Integral Transforms and Dual Integral Equations to Solve Heat Equation with Mixed Conditions

Transcription

1 Int J Open Probles Copt Math, Vol 7, No 4, Deceber 214 ISSN ; 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

2 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) 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

3 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 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, 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

4 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

5 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

6 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, ,Hkar Ltd [4] N Hoshan, Integral Transfor Method n Soe Mxed Probles Internatonal Journal of Matheatcal Foru, 29, V4, No 4, , Hkar Ltd

7 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 [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 [7] B Mandal, N Mandal, Advances n Dual Integral Equaton, London, CC1999 [8] M Ozsk, Heat Conducton, Wley & Sons NewYork, 22