IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Using Semi-Analytic Method to Decreasing Dangers of Lead L. N. M. Tawfiq and M. N. M. Tawfiq Departme nt of Mathematics, College of Education Ibn Al - Haitham , Unive rsity of Baghdad Received in : 7 November 2010 Accepte d in : 8 February 2011 Abstract The aim of this p aper is t o p resent a method for solving of sy st em of first order initial value p roblems of ordinary differential equation by a semi-analytic technique with constructing p olynomial solutions for decreasing dangers of lead. The original p roblem is concerned using two-p oint osculatory interp olation with the fit equals numbers of derivatives at t he end p oints of an interval [0 , 1]. Key word : Osculator interpolation , Initial value problem Introduction Lead is a metal found naturally in the earth’s crust. In nature , it is found more often in chemical comp ounds than as a p ure metal. When released into the air, it may travel long distances before settling to the ground , where it can contaminate water and soil . Lead has been used extensively throughout human history because it is easy both to extract and to work with. Where used in some Cosmetics and hair dy es, p aint houses , Plumbing and solder , Gut ters , window glazing , Leaded gasoline , cray ons , Printers ( ink ) , batt eries , p last ics and used in the manufacturing of p roducts others . [1] Where the deficiencies in some mineral nutrients, sp ecifically calcium, iron and zinc, where involvement in building bones, may increase the amount of lead absorbed . As a result, Children, Pregnant Women, old Women and old man are more efficient at absorbing lead, than adult's because they need more calcium. Additionally , the kidney s and the intest ine, less able to eliminate lead . Lead absorp tion rates vary ; the gastrointest inal tracts of adults ty p ically absorb 10 - 15 p ercent of ingested lead , while those of p regnant women and children can absorb up to 50 p ercent . Lead p oisoning occurs when there are adverse health effects due from lead in the body .Some of the more p rominent sy mptoms of lead p oisoning include headaches, irritability , abdominal p ain , vomiting , anemia ( general weakness) , weight loss, p oor att ention sp an, noticeable learning difficulty , Nervous sy st em and kidney damage. However, at very low exp osure levels, lead may not p roduce sp ecific sy mptoms, but can p roduce subt le adverse effects on children's develop ment . There is a medical test to show whether I've been exp osed to lead through a simp le blood test, there are treatments to remove lead from the body although medication that used CAedta intravenously must have the att ention into treatment because of p robably the metal exciding ZINC from body . Therefore, suggesting treatment app lication CaNa2 EDTA from mouth way over a p eriod of four weeks , if high levels of lead in blood between 90-100 mg / de , treatment app lication dimercaprol, also known as BAL the dose 500 and 1500 CAedta where inject in the time self ,but all cured ty p es used at case ,lead poisoning where thrown metal into out of ,but there metals desirable in body may thrown t herefore the safeguard best from the remedy IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Lead in the human body : A case study Rabinowitz , Wetherill and Kop p le in [2] made a carefully controlled st udy of the lead intake and exertion of a healthy volunteer in an industrial urban setting . The data from this st udy were used to est imate the rate const ants for the compartment model (1): ( Blood ) y'1 = – ( k01 + k21 + k31) y 1 + k12 y 2 + k13 y 3 + I1 ( Tissue ) y '2 = k21 y 1 - ( k02 + k12 ) y 2 ..…. (1) ( Bone ) y'3 = k31y 1 - k13 y 3 Lead is measured in micrograms and time in days. For examp le, the rate term 49.3 in sy st em (2) below is the ingestion rate I1 of lead in micrograms p er day, while the coefficient 0.0361 (day) -1 in the first rate equation of (2) is the sum of the three comp artment transfer coefficients k02 , k21 and k31 of lead from the blood into, resp ectively, the excretory sy st em, tissue and bones. T he full IVP is given by : y '1 = – 0.0361 y 1 + 0.0124 y 2 + 0.000035 y 3 + 49.3 , y 1(0) = 0 y '2 = 0.0111 y 1 + 0.0286 y 2 , y 2(0) = 0 …(2) y '3 = 0.0039 y 1 – 0.00035 y 3 , y 3(0) = 0 Assumed that initially there is no lead in the compartments . Now, we solve sy stem (2) using se mi-analy tic method ,i.e., use two-p oint osculatory interp olation [3]. Essentially this is a generalization of interpolation using Taylor polynomials and for that reason osculatory interp olation is sometimes referred to as two-p oint Tay lor interp olation. The idea is t o approximate a function y(x) by a polynomial P(x) in which values of y (x) and any number of its derivatives at given p oints are fitt ed by the corresp onding function values and der ivatives of P(x) . In this p ap er, we are p articularly concerned with fitt ing function values and derivatives at t he two end points of a fin ite interval, say [0 , 1] ,wherein a useful and succinct way of writing osculatory interpolant P2n+1(x) of degr ee 2n + 1 was given for example by Phillips [4] as : P2n+1(x) =   n j 0 { y )( j (0) q j (x) + (-1) j y )( j (1) q j (1-x) } ………….(3) q j (x) = ( x j / j!)(1-x) 1n    jn s 0        s sn x s = Q j (x) / j! ...………..( 4 ) so t hat ( 3 ) with ( 4 ) satisfies : y )( j (0) = )( 12 j nP  (0) , y )( j (1) = )( 12 j nP  (1) , j = 0, 1, 2,…, n . imp lying that P2n+1(x) agr ees with t he app rop riately truncated Tay lor series for y (x) about x = 0 and x = 1. The error on [0, 1] is given by : R2n+1 = y (x)-P2n+1(x) = )!22( )()1()1( )22(1)1(1    n yxx nnnn  where ε  (0, 1) and y )22( n is assumed to be continuous. Finally we observe that ( 3 ) can be writt en directly in terms of the Tay lor coefficients ai and bi about x = 0 and x = 1 resp ectively, as : P2n+1(x) =   n j 0 { a j Q j (x) + (-1) j b j Q j (1-x) } ….. (5) Now, the simple idea behind the use of two-p oint p oly nomials is to replace y (x) in p roblem by a P2n+1 ( equation(3) or (5) ) which enab les any unknown derivatives of y (x) to be comp uted . The first step therefore is to construct the P2n+1 . To do this we need the Taylor coefficients of y 1(x) and y2(x) resp ectively about x = 0 : IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 y 1 = a 0 + a 1 x +    2i a i x i ….…. ( 6a ) y 2 = b 0 + b 1 x +    2i b i x i …….. ( 6b ) where y1(0) = a0 , y 1 ' (0) = a1 , … , y 1 (i) (0) / i! = ai , i = 2, 3, …… and y2(0) = b0 , y 2 ' (0) = b1 , … , y 2 (i) (0) / i! = bi , i = 2, 3, …… then insert the series forms (6a) and (6b) r esp ectively into (2) and equate coefficients of p owers of x . Also ,we need Tay lor coefficients of y 1(x) and y2(x) about x = 1, resp ectively y 1 = c 0 + c 1 (x-1) +    2i ci (x-1) i …….. ( 7a ) y 2 = d 0 + d 1 (x-1) +    2i di ( x-1) i …….. ( 7b ) where y1(1) = c0 , y 1 ' (1) = c1 , … , y 1 (i) (1) / i! = ci , i = 2, 3, …… and y2(1) = d0 , y 2 ' (1) = d1 , … , y 2 (i) (1) / i! = di , i = 2, 3, …… then insert the series forms (7a) and (7b) r esp ectively into (2) and equate coefficients of p owers of ( x – 1 ) . The resulting system of equations can be solved using M ATLAB version 7.9 to obtain ai , bi , ci and di for all i ≥ 2, we see that ci sۥ and disۥ coefficients dep end on indicated unknowns c0 and d0 . The algebraic manip ulations needed for this p rocess .We are now in a position to construct a P2n+1(x) and 12 ~  n (x) from (6) and (7) of the form ( 3 ) by the following : P2n+1(x) =   n i 0 { ai Qi(x) + (-1) i ci Qi(1- x) } ………… (8a) and 12 ~   n (x) =   n i 0 { bi Qi(x) + (-1) i di Qi(1- x) } ………… (8b) Where Qi( x) defined in (4) , We see that (8) have only two unknowns c0 and d0 . Now, integrate equation (2) to obt ain : c0 – a0 =  1 0 f1( x, y 1, y 2 ) dx ………. (9a) d0 – b0 =  1 0 f2( x, y 1, y 2 ) dx ………. (9b) use P2n+1 and 12 ~   n as a rep lacement of y 1 and y2 resp ectively in (9) . Since we have only the two unknowns c0 and d0 to comp ute for any n we only need to generate two equations from this p rocedure as two equations are already supp lied by (9) and initial cond ition. Then solve this system of algebraic equations using M AT LAB version 7.9 to obtain c0 and d0 ,so insert it into (8) thus (8) rep resent the solution of (2) . Extensive computations have shown that this generally p rovides a more accurate p oly nomial representation for a given n . Now from equations (3) and (4) we have : P9 = 0.00000000000000137t 9 – 0.00000000000028t 8 + 0.00000000005t 7 – 0.00000000802t 6 + 0.00000109t 5 –0.00012516007t 4 + 0.0118401057t 3 – 0.889865000005t 2 + 49.3t IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 P ~ 9=– 0. 0000000000000009t 9 +0.00000000000019t 8 – 0.000000000035t 7 + 0.000000005387t 6 – 0.0000007071t 5 +0. 0000750482t 4 –0.0059009635t 3 + 0. 273615t 2 T9=–0.00000000000000012t 9 +0. 000000000000025t 8 –0.0000000000045t 7 + 0.000000000709t 6 – 0.00000009771t 5 +0.0000115542t 4 – 0.0011579461t 3 + 0.096135t 2 Table (1) give the results for n = 4 to different nodes in the domain . Figure (2) illustrate buildup of lead in the bones , tissue , blood using semi-analytic solution . Also , A comp arison between different methods are given in table (2) to illustrate the accuracy of suggested method . With sy st em (2) we can st udy the effect of changing the input rate I1 = 49.3 micrograms / day of lead into the bloodstream, or the effect of a medication that increases the diffusion coefficient k13 = 0.000035 (day) -1 of lead out of the bones. Figure (2) disp lay s t he buildup of lead in the body comp artments over a period of 800 days. Lead levels in the bloodst ream and the issue ap p ear to have nearly reached st eady st ate after the first 200 days, but the lead level in the bones is far from a steady st ate. The transfer coefficients k13 = 0.000035 (day) -1 of lead for bones back into the blood stream is so small that the skeleton acts like a st orage reservoir for lead . Now, sup p ose that after 400 days the subject is p laced in a comp letely lead-free environment (i.e., the term 49.3 in sy st em (2) becomes 0 for t > 400 ). Figure (3) shows that the lead levels in the blood and tissue p lunge dramatically , but the amount of lead in the bones, does not seem to drop very much, at least not in the next 400 days . Anot her way to remove lead from the bones is to administ er and anti lead medication that increases the rate at which lead leaves t he skeletal sy st em In p articular, sup p ose that the rate coefficient k13 = 0.000035 increases by an order of magnitude to 0.00035. Figure (4) shows t he very small effect this change has if the medication is administ ered from the 400th day onward. However, if the subject moves to a lead free environment and takes the medication from the 400th day onward, then there is a noticeable drop in the skeletal lead levels (Figure 5). Let us see what happ ens when a massive dose is given, so large that k13 increases from 0.000035 to 0.035. Sup p ose from the 400th day on that t he heavy dose is given and the subject is in a lead free environment. The lead exists the bones very quickly. Following a slight rise after the 400th day, lead exists the blood and tissue as well (Figure 6) of course that much medication is p robably more harmful than the lead . Why might blood lead levels be higher in wome n after menopause ? Aft er menop ause, women often develop ost eoporosis , which is a p rogressive and serious loss of bone mass . As a result of loss of bone mass, lead st ored in bone may be released into t he blood . What is lead "poisoning" compared to lead "e xposure" ? Lead exp osure refers to the entry of lead into the body , through ingestion, inhalation, the skin or the p lacenta . Lead p oisoning occurs when there are adverse health effects due to lead in the body . Recomme ndations Aft er above st udy ,we introduce some recommendations and advices which help to minimize harms of lead in society and we get some of this information from [5] . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 What role has the government played in reducing lead poisoning? 1. Using Benzene free of Lead which is c alled ( Green Benzene ) 2. Not renewing licenses for vehicles mechanically exp ired , with using transp ortation means that uses electricity , and the exp ansion in using collective transp ortation means such as ( M itro) with improving them 3. Surveillance on batt eries factories the p rivate as well as the governmental , which conduct the melting of Lead in addition to newsp apers and magazines p ressing , with transferring them to deserted regions 4. Pay ing the attention for not to construct new schools near to highway s t hat is crowded with vehicles 5. It was p roved through scientific st udy in the college of agriculture / Ain Shams university , that t he root p art absorbs the Lead more than the green sy st em , where there are many kinds of vegetables which the root p art is eaten , that is p lanted beside the highway s the fact that leaded to the Lead to go into the human as well as the animal's body , therefore the work must aim for vegetables and fruits not to be planted near highway s 6. Surveillance on all kinds of p aints factories also the imported p roducts with their insp ection to make sure that t hey are free of Lead subst ance What role has the family played in protecting children from lead poisoning ?[6] 1. Test y our home for p aint .if it content of lead or no 2. Hire a p erson with sp ecial training for correcting lead p aint p roblems to remove lead p aint from your home, someone who knows how to do this work safely. 3. Home Clean, window sills, and other surfaces weekly. 4. Don’t bring lead dust into y our home from work . 5. M ake sure y our children wash t heir hands before meals . 6. A child who gets enough iron and calcium will absorb less lead . 7. Don’t st ore food or liquid in lead cryst al glassware or imported or old p ottery. 8. Household water will contain more lead if it has sat for a long time in the pipes 9. .therefore; let it run for 15–30 seconds before drinking it or cooking with it . Re ferences 1. Lewis , J. (1985), Lead poisoning : a hist orical p ersp ective , EPA Journal. 2. Robert, L. B. and Courtney , S. C. (1996 ), Differential Equations A M odeling p ersp ective" , United States of America. 3. Girault ,V. and Scott ,L. R. (2002),Hermite Interpolation of Non smooth Functions Preserving Bound ary Conditions, M athematics of Comp utation, 71( 239): 1043-1074 . 4. Phillips , G .M . (1973), Exp licit forms for certain Hermite ap p roximations, BIT 13 , 177- 180 . 5. New England Journal of M edicin e, (2003), Ap ril, 17(348): 1517-1526 . 6. Agency for Toxic Subst ances and Disease Registry . Lead ToxFAQs sheet . August ( 2007), www.atsdr.cdc.gov/tfacts13.html , viewed November 14 , (2007). IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Table (1) : The result of the method for n = 4 P9 P ~ 9 T9 b10 48.42185102755592 48.42185102755592 48.42185102755592 b20 0.2677883829038516 0.2677883829038516 0.2677883829038516 b30 0.09498851115891924 0.09498851115891924 0.09498851115891924 t P9 P ~ 9 T9 0 0 0 0 0.1 4.92111317760058 0.00273025653425228 0.000960193208372153 0.2 9.82449992093787 0.0108975121431540 0.00383615488695559 0.3 14.7102308217005 0.0244666301613681 0.00862097880837015 0.4 19.5783761737962 0.0434026523504553 0.0153077862420084 0.5 24.4290059746451 0.0676707980599557 0.0238897258380576 0.6 29.2621899264675 0.0972364633922961 0.0343599735120251 0.7 34.0779974375664 0.132065220371507 0.0467117323297653 0.8 38.8764976236032 0.172122816115731 0.0609382323930065 0.9 43.6577593088695 0.217375172013507 0.0770327307253737 1 48.4218510275513 0.267788382903812 0.0949885111589079 IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Table( 2) : A comparison betwee n semi-analytic and other methods P9 by using Osculatory y 1:ABM solution y 1:RK solution t 0 0 0 0 4.92111317760058 4.9211131775747 7 4.92111317760100 0.1 9.82449992093787 9.8244999208060 5 9.82449992093843 0.2 14.7102308217005 14.710230821422 5 14.7102308217013 0.3 19.5783761737962 19.578376173370 2 19.5783761737973 0.4 24.4290059746451 24.429005974074 3 24.4290059746466 0.5 29.2621899264675 29.262189925928 3 29.2621899264696 0.6 34.0779974375664 34.077997437030 8 34.0779974375690 0.7 38.8764976236032 38.876497623070 6 38.8764976236066 0.8 43.6577593088695 43.657759308340 1 43.6577593088736 0.9 48.4218510275513 48.421851027025 1 48.4218510275559 1 T9 by using Osculatory y 3:ABM solution y 3:RK solution t 0 0 0 0 0.0009601932083 72153 0.0009601932106 90100 0.00096019320833 8816 0.1 0.0038361548869 5559 0.0038361548987 9100 0.00383615488692 262 0.2 0.0086209788083 7015 0.0086209788333 3617 0.00862097880833 768 0.3 0.0153077862420 084 0.0153077862802 708 0.01530778624197 67 0.4 0.0238897258380 576 0.0238897258893 462 0.02388972583802 68 0.5 0.0343599735120 251 0.0343599735605 295 0.03435997351199 56 0.6 0.0467117323297 653 0.0467117323780 034 0.04671173232973 76 0.7 0.0609382323930 065 0.0609382324410 489 0.06093823239298 09 0.8 0.0770327307253 737 0.0770327307732 064 0.07703273072535 09 0.9 0.0949885111589 079 0.0949885112065 369 0.09498851115892 05 1 P ~ 9 by using Osculatory y 2:ABM solution y 2:RK solution t 0 0 0 0 0.0027302565342 5228 0.0027302565510 2616 0.00273025653401 098 0.1 0.0108975121431 540 0.0108975122287 910 0.01089751214291 52 0.2 0.0244666301613 681 0.0244666303419 837 0.02446663016113 21 0.3 0.0434026523504 553 0.0434026526272 081 0.04340265235022 29 0.4 0.0676707980599 557 0.0676707984308 485 0.06767079805972 79 0.5 0.0972364633922 961 0.0972364637429 603 0.09723646339207 40 0.6 0.1320652203715 07 0.1320652207201 43 0.13206522037129 2 0.7 0.1721228161157 31 0.1721228164628 73 0.17212281611552 5 0.8 0.2173751720135 07 0.2173751723590 46 0.21737517201331 2 0.9 0.2677883829038 12 0.2677883832477 92 0.26778838290386 1 1 IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 0 0.1 0.2 0.3 0 .4 0 .5 0 .6 0.7 0.8 0.9 1 0 10 20 30 40 50 The s olu tion a t n=4 blood tissues bones Fig.( 1) : Buildup of lead in the bone s , tissue , blood 0 100 200 300 400 500 600 700 800 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 t(days) y1 ,y 2, y3 (m ic ro gr am s) blood tissues bones Fig.( 2) : Buildup of lead in the bone s(circle), blood(line), tissue(plus) 0 100 200 300 400 500 600 700 800 0 500 1000 1500 2000 2500 t(days) y1 ,y 2, y3 (m ic rog ra m s) blood tis sues bones Fig.( 3): lead intake stops on the 400 th day IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 0 100 200 300 400 500 600 700 800 0 500 1000 1500 2000 2500 3000 3500 4000 4500 t(days ) y1 ,y 2, y3 (m ic rog ra m s) blood ti ssues bones Fi g.( 4): An ti lead me di cation taken from the 400 th day on does not hel p mu ch 0 100 200 300 400 500 600 700 800 0 500 1000 1500 2000 2500 t(days) y1 ,y 2, y3 (m icr og ra m s) blood ti ssues bones Fig.( 5): lead-free environment and medication from the 400 th day on help a lot 0 100 200 300 400 500 600 700 800 0 500 1000 1500 2000 2500 t(days) y1 ,y 2, y3 (m ic ro gr am s) blood ti ssues bones Fig.( 6): lead-free environment and a heavy dose of antilead drug from the 400 th day on .Is the subject still alive? 2011) 3( 24مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد شبة التحلیلیة لتقلیل أخطار الرصاص طریقةاستخدام ال قلمى ناجي محمد توفیق و محمد ناجي محمد توفی جامعة بغداد ،كلیة التربیة ابن الهیثم ،قسم الریاضیات 2010تشرین الثاني 7 :استلم البحث في 2011 شباط 8 :قبل البحث في خالصةال القـیم االبتدائیــة الهـدف مـن البحــث هـو أیجــاد طریقـة لحـل منظومــة معـادالت تفاضــلیة اعتیادیـة مـن الرتبــة األولـى لمســائل .صاص التقنیة شبه التحلیلیة وذلك بإیجاد الحل بشكل متعددة حدود و ذلك للتقلیل من أخطار الر عمالباست النقطتین التي تتفق فیهـا الصـورة وعـدد متسـاو مـن المشـتقات المعرفـة ياالندراج التماسي ذ عمالالمسالة األصلیة تتعلق باست .مع البیانات المعطاة [1 , 0] مدةعند نقطتي نهایة ال االندراج التماسي، مسائل القیم االبتدائیة : الكلمات المفتاحیة