting speciesM2. The basic equations (1) for the concentrations1 Dissolved in water and solid phase transport participantsare given symbolsM andM , respectively.c
C. Cattani, E. Laserra, 2002 75
ISSN 1561 -9087 �ਪ« ¤ £÷¤à®¬¥å ÷ª . 2002. �®¬ 4 (76), N 3. �. 75 { 77c1; �c1; c2; c3; �c3 are [6, p. 1237]8>>>>>>>>>>>><>>>>>>>>>>>>: �@c1@t + �@�c1@t = Lc1;�@c2@t = Lc2;�@c3@t + �@�c3@t = Lc3;�@�c1@t + �@�c3@t = 0K13 = �c1c3c1�c3 (2)with K13 a given constant. The e subscript, repre-senting �Me, has been dropped from the �c functions,for them � is the anologous of � and represents theporous medium's bulk density (mass of the medium'ssolids/the medium's volume). In the heterogeneousreactions both the liquid and solid phases are involvedin the tenads. From (2)4 we get�c1 + �c3 = c (3)since the exchange capacity does not vary with x.Then it follows:� @@t (c1 + c3) = L (c1 + c3)that, with the change of variable u = c1+c3, becomesthe linear equation �@u@t = Lu;whose solution can be expressed in terms of wavelets[2].Deriving the condition (2)5 with respect to t andtaking into account equation (3) we obtain from (2)1;4the following nonlinear system [6]:�� + �pig � @ci@t � �pig @cj@t = Lci i; j = 1; 3pi def= K13 (c� ci) ; g def= K13c1 + c3;from which we obtain the non-linear equationa@g@t = gLg; L def= D @2@x2 �Q @@xwith a suitable constant a. In particular, we restrictourselves to the following one-dimensional initial-boundary problem:@g@t = gLg; L def= D @2@x2 � Q @@x; (a = 1)8>><>>: u(x; 0) = uB(x); 0 � x � 1; t = 0;u(0; t) = 0; x = 0; t > 0;u(1; t) = 0; x = 1; t > 0 (4)
and we assume as the initial functionuB(x) = 8<: 1; x 2 �1;�1; x 2 ��1;0; x 62 �1 [ ��1;�1 def= fx : 0 < x0 � x � x1 < 1g��1 def= fx : 0 < x1 � x � x2 < 1g (5)which corresponds to the realistic case of an impulsefunction when jx2 � x0j ! 0.2. MULTI{RESOLUTION ANALYSIS IN HAARBASISThe Haar scaling function �nk (x) def= 2n=2�(2nx�k)has a compact support on the dyadic intervalDnk def= � k2n ; k + 12n �;where its value is 1. The Haar family of wavelets nk (x) def= �2n2 (2nx� k) k;n2Z ; (6)is a complete orthonormal system for the L2(R) func-tions [5] nk (x) = 8>>>><>>>>: 1; x 2 � k2n ; k + 1=22n ��1; x 2 �k + 1=22n ; k + 12n �0; elsewhere : (7)Let Vn; n 2 Z be the subspace of L2(R) de�ned asthe set of the piecewise constant functions f(x) ofcompact support on Dnk ( n �xed) , and Wn theorthogonal subspace such that the axioms of multi{resolution (or multiscale) analysis [4, 5]are ful�lled,8<: L2(R) = Ln2ZWn = Vq � Lj�qWj; q 2 ZVn+1 = Vn �Wn; (8)being � the direct sum of orthogonal spaces. The setof functions f nkg (n 2 Z) represents an orthonormalbasis for L2(R).Fixing the resolution value N < 1, in (8), theL2(R) space is approximated by L2(R) �= NMn=0Wn;that is,f(x) �= �N+1f(x) def= �00 + NXn=0 nXk=0�nk nk (x); (9)76 C. Cattani, E. Laserra
ISSN 1561 -9087 �ਪ« ¤ £÷¤à®¬¥å ÷ª . 2002. �®¬ 4 (76), N 3. �. 75 { 77
�¨á. 1. Haar-wavelet representation of the wave solutionbeing �n a projection operator into Vn+1, so that�n : L2(R)! Vn+1.Choosing a number of dyadic nodesxk def= k2n ; (k = 0; : : : ; 2n� 1) the dyadic discretiza-tion is the operator rn : L2(R)! L2 (Z(2�n)), withL2 (Z(2�n)) � L2(R) being the set of L2(R) func-tions sampled at xk. The action ofrn on f(x) is suchthat rnf(x) = fn with fn = ff0; f1; : : : ; f2n�1g andnfk def= f(x) jx=xk ; 0 � k � 2n � 1o. The fast Haar-wavelet transform H of fN is the linear operator [1,4] H : L2 (Z(2�n))! Vn j fN 7! HfN == ��00; �nk�n=0;:::;2N�1k=0;:::;n ; (10)where �00; �nk are the wavelet coe�cients.According to the above de�nitions, the projectionoperator �n : L2(R) ! Vn+1 is factorized as �n =Hrn.A p-order Cardinal spline, is a Cp�1([0; 1)) di�er-entiable operatorSp : L2�Z(2�N )�! Cp�2�[0; 1)� : fN 7! s(x) def= SpfN ;such that for the di�erential operatorL : L2�Z(2�N )�! L2�Z(2�N )�it is LH = HLSp: (11)There follows that, given the set fN and computedthe spline of a su�ciently large order, the spline{derivative of HfN belongs to the same space of fN[1, 4].According to the above, using splines and waveletsup to the resolution N , the approximate solution of
the equation (4)is the vector uN (2 VN+1), i.e. as-suming the Euler formula for the time-derivative�N @u@t = uN+1 � uN+1�twe have from (4)uN+1 = uN +�tL�HuN�and, according to (11)uN+1 = �1 + �tHLSp�uN : (12)With the boundary condition (5),time step �t =0:01, and assuming in (4)1 Q = D = 1 and in(5)x0 = 14 ; x1 = 38 ; x2 = 12, after 3 time steps(t = 0:05) we obtain the evolving function of Fig.1. C. Cattani Haar Wavelet Spline // Journal of In-terdisciplinary Mathematics.{ 2001.{ vol.4, No. 1.{P. 35-47.2. C.Cattani and E.Laserra , // Transport Theory ofHomogeneous Reacting Solutes.{ Ukrainian Mathe-matical Journal.{ 2001.{ P. vol. 53, No. 8.1048-10523. C.Cattani and F.Passarella Nonsimple Porous mate-rials with Thermal Relaxation // Rend. Ist. Lombar-do Acc. di Scienze e Lett.,A.{ 1996.{ vol. 130, No 2.{P. 1-19.4. C.Cattani and M.Pecoraro Non Linear Di�erentialEquations in Wavelet Basis // Acoustic Bulletin.{2000.{ vol. 3, (4).{ P. 4-10.5. I.Daubechies Ten lectures on wavelets/CBMS-NSFRegional Conference Series in Applied Mathematics.{Philadelphia.:SIAM, 1992.6. J.Rubin Transport of Reacting Solutes in PorousMedia: Relation Between Mathematical Nature ofProblem Formulation and Chemical Nature of Reac-tions // Water Resour. Res.{ 1983.{ No 13.{ P. 1231-1252.7. D.Schweich and M.Sardin Absorption, partition, ionexchange, and chemical reaction in batch reactors orin columns // J. Hydrol.{ No 50.{ 1981.{ P. 1-33.C. Cattani, E. Laserra 77
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