Fourier-Bessel Expansions with Arbitrary Radial Boundaries

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1 Applied Mthemtics,,, - doi:./m.. Pulished Online My ( Astct Fouie-Bessel Expnsions with Aity Rdil Boundies Muhmmd A. Mushef P. O. Box, Jeddh, Sudi Ai E-mil: mmushef@yhoo.co.uk Received Jnuy, ; evised Feuy, ; ccepted Feuy, Seies expnsion of single vile functions is epesented in Fouie-Bessel fom with unknown coefficients. The poposed seies expnsions e deived fo ity dil oundies in polems of cicul domin. Zeos of the geneted tnscendentl eqution nd the eltionship of othogonlity e employed to find the unknown coefficients. Sevel numeicl nd gphicl exmples e explined nd discussed. Keywods: Fouie-Bessel Anlysis, Boundy Vlue Polems, Othogonlity of Bessel Functions. Intoduction Sevel oundy vlue polems in the pplied sciences e fequently solved y expnsions in cylindicl hmonics with infinite tems. Polems of cicul domin with ounded sufces often genete infinite seies of Bessel functions of the fist nd second types with unknown coefficients. In this cse, the intention is to find the seies coefficients which should stisfy the oundy conditions. The suject of Fouie-Bessel seies expnsions ws investigted nd exmined in mny texts [-]. Nely ll of them hs deived cylindicl hmonics expnsions in J () fo the intevl [, ] only, whee J () is the Bessel function of the fist kind with ode zeo nd gument []. The existence of the oigin point excludes Y (), Bessel function of the second kind with ode zeo nd gument, ecuse it goes to negtive infinity s ppoches zeo []. Both J () nd Y () e shown plotted in Figue. In mny othe polems in the pplied sciences, the intevl of expnsion is found to e [, ] such tht, R. An exmple of this could e hollow cylinde in het conduction polems o cicul nd in vitions nlysis solved in the cylindicl coodinte system. In this cse, cylindicl hmonics expnsions in oth J () nd Y () e necessy. In this ppe, the deivtion of cylindicl hmonics expnsion of single vile function in [, ] in oth J () nd Y () is solved. In ccodnce with the oundies t = nd =, zeos of the otined tnscendentl eqution e fist clculted. As shown in Figue, the solution egion is fo whee the desied seies expnsions e foced to e zeo t = nd = espectively. Unknown coefficients e then found nd the complete seies expnsion cn e chieved. J() nd Y() Figue. Eqution (), J (), Y (). Solution Region Figue. The solution egion in dil oundies. Copyight SciRes.

2 M. A. MUSHREF. Fomultion nd Solution The Bessel diffeentil eqution of ode zeo is well known s [, ]: d d f ( ) + f ( ) + α f ( ) = () d d α nd R nd. The genel solution to Eqution () fo el vlues of αα is known to e [, ]: f ( ) = n= An J ( α ) + BnY ( α) () As in Eqution (), the ssumed oundy conditions t = nd = e of Diichlet type s f() = nd f() = espectively. Both A n nd B n e then elted s: Y ( α) An = B n () J ( α) A Y ( α) = () n B n J ( α) Going fte the elimintion method, the tnscendentl eqution cn e otined s: J α ) Y ( α) J ( α) Y ( α) () ( = In ode fo Eqution () to e stisfied, thee exist mny zeos o vlues of αα to e clculted. Thus, in ll fome nd coming equtions αα cn e eplced y αα n which e the zeos otined fom the tnscendentl eqution n I. Tht is: J α ) Y ( α ) J ( α ) Y ( α ) () ( n n n n = The othogonlity fetue of Bessel functions cn e pplied to Eqution () y multiplying oth sides y [ Am J( α m) + BmY ( α m) ] nd integting it ove ll possile vlues of fom to s: whee, n ( ) Cm ( ) d = n= C C C ( ) f ( ) d m ( m m m m n ( ) An J ( α n) + BnY ( α n) C m () ) = A J ( α ) + B Y ( α ) () = () The tems unde the summtion in the left side of Eqution () e zeos fo ll vlues of m n [,, ]. Hence, Eqution () cn e simplified to: n ) { Cn ( ) f ( ) } d = C ( () Eithe Eqution () o () cn help. Using Eqution () we cn otin the B n coefficients s: B n = S ( α ) f ( ) d n [ S ( α ) ] n d () whee, S (αα n ) is given y: Y ( αn) S( αn) = Y ( αn) J( αn) () J ( α ) By Eqution () o (), the A n coefficients cn lso e found. Once the coefficients A n nd B n e clculted, the function f() cn e expnded s in Eqution ().. Numeicl Exmples The tnscendentl expession in Eqution () shows gdul decy s αα inceses which men smll mgnitudes etween high zeos. This leds to the convegence of the seies in Eqution () ove s n inceses. As consequence, finite nume of tems in Eqution () cn e sufficient fo numeicl ppoximtions. The zeos e fist evluted using the tnscendentl coss poduct Bessel functions eqution fo the intevl [, ]. A gph of Eqution () is shown in Figue fo the solution egions [.,.] nd [., ]. Tle shows the fist zeos of Eqution () fo =. nd =.. Zeos otined fom the tnscendentl eqution chnges ccoding to the vlues of nd ssumed fo the solution egion. The dt pesented in Tle indictes tht the clculted zeos e not peiodic nd should e clculted using pope numeicl technique. Let s ssume tht the function f() to e expnded s in Eqution () is sin() with dil solution egion in [.,.]. The coefficients B n cn e evluted fom Eqution () nd the A n coefficients e then otined y Eqution (). Both coefficients e shown in Tles nd espectively fo n = to. J(αn)Y(αn) J(αn)Y(αn) Figue. Eqution (), [.,.], [., ]. α n Copyight SciRes.

3 M. A. MUSHREF Mny vitions cn e noticed fo the numeicl vlues of A n nd B n with genel solute scle of < except fo B =.. Some coefficients e in the ode of - mening tht thei ssocited tems e vey smll such s B nd A in Tles nd espectively. The function sin() nd its ppoximte expnsions e plotted in Figue. Summtion ove the fist tems poduced n cceptle estimtion in the intevl [.,.] with some ppent oscilltions ound the exct function. An impoved ppoximte expnsion is lso plotted fo n = to with less fluctutions in the sme dil domin. Tle. Fist fifty zeos of Eqution () in [.,.]. n α n n α n n α n n α n n α n Tle. Fist fifty B n fo f() = sin() in [.,.]. n B n n B n n B n n B n n B n.....e e e Tle. Fist fifty A n fo f() = sin() in [.,.] E Tle. Fist fifty B n fo f() = cos() in [.,.]. n B n n B n n B n n B n n B n.....e e In ddition, f() = cos() is expnded s in Eqution () nd the fist fifty coefficients e listed in Tles nd fo the B n nd A n espectively. Simil to the sin(), the cos() coefficients go though sevel vitions with genel solute scle of < except A =.. Also, only fou coefficients e in the ode of - implying tht thei elted tems in the seies e extemely smll such s B nd A in Tles nd espectively. Copyight SciRes.

4 M. A. MUSHREF Tle. Fist fifty A n fo f() = cos() in [.,.] E E sin() Figue. sin(), Eqution () with n = to, Eqution () with n = to. cos() with n solute level of > o <. The getest vlues in Tles nd e found s B =. nd A =.. In ddition, no coefficients e clculted in the ode of - implying tht ll coefficients e to e included in the seies expnsion. The function exp() nd its estimted expnsions e shown plotted in Figue in [.,.]. A stisfctoy estimtion of finite summtion ove the fist tems e geneted with sevel oscilltions close to the exct function. A good ppoximted expnsion is lso plotted fo n = to with fewe vitions in the sme solution egion. The lst numeicl exmple to e discussed is the sque function expessed s:.. f ( ) = () othewise The clculted B n nd A n coefficients fo this function e shown in Tles nd espectively. Simil to fome expnsions, oth coefficients vy out the exct vlues of Eqution (). The B n coefficients hve genel solute level of < except B, B, B, B nd B tht hve n solute scle of >. Futhemoe, the A n coefficients show n solute level of < except the solute vlues of A, A, A nd A tht e >. Some B n nd A n coefficients e clculted in the ode of - like A o Figue. cos(), Eqution () with n = to, Eqution () with n = to. The function cos() nd its estimted expnsions e shown plotted in Figue. Finite summtion ove the fist tems geneted stisfctoy estimtion in the intevl [.,.] with sevel ovious oscilltions close to the exct function. A ette ppoximte expnsion is lso plotted fo n = to with less fluctutions in the sme solution egion. The clculted coefficients fo the function e e lso shown in Tles nd fo B n nd A n espectively. Appently, the coefficients swing ound the exct vlues exp() Figue. exp(), Eqution () with n = to, Eqution () with n = to. Copyight SciRes.

5 M. A. MUSHREF Tle. Fist fifty B n fo f() = exp() in [.,.]. n B n N B n n B n n B n n B n Tle. Fist fifty A n fo f() = exp() in [.,.] Tle. Fist fifty B n fo Eqution () in [.,.]. n B n n B n n B n n B n n B n.... E E E-.... E-.. E-.. E E-. E E E-.. Tle. Fist fifty A n fo Eqution () in [.,.]. E E-.E E E E- E-. in the ode of - such s B indicting tht thei ssocited tems in the seies e vey smll. The function expessed y Eqution () nd its ppoximte expnsions e plotted in Figue. Summtion ove the fist tems poduced n cceptle estimtion in the intevl [.,.] with some noticele oscilltions ound the exct function. A ette ppoximte expnsion is lso plotted fo n = to with less fluctutions in the sme dil domin. In ll gphicl plots peviously shown, the cuves etun to zeo t the ssumed oundies =. nd =.. In ddition, ccucy of the expnded cuves my ppe ette s n inceses due to lge nume of tems involved in the seies nd less fluctutions seen ound the exct vlues.. Conclusions Functions wee expnded s Fouie-Bessel seies summtion in oth J () nd Y (). A finite seies expn- Copyight SciRes.

6 M. A. MUSHREF f() Figue. Eqution (), Eqution () with n = to, Eqution () with n = to. sion ws otined fo ity dil oundies in [, ]. Coefficients wee found y clculting the zeos of the tnscendentl eqution nd y employing the eltionship of othogonlity. A nume of exmples wee numeiclly nd gphiclly discussed.. Refeences tions, Cmidge Univesity Pess, United Kingdom,. [] H. F. Dvis, Fouie Seies nd Othogonl Functions, Dove Pulictions, Inc., New Yok,. [] W. E. Byely, An Elementy Tetise on Fouie s Seies, Dove Pulictions, Inc., New Yok,. [] F. Bowmn, Intoduction to Bessel Functions, Dove Pulictions, Inc., New Yok,. [] N. N. Leedev, Specil Functions nd Thei Applictions, Dove Pulictions, Inc., New Yok,. [] E. A. Gonzlez Velsco, Fouie Anlysis nd Boundy Vlue Polems, Acdemic Pess, Sn Diego,. [] A. Bomn, Intoduction to Ptil Diffeentil Equtions: Fom Fouie Seies to Boundy Vlue Polems, Dove Pulictions, Inc., New Yok,. [] P. V. O Neil, Advnced Engineeing Mthemtics, Wdswoth Pulishing Compny, Clifoni,. [] H. Sgn, Boundy nd Eigenvlue Polems in Mthemticl Physics, Dove Pulictions, Inc., New Yok,. [] N. N. Leedev, I. P. Sklsky nd Y. S. Uflynd, Woked Polems in Applied Mthemtics, Dove Pulictions, Inc., New Yok,. [] G. N. Wtson, A Tetise on the Theoy of Bessel Func- Copyight SciRes.