Int J Open Probles Copt Math, Vol 7, No 4, Deceber 214 ISSN 1998-6262; Copyrght ICSS Publcaton, 214 www-csrsorg Integral Transfors and Dual Integral Equatons to Solve Heat Equaton wth Mxed Condtons Naser Hoshan Departent of Matheatcs, Tafla Techncal Unversty, PO Box 179, Fax 22533, Tafla Jordan e-al: dr_naserh@yahooco eceved 28 Aprl 214; Accepted 2 October 214 Abstract The paper s devoted to deterne the soluton of the non-statonary heat equaton n a non axal syetrc cylndrcal coordnates subject to xed dscontnuous boundary condtons of the second knd and thrd knds, wth the ad of a fnte Fourer transfor and dual ntegral equatons ethod The soluton of the gven xed proble s ntroduced to a Fredhol ntegral equaton of the second knd Keywords: Integral transfors, dual ntegral equatons, xed boundary Condtons, heat equaton 1 Introducton Integral transfor ethod s wdely used to solve several probles n heat transfer theory wth dfferent coordnate systes for unxed boundary condtons [1,8] In onographs [3-6] Hankel and Laplace transfors were effectvely used to nvestgate exact solutons for Helholtz and heat equaton subject to xed boundary condtons of the frst,the second and of the thrd knds
Naser Hoshan 16 for cylnder In ths paper we propose the soluton of three-densonal nonstatonary heat equaton n a non axally syetrcal cylndrcal coordnates wth dscontnuous xed boundary condtons of the second and thrd knd on the level surface of a se-nfnte sold cylnder Exact soluton of the gven xed boundary value proble s obtaned wth the use of fnte Fourer, Hankel ntegral transfors separaton of varables and based on the applcaton of dual ntegral equatons ethod In ths paper we apply a fnte Fourer ntegral transfor and then Hankel ntegral transfors wth respect to coordnate varables and r, oreover, an ntal xed boundary value proble s transfored to a Helholtz boundary proble n cylndrcal coordnates, next,applcaton of xed boundary condtons yelds new for of a dual ntegral equaton wth Bessel functon of the frst knd of order n as a kernel, weght and unknown functons depend on paraeters The soluton of the obtaned dual ntegral equatons s ntroduced to a Fredhol ntegral equaton of the second knd wth kernel and free ter gven n for of proper ntegrals 2 Forulaton of the Proble The an goal n ths paper s to solve the non-statonary heat equaton for se-space n cylndrcal coordnates wth a non-axally syetry (21) 1 1 1 2 r r r r z a T T T T T where T T ( r, z,, ) s the teperature dstrbuton functon, r, z, 2, are the correspondng cylndrcal coordnates a s the teperature dffusvty coeffcent (constant) The boundary condtons (22) T (, z,, ) T (, z,, ) T ( r,,, ), T () T (2 ) Subject to xed dscontnuous boundary condtons of the second knd and of the thrd knd (23) T ( r,,, ) / z f 1( r,, ), r S (2,4) T ( r,,, ) / z T ( r,,, ) f 2( r,, ), r S where S (, ), S (, ), on a surface z,, constants The ntal condton s (2,5) T ( r, z,,) ( r, z,,) U
17 Integral Transfors and Dual Integral Where U s the ntal teperature (constant), the known functons f, 1,2 n (23),(24) contnuous and have the lted varaton wth respect of each varables r and,wth respect to r, and perodc wth perod 2 The physcal sgnfcance of the gven xed boundary value proble forulated such that, fnd the teperature dstrbuton functon T for a senfnte cylnder f the nsde dsk S a heat flux s gven accordng to Fourer low, whereas on the outsde dsk S a heat exchange obey Newton's low of coolng, on the lne of dscontnuty r no boundary condtons were gven As r z the teperature s vanshed; T 2 2 3 Soluton of the Proble Use a fnte Fourer transfor for a functon T wth respect to, to the ntal boundary value proble (21)-(25), we have T 2 TF d, K() K(2 ), Where cos, j 2 Fj sn, j 2 1 The nverse Fourer ntegral transfor s gven by the forula 1 T T ( T 2 cos T 21 sn ) 2 1 Where 2 T T () Td, 2 2 1 21 j j 2 T T (2 ) T F d T T (2 1) T F d 2 2 j Equaton (21) n a Fourer transfor range s T 1 T T 1 T (31) T r r r r z a To splfy the proble entoned above,use well known transforaton [1] (32) T ( r, z,, t ) exp( ) u( r, z, ) for (21)-(25)after applcaton Fourer transfor, s constant, we obtan a Helholtz equaton n cylndrcal coordnates u 1 u u (33) u u r r r r z a Separatng varables n (33), the general soluton of the Helholtz boundary value proble s obtaned n for of proper ntegral
Naser Hoshan 18 (34) u ( r, z, ) A ( p, )exp( h( p, )) J ( pr ) dp 2 where h( p, ) p / a, J ( pr ) s the Bessel functon of the frst knd of order, p s the paraeter of separaton of varables, A( p, ) unknown functon Applyng a xed boundary condtons (23) and (24) to (34), we obtan the dual ntegral equatons to deterne the unknown functon A( p, ) (35) A( p, ) J ( pr) h( p, ) dp f 1( r, ), r S (36) A( p, ) J ( pr )( h( p, ) ) dp f 2( r, ), r S 2 (, ) f (, ) j f r e r F d To solve the dual equatons (35),(36), rewrte the equatons n the standard for (37) B ( p, ) J ( pr) g ( p, ) dp f 1( r, ), r S (38) B ( p, ) J ( pr ) dp f 2( r, ), r S where h( p, ) w ( p, ), l w ( p, ) 1/, h( p, ) p B ( p, ) ( h( p, ) ) A( p, ), g ( p, ) w ( p, ) 1/ h( p, ) ewrte (38) n for (39) ( r, ), r S B ( p, ) J ( pr ) dp f 2( r, ), r S where ( r, ) s unknown contnuous functon defned outsde the dsk n S Applyng to (39) the nverse Hankel ntegral transfor [7] n the nterval S S we have (31) B ( p, ) ypj ( py ) ( y, ) dy ypj ( py ) f 2( y, ) dy Substtutng (31) nto (37), then nterchangng the order of ntegraton, we get a Fredhol ntegral equaton of the second knd for deternaton the unknown functon ( r, ) (311) ( r, ) ( y, ) K ( r, y, ) dy F ( r, ), r S
19 Integral Transfors and Dual Integral wth kernel (312) and free ter K ( r, y, ) pyj ( py ) J ( pr ) g ( p, ) dp (313) F ( r, ) f 2( r, ) p y f 2( y, ) J ( pr) J ( py ) g ( p, ) dp dy Integral equaton (311) should be solved wth the use of nuercal ethods for soe choces of f 1, f 2,,,,1,2, by usng soe software packages such atheatca or atlab The kernel gven n (312) contnuous and quadratc ntegrable n the square : r, y, for soe certan nuercal values of furtherore, the free ter (313) s ntegrable and bounded n the nterval y [2]Fnally the general soluton T ( r, z,, ) n the Fourer transfor doan s gven by the expresson 1 T ( r, z,, ) e h( p, ) (314) ypj ( py ) ( y, ) dy ypj ( py ) f 2( y, ) dy h( p, ) J ( pr) dp Put the value of the general soluton (314) nto the nverson forula of the nverse Fourer transfor, the general soluton of the ntal xed boundary value proble (21)-(25) T ( r, z,, ) (315) 1 T ( r, z,,) ( T ( r, z,,2 )cos T ( r, z,,2 1)sn ) 2 1 If, the soluton (315) reduced to the soluton of an axal syetry heat equaton wth xed condtons[6],oreover, f h( p, ) p, the above soluton s reduced to the soluton of the Laplace's' equaton wth xed condtons n dfferent engneerng and physcal applcatons[3-7] Theory entoned above of the applcatons of a ntegral transfors nvolvng exact soluton of the xed ntal boundary value proble can be used wdely to solve varous xed boundary probles for a non-statonary heat equaton n an nfnte or fnte cylnder, unsyetrcal cylndrcal coordnates for unbounded plate, sphercal coordnates and other xed probles 4 Concluson Fnally the above technque nvolvng applcaton of ntegral transfors for solvng xed boundary value probles can be used for nvestgatng several
Naser Hoshan 2 hoogeneous probles (heat equaton, Helholtz equaton and Laplace equaton) n dfferent coordnate syste and any areas of applcatons n techncal and physcal scences under xed boundary condtons of the frst, the second and of the thrd knds 5 Open Proble We consder an ntal xed boundary value proble (21) (22) for an nfnte plate of hgh h, n cylndrcal coordnates wth non axally syetry, subject to dscontnuous nhoogeneous xed boundary condtons of the second and of the thrd knd on a surface z (316) T ( r,,, ) / z f 1( r,, ), r S, (317) T ( r,,, ) / z T ( r,,, ) f 2( r,, ), r S On a level surface boundary condtons (318) T( r, h,, ) / z T ( r, h,, ) f3( r,, ) z h, located a thrd knd lnear nhoogeneous Where f, 1, 2,3 known functons, S (, ), S (, ),,,,, constants The above xed proble (316)-(318), ntroduced to soe type of dual ntegral equatons, however no one n the world solve ths proble, snce the boundary condton (318) coplcates soluton of the gven proble furtherore, known ethods concernng dual ntegral equatons ay be dffcult to use eferences [1] A Galtsyn, A Zhukovsk, Integral Transfors and Specal Functons n Heat Probles, Kev, Duka1976 [2] W Hackbusch, Integral Equaton, Theory and Nuercal Treatent Brkhäuser Verlag, Boston,1995 [3] N Hoshan, Dual Integral Equatons and Sngular Integral Equatons for Helholtz Equaton Internatonal Journal of Contep Math Scences, 29, V4, No 34,1695-1699,Hkar Ltd [4] N Hoshan, Integral Transfor Method n Soe Mxed Probles Internatonal Journal of Matheatcal Foru, 29, V4, No 4, 1977-198, Hkar Ltd
21 Integral Transfors and Dual Integral [5] N Hoshan, The dual ntegral equatons ethod nvolvng heat equaton wth xed boundary condtons, Engneerng Matheatcs Letters, No2 213, pp 137-142 [6] N Hoshan, The Dual Integral Equatons Method for Solvng Hlholtz Mxed Boundary Value Proble, Aercan Journal of Coputatonal and Appled Matheatcs, 213, 3(2), pp 138-142 [7] B Mandal, N Mandal, Advances n Dual Integral Equaton, London, CC1999 [8] M Ozsk, Heat Conducton, Wley & Sons NewYork, 22