Mathematical Modelling of the Solutions of Partial Differential Equation with Physical Implications

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1 IOSR Jornal of Mahemais (IOSR-JM) e-issn: , p-issn: X. Volme 4, Isse 3 Ver. II (May - Jne 8), PP Mahemaial Modelling of he Solions of Parial Differenial Eqaion wih Physial Impliaions Ben Obakpo Johnson, Salim Sa ad Hama adama, Okorie hariy Ebelehk 3,,3 Deparmen of Mahemais and Saisis, Federal Universiy, kari, Taraba Sae, Nigeria *orresponding Ahor: Ben Obakpo Johnson, Absra: This work aimed a eamining he mahemaial modelling of he solions of parial differenial eqaion wih physial impliaions. In parilar, we looked a he onsiseny and well-posedness of he solion of homogeneos one-dimensional wave. The general solion of he wave eqaion was derived by he mehod of hange of variable and finally he general solion derived leads s o he niqe solion of he problem, alled he d Alember s formla. e hen proof ha he d Alember s formla obained eiss, i is niqe and sable. There afer we sed he solion obained o analyze resl and show he behavior of or resl in a able and onlde ha he noion of a well-posed problem is imporan in applied mahemais. Keyword: Parial Derivaives; Parial Differenial Eqaion, ell-posedness, Modelling Dae of Sbmission: Dae of aepane: I. Inrodion Generally, finding he ea solion of problems in parial differenial eqaion may be impossible or a leas poses praial hallenges o obain i. The sdy of parial differenial eqaions sared as a ool o analyze he models of physial siene. The PDE s sally arise from he physial laws sh as balaning fores (Newon s law), momenm and onservaion laws e. (Srass, 8). In his work he eqaion of moion for he sring nder erain assmpion has been derived whih is in he form of seond order parial differenial eqaion. The governing parial differenial eqaion represens ransverse vibraion of an elasi sring whih is known as one dimensional wave eqaion (King and Billingham, ). The analyial solion has been obained sing mehod of hange of variable. The solion of wave eqaion was one of he major mahemaial problems of he mid eigheenh enry. The wave eqaion was firs derived and sdied by D Alember in 746. He inroded he one dimensional wave eqaion This also araed he aenion of Merwin, (4), Marel, (4), Shiesser and Griffhs, (9), and Broman, (989). The wave eqaion was hen generalized o wo and hree dimensions, i.e. Δ= (, here 3 i i Δ (, The solion of wave eqaion was obained in several differen forms in series of works (Benzoni- Gavage and Serre, 7). Parial differenial eqaions are ofen sed o onsr models of he mos basi heories nderlying physis and engineering (Lawrene, ). Unlike he heory of ordinary differenial eqaions, whih relies on he fndamenal eisene and niqeness heorem, here is no single heorem whih is enral o he sbje. Insead, here are separae heories sed for eah of he major ypes of parial differenial eqaions ha ommonly arise. I is worh poining o ha he preponderane of differenial eqaions arising in appliaions, in siene, in engineering, and wihin mahemais iself, are of eiher firs or seond order, wih he laer being by far he mos prevalen. ave eqaion is a seond-order linear hyperboli PDE ha desribes he propagaion of a variey of waves, sh as sond or waer waves. I arises in differen fields sh as aosis, eleromagneis, or flid dynamis (Sajjadi and Smih, 8). DOI:.979/ Page

2 Mahemaial Modelling Of he Solions of Parial Differenial Eqaion wih Physial Impliaions Parial differenial eqaion (PDE) onains parial derivaive of he independen variable, whih is an nknown fnion in more han one variable., y,,..., y y & e an wrie he general firs order PDE for, as F,, (,, (,, (, F(,,,, ) (.) (Srass, 8). Alhogh one an sdy PDEs wih many independen variables as one wish, b in his researh work we will be primarily onerned wih PDEs in wo independen variables. So many mehods have been sed in solving parial differenial eqaion for eample: I. Mehod of separaion of variable, II. Mehod of hange of variable, III. Laplae Transform mehod, IV. Nmerial mehod and V. Mehod of haraerisis, e. B in his paper, we obain d Alember s general solion by he mehod of hange of variable. Saemen of he problem In hese researh work, we are mainly onerned wih impliaions of a solion o a given parial differenial eqaion, and he good properies of ha solion. e are ineresed in he mahemaial modelling of he onsiseny and well-posedness of solion( o erain PDEs, in parilar he homogeneos onedimensional wave eqaion. Varios physial qaniies will be measred by some fnion = (,y,z, This old depend on all spaial variables and ime, or some sbse (Go and Zhang, 9). The parial derivaives of will be denoed wih he following ondensed noaion: y y,, y,, e. Aim The aim of his work is o presen he mahemaial modelling of parial differenial eqaions wih impliaions. Objeives. To obain a solion, eamine he onsiseny and well-posedness of he solion of he homogeneos onedimensional wave eqaion via he mehod of hange of variable. To presen he resl in a ablar form II. METHODS e shall onsider erain ype of problem ha is assoiaed wih linear hyperboli parial differenial eqaions. e shall onsider his problem in onneion wih he homogeneos one-dimensional wave eqaion of he form here and are he independen variables and is a onsan. This eqaion, alled he homogeneos one-dimensional wave eqaion, serve as he prooype for a lass of hyperboli differenial eqaion. Hyperboli eqaions arise in physial appliaion as models of waves, sh as aosi, elasi, eleromagnei or graviaional waves. The qaliaive properies of hyperboli PDEs differ sharply from hose of paraboli and ellipi PDEs. The wave eqaion is erainly one of he mos imporan lassial eqaions of mahemaial physis. e look a he mahemaial modelling of parial differenial eqaions as applied o he homogeneos onedimensional wave eqaion. In parilar, we eamine he onsiseny and well-posedness of he solion (Go and Zhang, 7). The d Alember s general solion is o be derived by he mehod of hange of variable whih will lead s finally o he d Alember s formla for he solion of he wave eqaion.. Mahemaial Formlaion The wave eqaion has many physial appliaions from sond waves in air o magnei waves in he Sn's amosphere. However, he simples sysems o visalize and desribe are waves on a srehed elasi sring. DOI:.979/ Page

3 Mahemaial Modelling Of he Solions of Parial Differenial Eqaion wih Physial Impliaions Iniially he sring is horizonal wih wo fied ends say a lef end L and a righ end R : Then from end L we shake he sring and we noie a wave Propagae hrogh he sring. The aim is o ry and deermine he verial Displaemen from he ais of he sring, (, as a fnion of posiion And ime : Tha is, (, is he displaemen from he eqilibrim a Posiion and ime : A displaemen of a iny piee of he sring beween Poins P and Q is shown in; here (, T(, () is he angle beween he sring and a horizonal line a posiion and ime is he ension in he sring a posiion and ime ; is he mass densiy of he sring a posiion : To derive he wave eqaion we need o make some simplifying assmpions: () The densiy of he sring, is onsan so ha he mass of he sring beween P and Q is simply imes he lengh of he sring beween P and Q where he lengh of he sring is s given by s ( ) ( ) () The displaemen, (, and is derivaives are assmed small so ha s and he mass of he porion of he sring is (3) The only fores aing on his porion of he sring are he ensions T(, a P and T(, a Q (In physis, ension is he magnide of he plling fore eered by a sring). The graviaional fore is negleed. (4) Or iny sring elemen moves only verially. Then he ne horizonal fore on i ms be zero. Ne, we onsider he fores aing on he ypial sring porion shown above. These fores are: (i) Tension plling o he righ, whih has magnide T(, and as a an angle (, above he horizonal. (ii) Tension plling o he lef, whih has magnide T (,, and as a an angle (,, above he horizonal. Now we resolve he fores ino heir horizonal and verial omponens.- Horizonal: The ne horizonal fore of he iny sring is T(, os (, T(, os (,. Sine here is no horizonal moion, we ms have T(, os (, T(, os (, T. Verial: A P he ension fore is T(, sin (, where as a Q he Fore is T(, sin (,. Then Newon's Law of moion Mass aeleraion Applied Fores Gives DOI:.979/ Page

4 Mahemaial Modelling Of he Solions of Parial Differenial Eqaion wih Physial Impliaions T(, sin (, T(, sin (,. T (, sin (, T (, sin (, T (, sin (, T (, os (, an (, an (, B an (, lim Likewise, (,. an (, (, Hene, we ge (, (, (, T Dividing by and leing we obain (, (, T Or (, (, T here, e all he wave speed. (hp://hyperphysis.phyasr.gs.ed/hbase/waves/waveq.hml) This is he parial differenial eqaion giving he ransverse vibraion of he sring. I is also alled he one dimensional wave eqaion.. Homogeneos one-dimensional wave eqaion e solve he wave eqaion on he whole real line < < +. Real physial siaions are sally on finie inervals. e are jsified in aking on he whole real line for wo reasons. Physially speaking, if yo are siing far away from he bondary, i will ake a erain ime for he bondary o have a sbsanial effe on yo, and nil ha ime he solions we obain in his haper are valid. Mahemaially speaking, he absene of a bondary is a big simplifiaion. The mos fndamenal properies of he PDEs an be fond mos easily wiho he ompliaions of bondary ondiions. = for < <, > () (,) = f() for < < () (,) = g() for < < (3).3 Solion via hange of variable Sine he eqaion is hyperboli we inrode he new variable Then, by he hain rle, we have.. ( ) Faoring o, we have, defined by DOI:.979/ Page

5 Mahemaial Modelling Of he Solions of Parial Differenial Eqaion wih Physial Impliaions DOI:.979/ Page Therefore, Also.. From he oordinae eqaions above, we have =.. = Therefore, Le ransform,, sh ha ), ( ), ( U Hene U U Beomes ) ( ) ( 4 Therefore, he wave eqaion (*) ransforms ino. OR w (4) I is easy o find he general solion of he eqaion (4) by inegraing i wie. Firs sppose yo inegrae wih respe o and noie ha he onsan of inegraion ms depend on o ge, G() w Now inegrae wih respe o and noie ha he onsan of inegraion depends on.

6 Mahemaial Modelling Of he Solions of Parial Differenial Eqaion wih Physial Impliaions w G( ) d F( ) Therefore, le G( ) d G( ) So ha w G( ) F( ) Reall ha we ransformed U(, w(, ) Therefore U(, F( G( (5) This is d Alember s general solion of he wave eqaion, where f, g are arbirary fnions. Along wih he wave eqaion (), we ne onsider some iniial ondiions, o single o a parilar physial solion from he general solion (5). The eqaion is of seond order in ime, so onsidering f() and g() whih are speified boh for he iniial displaemen (,), and he iniial veloiy (,) respeively, where f and g are arbirary fnions of single variable, and ogeher are alled he iniial daa of he eqaion (). Now we deermine he fnion F and G so ha he general eqaion (5) may saisfy eqaion () above. Given; (,) = f() for < < (,) = g() for < < Firs, seing = in (5), we ge U(,) f ( ) F( ) G( ) U(,) F( ) G( ) f ( ) (6) Then, sing he hain rle, we differeniae (5) wih respe o and p = o ge U (,) F'( ) G'( ) (The primes here mean differeniaion wih regard o he single variable he fnion depends on.) F' ( ) G'( ) g( ) g( ) F' ( ) G'( ) (7) Inegraing (ii) g( F( ) G( ) ds k (8) Adding (i) and (ii) g( F( ) ds k f ( ) k f ( ) F( ) g( ds (9) Also sbraing (i) from (ii) yield G( ) g( ds k f ( ) k f ( ) G( ) g( ds () Reall ha U(, F( G( Therefore from (IV) DOI:.979/ Page

7 Mahemaial Modelling Of he Solions of Parial Differenial Eqaion wih Physial Impliaions F( f ( k g( ds () And f ( k G ( g( ds () Hene, adding (vi) and (vii) yield. U(, f ( f ( g s ds ( ) (3) This is known as d Alember s formla for he solion of he iniial vale problem above. Or derivaion of d Alember s formla shows ha any solion of (),() and (3) ha is wie oninosly differeniable ms have he represenaion (3), hene he solion is niqely deermined by he iniial daa f and g. Ths d Alember s formla represens he niqe solion of (), () and (3)..3 Proof of he well-posedness of he solion e prove he well-posedness of he ahy problem (), () and (3), above by proving (a) The eisene of he solion (b) The niqeness of he solion () The sabiliy of he solion.3. learly he solion eis from (3), hene proved Uniqeness of he Solion In his seion, we define an energy fnion E( for he wave eqaion, show ha energy is onserved for he ahy problem (), () and (3), and se his propery o esablish niqeness of solions of he ahy problem (), () and (3), above. Le s assme ha = (, To be a smooh solion of he ahy problem and he derivaives (, and (, Are sqare inegrable for eah. Then he oal energy defined by E ( ) ( ) d is finie, noe ha E( is he sm of kinei and poenial energy. The poenial energy PE( d Is he energy sored in he sring de o ension and he kinei energy KE ( ) d Is he form of mv in lassial mehanis of a rigid body wih mass m and veloiy v. To see how E( is onneed o he one-dimensional wave eqaion Le s mliply he PDE by and inegrae by pars. Mliplying Inegraing by pars d d d d DOI:.979/ Page d

8 Mahemaial Modelling Of he Solions of Parial Differenial Eqaion wih Physial Impliaions ' Tha is E ( ( ) d d d Therefore, we have onservaion of oal energy: E( = onsan, from whih we dede E E ' ( ) () ( ( ) ( ) ) d, o This ideniy is an imporan ool for eisene and reglariy of solions, b also for niqeness of solions, as we now dissses..4 Proof of he niqeness of Solion To prove niqeness, we show. Le define (, (. (,. Then saisfies he homogeneos wave eqaion, wih zero iniial daa: (,), (,) Sine E( = E() = for his problem, we have E( Therefore (, And ) d Ths, is onsan in and. B (,) =, so he onsan is zero. Hene.5 Proof of oninos dependene (sabiliy) Le and Be solions of problem (), () and (3) wih iniial daa φ k, ψ k, where k =,,, Those are bonded, and niformly lose in he sense of oninos fnions ( ) ( ) ; ) ( ), here is small. From (5), (. (, ( )( ( )( ( ( ( )( ( )( ( )( y) dy ( )( y) dy I follows ha if φ φ and ψ ψ are niformly small, hen (, (, is small a eah, <. Ths he small hange in he iniial daa leads a small hange in he solion a any posiive ime. One epes his, sine he iniial vale problem is well-posed for he wave eqaion. Ths, he iniial vale problem (), () and (3) is well posed. DOI:.979/ Page III. Resl

9 Mahemaial Modelling Of he Solions of Parial Differenial Eqaion wih Physial Impliaions onsidering eqaion (5) obained ha is, U(, f ( f ( g s ds ( ) and he sandard wave eqaion; = for < <, > () (,) = f()for < < () (,) = g() for < < (3) Le solve some problems; 3. Problem () U 5U `,, f ( ) sin, o U, U,, U Solion omparing wih he sandard wave eqaion U U `,, f ( ), o g( ) U, U Therefore from he above qesion 5 5 f ( ) sin g ( ) Therefore, sing U, f f g ds ( ) e have U, sin sin ds U, sin sin U, sin 5 sin 5 () Reall ha sin A B A B B sina B sin os A () Therefore, ping () ino () yield U (, sin os Hene, sin os U 5 Table showing he vales of (, a varying and S/N (, DOI:.979/ Page

10 Mahemaial Modelling Of he Solions of Parial Differenial Eqaion wih Physial Impliaions Remark; A varying and as inreases (, also inrease nil maimm poin is aain (, hen sar dereasing. Table Showing resl of (, a fied and varying S/N (, ) Fied Varying Remark; From iniial poin a fied as inreases (, derease. Table3 Showing resl of (, a fied and varying S/N (, ) Varying Fied Remark; A fied as inreases (, also inreases. 3. Problem () U 9U `, sin o os U,, U, 3 Solion omparing wih he sandard wave eqaion U U `,, f ( ), o g( ) U, U here 9 3 f ( ) sin g( ) os3 Therefore, sing he formla U, f f g ( ) ds DOI:.979/ Page

11 Mahemaial Modelling Of he Solions of Parial Differenial Eqaion wih Physial Impliaions e have U, sin 3 sin 3 os3sds 3 B sin os3 U, sin 3 sin 3 Therefore sin os3 os3s 6 sin os3 6 3sin3s 3 sin os3 sin sin os3 sin3( 3 sin3( 3 8 sin os3 sin3 os 8 Hene U(, sin os3 sin3 os 9 Table 4 showing he vales of (, a varying and S/N (, Remark; A a poin when is a iniial, (, aain is highes poin, b when inrease o, (, hen dereases o is lowes poin. Then boh keep on inreasing nil maimm poin is aain, hen (, sar dereasing again. Table 5 Showing resl of (, a fied and varying S/N (, ) Varying Fied Remark; A fied as inreases (, also inreases Table 6 showing resl of (, a fied and varying DOI:.979/ Page

12 Mahemaial Modelling Of he Solions of Parial Differenial Eqaion wih Physial Impliaions S/N (, Fied Varying Remark; A fied as inreases (, also inreases nil, when is (, drasially rede o.77. IV. Disssion e have fosed on obaining solion, eamining is onsiseny and is well-posedness of he wave eqaion via hange of variable mehod. Using he mehod, he general solion of he wave eqaion was derived. Then, he general solion derived led s o he niqe solion of he problem, alled he d Alember s formla, we hen disssed he well-posedness of he solion and proved he niqeness, eisene and he dependeny of he solion on he iniial ondiions whih helped s o analyze and presen resl and showed he behavior of or resls in a ablar form. V. onlsion Finally, we onlde ha he noion of a well-posed problem is imporan in applied mahemais. Thogh he lassial heory of parial differenial eqaions deals almos ompleely wih he well-posed, illposed problems an be mahemaially and sienifially ineresing. VI. Reommendaion I s highly reommended ha, oher mehods for solving parial differenial eqaion sh as a. Mehod of separaion of variable, b. Laplae Transform mehod,. Nmerial mehod, e. Shold be arried o, his will enable readers o ompare any of he above mehod wih he mehod sed here. Referenes []. Benzoni-Gavage S. and Serre D. (7). Mlidimensional hyperboli parial differenial eqaions, Firs order sysems and appliaions. Oford Mahemaial Monographs. The larendon Press Oford Universiy Press, Oford. []. Broman A. (989). Inrodion o Parial Differenial Eqaions, Dover Pbliaions, In, New York. [3]. Go B. Z. and Zhang Z. X. (7). On he well-posed and reglariy of he wave eqaion wih variable oeffiiens. ESAIM: onrol, Opimizaion and alls of Variaions, 3: [4]. Go B. Z. and Zhang Z. X. (9). ell-posedness of sysems of linear elasiiy wih Dirihle bondary onrol and observaion. SIAM J. onrol Opim., 48: [5]. King A.. And Billingham J. (). ave moion. ambridge Tes in Applied Mahemais. ambridge Universiy Press, ambridge. [6]. Lawrene,. E. (). Parial differenial eqaions, volme 9 of Gradae Sdies in Mahemais. Amerian Mahemaial Soiey, Providene, RI, seond ediion. [7]. Marel B. F. (4). A Firs orse in Qasi-Linear Parial Differenial Eqaions for Physial Sienes and Engineering. Arkansas Teh Universiy. [8]. Merwin, K. (4). "Analysis of a Parial Differenial Eqaion and Real orld Appliaions Regarding aer Flow in he Sae of Florida," MNair Sholars Researh Jornal: Vol., Arile 8. [9]. Sajjadi, S. G. and Smih, T. A. (8). Applied Parial Differenial Eqaions. Boson, MA: Hffingon Mifflin Haror Pblishing ompany. []. Shiesser,. E. and Griffhs, G.. (9). A ompendim of Parial Differenial Eqaion Models: Mehod of Lines Analysis wih Malab, ambridge Universiy Press. []. Srass,. A. (8). Parial Differenial Eqaions, An Inrodion, John iley & Sons, seond ediion, Brown Universiy. Ben e al. "Mahemaial Modelling Of he Solions of Parial Differenial Eqaion wih Physial Impliaions. IOSR Jornal of Mahemais (IOSR-JM) 4.3 (8): DOI:.979/ Page