Research Article Mixed Initial-Boundary Value Problem for Telegraph Equation in Domain with Variable Borders

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1 Advnces in Mthemticl Physics Volume 212, Article ID 83112, 17 pges doi:1.1155/212/83112 Reserch Article Mixed Initil-Boundry Vlue Problem for Telegrph Eqution in Domin with Vrible Borders V. A. Ostpenko Dnepropetrovsk Ntionl University, Fculty of Mechnics nd Mthemtics, Pr. Ggrin 72, Dnepropetrovsk 491, Ukrine Correspondence should be ddressed to V. A. Ostpenko, Received 13 Mrch 212; Revised 6 April 212; Accepted 11 April 212 Acdemic Editor: Burk Polt Copyright q 212 V. A. Ostpenko. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Mixed initil-boundry vlue problem for telegrph eqution in domin with vrible borders is considered. On one prt of domin s border re the boundry conditions of the first type, on other prt of the boundry re set boundry conditions of the second type. Besides, the sies of re re vrible. The solution of such problem demnds development of specil methods. With the help of consecutive ppliction of procedure of construction wves reflected from borders of domin, it is possible to obtin the solution of this problem in qudrtures. In ddition, for construction of the wves reflected from mobile border, it is necessry to pply the procedure specilly developed for these purposes. 1. Introduction Mixed initil-boundry vlue problems for telegrph eqution in domin with vrible borders rise in mny pplictions. In prticulr, such problem rises in problem bout clcultion of field of stress in ropes of elevting devices. At lifting of lod the rope reels up on drum or reels off drum t lowering lod. Therefore, the length of tht prt of rope, which is reeled up on drum, chnges. If to tke into ccount friction of rope on drum, elstic displcements in rope cn be described by the telegrph eqution 1. In 1 it is shown, tht it occurs t dry friction. However, it is possible to show, tht the telegrph eqution describes elstic displcements nd t viscous friction s well. Thus, there is initilboundry vlue problem for the telegrph eqution in domin with vrible border. Here the telegrph eqution of generl view is considered. Regrding rope which hngs down from drum, elstic displcements there re described by the wve eqution. This circumstnce

2 2 Advnces in Mthemticl Physics induces to consider seprtely problems bout elstic displcements to vrious prts of rope. Initil-boundry vlue problem on elstic displcements to tht prt of rope which is reeled on drum is considered in the present pper. The rope is considered s flexible string. One end of rope is ttched to drum nd goes together with drum. The extreme point of contct of rope with drum is ccepted s the second end of rope. Elstic stresses re set in this point. Chnge of length of rope, which is reeled up on drum, is tken into ccount s follows. Portble movement of system is understood s rottion of drum nd rope s perfectly rigid body. Then the reltive movement of rope will be submitted s motionless in ll points of rope, except for n extreme point of contct of rope with drum. This lst point in reltive movement will mke the moving equl v t, where v t is movment of the centrl xis of rope together with drum. In reltive movement ll points of rope mke only elstic displcements. The xis x is directed lengthwys towrd conditionlly strightened rope nd its beginning is locted in point of ttching of rope to drum. The initil length of rope, which is reeled up on drum, is equl to l. At the solution of initil-boundry vlue problems for the telegrph eqution the vrious exct nd pproched methods were used. It is necessry to notice tht exct solutions were obtined only for the limited number of boundry problems. Recently for the solution of boundry problems even for frctionl telegrph equtions it is ctively used differentil trnsformtion method 2 4. This method is n improved version of power series method or its modifictions. In this cse the sme is; nd in power series method, representtion of the solution of problem in the form of type Tylor s series in neighborhood of some point or curve is used. Thnks to the developed procedure of use t n erly stge of clcultions of boundry conditions, clcultion of expnsion coefficients becomes essentilly simpler. However representtion of the solution in the form of Tylor s series type demnds fulfillment of dditionl conditions. Following conditions concern them. It is necessry tht ll fctors in the differentil eqution were nlyticl functions of the rguments. It is necessry lso tht the series obtined s the solution of the problem nd especilly series of derivtives converged uniformly. But the most importnt feture consists s such expnsion of solutions provides good pproch only in some neighborhood of n index point or curve. On the essentil distnce from such neighborhood the vlues of higher degree terms become dominting. Therefore preservtion in expnsion of finl number of the terms cn led to considerble errors. Besides, t gret distnce from re of initil or boundry vlues series cn pper divergent. Therefore differentil trnsformtion method yields stisfctory results only on smll intervls of chnge of sptil vribles nd time. Besides, for ppliction of this method it is necessry tht ll functions included in the eqution nd boundry conditions supposed the sme expnsion tht is used for solution representtion. This condition cnnot be executed in generl cse. Necessity to be limited to clcultion of finl number of terms of series leds to tht the solution ppers pproximte. At the sme time, if ll given problems cn be presented in the form of finite series, the solution of such problem cn be obtined exctly. For the telegrph equtions, including frctionl ones, it is possible to solve some initil-boundry vlue problems by mens of method of seprtion vribles 5. For ppliction of this method it is necessry tht the domin of serch of the solution possessed specil symmetry, nd initil nd boundry functions s well s the right prts of the equtions, supposed expnsion on eigen functions of boundry problem. It is cler tht these conditions cn be executed not lwys.

3 Advnces in Mthemticl Physics 3 Existing methods of the solution of initil-boundry vlue problems cnnot be pplied to the telegrph eqution in cses when the domin in which the solution is found is vrible. In prticulr, use for this purpose of method seprtion of vribles is impossible, s in cse of mobile borders the corresponding problem of Sturm-Liouville hs only trivil eigenfunctions. In present pper the method especilly developed in 6 12 for solution of problems bout movement of wves in domin with mobile borders is used. 2. Sttement of the Problem The following initil-boundry vlue problem is considered: in the domin <x<l v t, t> to obtin the solution of the telegrph eqution 2 u x, t x u x, t t 2 u x, t u x, t D B Cu x, t, t x 2.1 which stisfies initil conditions u x, ; u t x, 2.2 nd mixed boundry conditions u x l v t,t γ t ; u,t, t >. 2.3 Concerning function v t, describing displcement of the bottom end of rope, it is supposed v nd from condition of preservtion of integrtion re of n initil-boundry vlue problem follows tht v t > l t t>. The solution of the problem put here cnnot be crried out by existing methods becuse domin, in which the solution is found, is vrible. Therefore for the solution of such initil-boundry problems connected with the equtions of hyperbolic type in 6 12 specil method developed. This method hs three prominent fetures. The first of them consists in integrted representtion of solutions of the telegrph eqution in the form of extending wves for n extensive clss of boundry conditions 6. Such representtion is obtined by use of Riemnn s method. Use of the given integrted representtion of solutions demnds performnce of continution of initil nd regionl functions in the domin of ny vlues of their rguments. This continution should be crried out tking into ccount ll conditions of sttement of problem. In it the second feture of method consists. At lst, the third feture consists in working out of method of construction of the wves reflected from mobile border. The given method reduces reflexing problem to the solution of n uxiliry initil-boundry vlue problem with the initil conditions set t the moment of rrivl of forwrd front of the flling wve on mobile border. This method is used for the solution of stted problem.

4 4 Advnces in Mthemticl Physics 3. The Solution of the Problem For the solution of this problem the continution of function γ t on ll xis t is introduced s γ t, t > ; Γ t, t <, 3.1 nd the first boundry condition 2.3 is considered s continued on ll xis t: u x l v t,t Γ t. 3.2 At the first stge the solution of the given problem is serched s t x/ u x, t 2e B/2 x J e D2 /2 t η ( Γ η dη, 3.3 with unknown function Γ. Here J, J 1 -Bessel s functions s the ero nd first order, respectively, c 1 [ x 2 2( t η 2 ] ; c 1 C D2 2 4 B In 1 it is shown tht function 3.3 stisfies 2.1 t rbitrry function Γ. In the sme plce it is shown tht for solution of boundry problems with boundry conditions of the second type it is the most expedient to pply the form of the solution of kind 3.3. Hving substituted the form of the solution 3.3 in boundry condition 3.2 we obtin ( 2 e B D /2 l v t Γ t l v t t l v t / [ 3.5 B 2e B/2 l v t 2 J c 1 l v t J 1 ]e D2 /2 t η ( Γ η dη Γ t. In 3.5 c 1 [ v t l 2 2( t η 2 ]. 3.6 Thus, if function Γ is the solution of the integrl eqution 3.5, function 3.3 will stisfy the first boundry condition 2.3.

5 Advnces in Mthemticl Physics 5 With the purpose of obtining n opportunity to represent function Γ with vrious rguments we shll introduce into 3.5 such trnsformtion of vrible t: τ t v t l. 3.7 In order tht execution of trnsformtion 3.7 in the integrl eqution 3.5 ws possible, it is necessry tht the function t inverse to τ exists. Here the cse is considered when the mobile end moves with subsonic speed. Tht mens the next condition is stisfied v t <. 3.8 Then from 3.7 follows dτ dt 1 v t >. 3.9 It mens tht 3.7 will be strictly monotonously growing nd consequently there will be n inverse to them, function t, lso strictly monotonously growing. Thus s τ l/, we obtin tht t l/. From 3.9 follows tht τ > l/ t t >. As function v t is determined only t t>, function τ t is determined lso only t t>. Accordingly function t τ will be determined only t τ>l/. At the sme time during construction of the solution of considered initil-boundry vlue problem there is necessity of knowledge of function Γ τ behvior s well t vlues of rgument Γ τ, smller thn l/. With this purpose it is necessry to execute continution of function v t on ll xis t. It ppers tht continution of function v t on ll xes t cn be executed by rbitrry wy, hving demnded only existence of derivtive of this continution on ll xes t nd performnce on ll xes t condition 3.8. We shll designte this continution of function v t through v 1 t. Then on ll xes t such function will be determined: v t, t > ; N t v 1 t, t <. 3.1 Continued on ll xes t of function τ t, we shll designte T t nd we shll determine it by expression T t t N t l From this expression nd 3.1 it is cler tht t t>, T t τ t. As function N t stisfies to n inequlity N t < 3.12 t ll t, function T t will be strictly monotonously growing nd s τ l/, tt< it will be vlid such inequlity: T t <l/.

6 6 Advnces in Mthemticl Physics As function T t is strictly monotonous t ll t, there exists inverse to this the function T T, ndtt l/, T T T τ nd T T will be strictly monotonously growing function. Thus, function T T stisfies condition t τ >, T > l ; T T, T l ; 3.13 <, T < l. Now fter trnsformtion 3.11 integrl eqution 3.5 will become 1 e B D /2 l N T T Γ T T [ e B/2 l N T T B 2 J c 1 l N T T J ] 1 e D2 /2 t η ( 1 Γ η dη 2 Γ t In integrl eqution 3.14 becomes c 1 [ l N T T 2 2( T T η 2 ] From the integrl eqution 3.14, the qulity 3.1 of function Γ t nd n equlity 3.13 follow, tht is Γ T, T < l In turn, from the qulity 3.16 of function Γ T follows 3.3 stisfing initil conditions 2.2. Relly, from the formul 3.3 directly follows tht t t upper limit of integrtion becomes equl to x/. But t t,x <lnd on the bsis of the qulity 3.16 of function Γ T follows tht t t function 3.3 will be equl to ero. Tht mens it stisfies the first initil condition 2.2. Hving differentited function 3.3 on t, we shll obtin u x, t t 2 ( /2 x e B D2 Γ t x t x/ [ 3.17 D 2 2 e B/2 x 2 J ( J 1 c 1 t η ]e D2 /2 t η ( Γ η dη. If in the formul 3.17 we set t, then rgument of function Γ nd lso the upper limit of integrtion t x<lbecome smller thn l/. Therefore on the bsis of function s Γ

7 Advnces in Mthemticl Physics 7 qulity 3.16 the derivtive 3.17 t t will equl to ero. And it mens tht function 3.3 will stisfy lso the second initil condition 2.2. Thus, function 3.3 stisfies ll conditions of sttement of n initil-boundry vlue problem, except for the second boundry condition 2.3. With the purpose to check this condition we shll clculte from 3.3 u,t 2 t J e D2 /2 t η Γ ( η dη From the formul 3.18 on the bsis of function s Γ qulity 3.16 follows the function 3.3 t t < l/ stisfing the second boundry condition 2.3 s well. With the purpose of stisfction of the second boundry condition 2.3 t t>l/, the solution of n initilboundry vlue problem we hve to serch s the sum of two functions is u x, t u x, t u 1 x, t, 3.19 where t x/ u 1 x, t 2e B/2 x J e D2 /2 t η ( Γ η dη. 3.2 Function 3.2 stisfies 2.1 with rbitrry function Γ. It is obvious the function 3.19 stisfies the second boundry condition 2.3 t ll t. In the sme wy s for function u x, t it is possible to check up tht the function 3.19 stisfies initil conditions 2.2. In order tht 3.19 stisfies the first boundry condition 2.3 it is necessry the function u 1 x, t stisfies boundry condition u 1,x l v t,t, t > Hving clculted vlue of function u 1,x x, t t point x l v t,weobtin 2 e B D /2 l v t Γ ( t l v t t l v t / [ 3.22 B 2e B/2 l v t 2 J c 1 l v t J 1 ]e D2 /2 t η ( Γ η dη. In formul 3.22 c 1 [ v t l 2 2( t η 2 ] In order tht equlity 3.22 ws vlid, on the bsis of function s Γ qulity 3.16, itis necessry tht the rgument of this function in equlity 3.22 stisfies condition t l v t < l. 3.24

8 8 Advnces in Mthemticl Physics Thus, function 3.19 will be the solution of problem t t< 2l v t /.Att> 2l v t / for stisfction of first boundry condition 2.3 in the solution 3.19 it is necessry to enter the mendment. To enter such mendment by the methods used for domins with motionless borders, s shown in 7 12 for cse of the wve eqution, is impossible. Therefore for corrective ction the pproch developed in 7 12 is used. With this purpose we shll notice tht during n intervl of time determined by condition t v t 2l /, forwrd front of the wve, rdited on the mobile end, strting from the moment of time t, will rech the end x, will be reflected from it, nd will meet the mobile end. The length of this intervl of time will be determined s the less positive root τ 1 of eqution t v t 2l Theleftprtof 3.25 t t is less thn the right prt. At the sme time on the bsis of condition 3.8 t t> left prt of this eqution grows fster thn right prt. Hence, the positive root of 3.25 exists. From the crried out resoning it is cler lso tht the condition 3.22 will be vlid t t<τ 1. Relly, t t inequlity t< v t 2l / is vlid, s v. At the sme time τ 1 is the less positive number t which this inequlity turns into equlity Therefore correction function u 2 x, t, being in essence of wve reflected from the mobile end, is under construction s the solution of such n uxiliry initil-boundry vlue problem: in the domin <x>l v t, t>τ 1 to obtin the solution of the telegrph eqution 2.1 stisfying initil conditions u x, τ 1 ; u t x, τ 1 ; <x<l v τ 1, 3.26 nd boundry conditions u x l v t,t u 1,x l v t,t ; u,t, t > τ The solution of this uxiliry initil-boundry vlue problem is constructed s function stisfying 2.1 t rbitrry function Γ 2 : t x/ u 2 x, t 2e B/2 x J e D2 /2 t η ( Γ 2 η dη Hving substituted function 3.28 in the first boundry condition 3.27, weobtin ( 2 Γ 2 t v t l e D B /2 l v t t v t l / [ B 2e B/2 l v t u 1,x l v t,t. 2 J l v t c 1 ] J 1 e D2 /2 t η ( Γ 2 η dη 3.29

9 Advnces in Mthemticl Physics 9 Thus, if function Γ 2 will be the solution of the integrl eqution 3.29, the function u x, t u x, t u 1 x, t u 2 x, t, 3.3 will stisfy the second boundry condition 2.3 t ll t. Hving executed in 3.29 trnsformtion 3.11, we shll obtin 2Γ 2 T e D B /2 l v T T T [ 2e B/2 l v T T B u 1,x l v T T,T T. 2 J l v T T c 1 ] J 1 e D2 /2 T T η ( Γ 2 η dη 3.31 As the right prt of the integrl equtions 3.29 nd 3.31 is equl to ero t t<τ 1, function Γ 2 T lso will be equl to ero t t<τ 1. We shll find out, which dditionl vlues of T function Γ 2 T will be equl to ero. As function T t strictly monotonously grows, such inequlity will be vlid T t <T τ 1, t t<τ Using in this inequlity the definition 3.11 of function T t nd vlue τ 1 from 3.25, we shll obtin T<τ 1 v τ 1 l 3l 2v τ Thus, function Γ 2 T possesses the following qulity: Γ 2 T, T < 3l 2v τ If now to put in equlity 3.28 t, the upper limit of integrtion in this formul will ccept vlue x/. Tking into ccount tht t t it is vlid x<l, we shll hve tht x/ < l/.but s l>v τ 1, we shll obtin tht l < 3l 2v τ It mens tht t t function 3.28 will equl to ero, tht is, will stisfy the first initil condition 2.2. Hving clculted derivtive of function 3.28 on t, we shll obtin u 2 x, t t 2 ( /2 x e B D2 Γ 2 t x t x/ [ 3.36 D 2 2 e B/2 x 2 J ( J 1 c 1 t η ]e D2 /2 t η ( Γ 2 η dη.

10 1 Advnces in Mthemticl Physics At t rgument of function Γ 2 nd lso the upper limit of integrtion in the formul 3.36 will ccept vlue x/. Hence, s shown bove, in the domin of serch of the solution t such vlues of rgument, the function Γ 2 will be equl to ero. Therefore, the derivtive of function u 2 x, t on t t t will be equl to ero. And it mens tht function u 2 x, t stisfies s well the second initil condition 2.2. Thus, function 3.3 stisfies ll conditions of sttement of the bsic initil-boundry vlue problem, except for the second boundry condition 2.3. In order tht this boundry condition ws crried out, it is necessry tht there ws vlid n equlity u 2,x,t, t > Hving substituted function 3.28 in the left prt of boundry condition 3.37, weshll obtin u 2,x,t 2 t Γ B 2 t 2 2 J e D2 /2 t η ( Γ 2 η dη As follows from function s Γ 2 qulity 3.34, expression 3.38 will equl ero, tht is, will stisfy boundry condition 3.37 only t vlidity of n inequlity t< 3l 2v τ 1 /. For stisfction of the second boundry condition t big t in the solution 3.3, the mendment u 3 x, t is entered tht is, the solution is represented s u x, t u x, t u 1 x, t u 2 x, t u 3 x, t, 3.39 where t x/ u 3 x, t 2e B/2 x J e D2 /2 t η ( Γ 2 η dη. 3.4 Function u 3 x, t stisfies 2.1 t rbitrry function Γ 2 nd should provide performnce of boundry condition u 2,t u 3,t, t > Fct tht functions 3.28 nd 3.4 stisfy boundry condition 3.41 is prcticlly obvious. With the sme wy, s it is mde for function u 2 x, t, it is possible to show tht function u 3 x, t will stisfy initil conditions 2.2. Thus, function 3.39 stisfies ll conditions of sttement of the bsic initil-boundry vlue problem, except for the first boundry condition 2.3. In order tht this boundry condition ws crried out, it is necessry tht there ws vlid n equlity u 3,x l v t,t, t >. 3.42

11 Advnces in Mthemticl Physics 11 Hving clculted vlue of derivtive function 3.4 on x in point x l v t, we shll obtin u 3,x l v t,t 2 e D B /2 l v t Γ 2 ( t v t l t v t l / [ B 2e B/2 l v t 2 J l v t c 1 ] J 1 e D2 /2 t η ( Γ 2 η dη On the bsis of function s Γ 2 qulity 3.34 one cn conclude tht the right prt of equlity 3.43 will be equl to ero if rgument of function Γ 2 nd the upper limit of integrtion in the formul 3.43 will stisfy n inequlity whence follows t l v t < 3l 2v τ 1, 3.44 t< 4l 2v τ 1 v t Hence, s t stisfies n inequlity 3.45, the boundry condition 3.42 will be crried out. The inequlity 3.45 is inconvenient for use s its left nd right prts depend on t. Withthe purpose of more convenient use of this inequlity we shll consider the eqution t 4l 2v τ 1 v t Also we shll designte s τ 2 the less positive root of this eqution. At t right prt of 3.46 is more thn the left prt. At the sme time by virtue of condition 3.8 right prt of 3.46 grows fster thn its left prt; therefore, the positive root of 3.46 exists. Hence, number τ 2 is the less positive number t which the inequlity 3.45 terns into equlity. Therefore the inequlity 3.45 is equivlent to the inequlity t< 4l 2v τ 1 v τ 2 τ Let us notice tht on physicl sense of n initil-boundry vlue problem t the moment of time t τ 2 forwrd front of the wve rdited with the mobile end, hving twice reflected from the motionless end nd hving once reflected from the mobile end, it will meet gin the mobile end. At t>τ 2 function u 3 x, t 3.4 will not stisfy ny more to boundry condition Therefore t t>τ 2 solution of the bsic initil-boundry vlue problem is serched s u x, t u x, t u 1 x, t u 2 x, t u 3 x, t u 4 x, t, 3.48

12 12 Advnces in Mthemticl Physics where function u 4 x, t is under construction s the solution of the following uxiliry initilboundry vlue problem. In the domin <x<l v t, t>τ 2 to obtin the solution of the telegrph eqution 2.1 stisfying initil conditions u x, τ 2 ; u t x, τ 2 ; <x<l v τ 2, 3.49 nd boundry conditions u x l v t,t u 3,x l v t,t ; u x,t, t > τ The solution of this uxiliry initil-boundry vlue problem is under construction s the function being the solution of 2.1 t rbitrry function Γ 4 is t x/ u 4 x, t 2e B/2 x J e D2 /2 t η ( Γ 4 η dη Hving substituted function 3.51 in the first boundry condition 3.5, we shll obtin ( 2 Γ 4 t v t l e D B /2 l v t t v t l / [ B 2e B/2 l v t u 3,x l v t,t. 2 J l v t c 1 ] J 1 e D2 /2 t η ( Γ 4 η dη 3.52 Thus, if function Γ 4 will be the solution of the integrl eqution 3.52 the function 3.48 will stisfy the second boundry condition 2.3 t ll t. Hving executed in 3.52 trnsformtion 3.11, we shll obtin 2Γ 4 T e D B /2 l v T T T [ 2e B/2 l v T T B u 3,x l v T T,T T. 2 J l v T T c 1 ] J 1 e D2 /2 T T η ( Γ 4 η dη 3.53 As the right prt of the integrl equtions 3.52 nd 3.53 is equl to ero t t<τ 2, function Γ 4 T lso will be equl to ero t t<τ 2. We shll find out t wht dditionl vlues T function Γ 4 T will be equl to ero. As function T t strictly monotonously grows, the next inequlity will be vlid T t <T τ 2, t t<τ

13 Advnces in Mthemticl Physics 13 Using in this inequlity definition 3.11 of function T t nd vlue τ 2 from 3.52, we shll obtin T<τ 2 v τ 2 l 5l 2v τ 1 2v τ Thus, function Γ 4 T possesses the following qulity: Γ 4 T, T < 5l 2v τ 1 2v τ Precisely the sme s it is mde for function u 2 x, t, it is possible to show tht function u 4 x, t will stisfy initil conditions 2.2. Thus, function 3.48 stisfies ll conditions of sttement of the bsic initil-boundry vlue problem, except for the second boundry condition 2.3. This boundry condition function u 4 x, t will stisfy only t the some of vlues of t > τ 2. To obtin the solution of the bsic initil-boundry vlue problem t ll t>τ 2, 3.48 it is necessry to introduce the dditionl mendment into function Hving continued process of corrective ctions in the solution, we shll obtin tht function t x/ u x, t 2e B/2 x J e D2 /2 t η ( Γ 2n η dη n t x/ 2e B/2 x J e D2 /2 t η ( Γ 2n η dη n 3.57 will be the solution of considered initil-boundry vlue problem. Here function Γ is the solution of the integrl eqution 3.14, nd other functions Γ 2n re solutions of the following integrl equtions: 2Γ 2n T e D B /2 l v T T T [ 2e B/2 l v T T B u 2n 1,x l v T T,T T. 2 J l v T T c 1 ] J 1 e D2 /2 T T η ( Γ 2n η dη 3.58 Here u 2n 1,x l v t,t 2 e D B /2 l v t Γ 2n 2 ( t t v t l / [ B 2e B/2 l v t v t l 2 J l v t c 1 ] J 1 e D2 /2 t η ( Γ 2n 2 η dη. 3.59

14 14 Advnces in Mthemticl Physics Thus functions Γ 2n possess the following qulities: Γ 2n t, t < 2n 1 l 2 n i 1 v τ i,n, 1,..., 3.6 where τ n is less positive root of eqution t 2nl 2 n 1 i 1 v τ i v t By virtue of these qulities, t everyone fixed t H in the formul 3.57 will be only finl number of terms distinct from ero. Relly, in the sums of the formul 3.57 ech term under conditions 3.56 becomes equl to ero, if the upper limit of integrtion is less thn the right prt of n inequlity 3.6. For the first sum of the formul 3.57 such condition t t H looks like H x < 2n 1 l 2 n i 1 v τ i, 3.62 whence follows n> 1 ( n H x l 2 v τ i l i 1 And s in the domin of obtining the solution, next inequlity is vlid: <x<l v H, 3.64 nd we obtin tht t ll n, stisfying condition n> 1 ( n H v H 2 v τ i, l ll terms in the first sum of formul 3.57 will be equl to ero. Differently, summtion in the first sum of the formul 3.57 needs to be mde in this cse not up to infinity, but up to N 1, where N is the less nturl number stisfying n inequlity For the second sum of the formul 3.57, condition tht the upper limit of integrtion is less thn right prt of inequlity 3.6 t t H looks like i 1 H x < 2n 1 l 2 n i 1 v τ i, 3.66 from which follows n> 1 ( n H x l 2 v τ i l i 1

15 Advnces in Mthemticl Physics 15 Therefore on the bsis of n inequlity 3.64 it is obtined tht t ll n, stisfying condition n> 1 ( n H l 2 v τ i, l i 1 ll terms in the second sum of formul 3.57 will be equl to ero. Differently, summtion in the second sum of the formul 3.57 needs to be mde in this cse not up to infinity, but up to N 1 1, where N 1 is the less nturl number stisfying inequlity All terms in the formul 3.57 re solutions of 2.1. And s for everyone fixed t number of terms in the formul 3.57 is finite, differentition in the formul 3.57 is possible to crry out term by term. Therefore function 3.57 is the solution of 2.1. From the formul 3.57 directly follows tht t t nd<x<lthe upper limits of integrtion of ll integrls become smller, thn l/. It mens tht on the bsis of function s Γ k qulities, 3.6 from 3.57 follows u x,. Thus, function 3.57 stisfies the first initil condition 2.2. Hving differentited function 3.57 on t we obtin u x, t t { 2 /2 x e B D2 Γ 2n (t x n t x/ [ } D 2 2 e B/2 x 2 J ( J 1 c 1 t η ]e D2 /2 t η ( Γ 2n η dη { 2 e D B /2 x Γ 2n (t x n t x/ [ D 2 2 e B/2 x 2 J t η c 1 } J 1 ]e D2 /2 t η ( Γ 2n η dη From the formul 3.69 it is obtined tht t t nd<x<lthe upper limits of integrtion of ll integrls nd s well rguments of functions Γ k become smller, thn l/. It mens, on the bsis of function s Γ k qulities 3.6, thtu t x,. Thus, function 3.57 stisfies lso the second initil condition 2.2. Hving clculted vlue of derivtive of function 3.57 on x in point x l v t, we obtin u x l v t,t { 2 Γ 2n n ( t l v t e D B /2 l v t t l v t / [ B 2e B/2 l v t 2 J l v t c 1 } J 1 ]e D2 /2 t η ( Γ 2n η dη

16 16 Advnces in Mthemticl Physics { ( 2 l v t e D B /2 l v t Γ 2n t n t l v t / [ } B 2e B/2 l v t 2 J l v t c 1 J 1 ]e D2 /2 t η ( Γ 2n η dη. 3.7 In the formul 3.7 we shll write down the first term of the first sum t n seprtely, nd in the second sum we shll replce n index of summtion n on s n 1. We shll obtin u x l v t,t 2 ( Γ l v t t e D B /2 l v t t l v t / [ B 2e B/2 l v t { 2 Γ 2n n 1 ( t l v t 2 J l v t c 1 e D B /2 l v t t l v t / [ B 2e B/2 l v t { ( 2 e D B /2 l v t Γ 2s 2 t s 1 t l v t / [ B 2e B/2 l v t 2 J l v t c 1 l v t 2 J l v t c 1 ] J 1 e D2 /2 t η ( Γ η dη } J 1 ]e D2 /2 t η ( Γ 2n η dη } J 1 ]e D2 /2 t η ( Γ 2s 2 η dη First two summnds in this formul represent the left prt of the integrl eqution 3.5 nd consequently re equl to Γ t. Summnds in the first sum of lst formul represent the left prts of the integrl equtions 3.58 divided on. It mens they re equl to u 2n 1 x l v t,t. Summnds in the second sum represent functions u 2n 1 x l v t,t. Therefore ll terms under signs Σ in lst formul will equl ero. Hence u x l v t,t Γ t γ t, t > And it mens tht function 3.57 t t> stisfies the first boundry condition 2.3. The fct tht function 3.57 stisfies the second boundry condition 2.3 is obvious. Thus, it is shown tht function 3.57 stisfing ll conditions of sttement of the bsic initil-boundry vlue problem consequently is its solution.

17 Advnces in Mthemticl Physics Conclusion The exct solution of the mixed initil-boundry vlue problem for the telegrph eqution in domin with mobile borders is obtined. The solution is obtined s superposition of n initil wve nd wves of reflection from the borders of domin. It is necessry to note tht the form of the solution in ccurcy corresponds to those nturl phenomen which, in prticulr, occur in rope during its loding. The solution of problem represents vlue of field of elstic displcements in rope. To obtin field of pressure in rope, it is enough to differentite field of displcements on x nd to increse this result on the module of elsticity of rope. Here, the developed method of the solution of evolutionry initil-boundry vlue problems with mobile borders of domin is suitble for serch of solutions of wide clss of similr problems. References 1 O. A. Goroshko nd G. N. Svin, Introduction in Mechnics of One Dimensionl Deformble Bodies of Vrible Length, Nukov Dumk, Kiev, Ukrnine, S. Momni, Anlytic nd pproximte solutions of the spce- nd time-frctionl telegrph equtions, Applied Mthemtics nd Computtion, vol. 17, no. 2, pp , J. K. Zhou, Differentil Trnsformtion nd Its Applictions for Electricl Circuits, Huhong University Press, Wuhn, Chin, J. Bir nd M. Eslmi, Anlytic solution for Telegrph eqution by differentil trnsform method, Physics Letters A, vol. 374, no. 29, pp , J. Chen, F. Liu, nd V. Anh, Anlyticl solution for the time-frctionl telegrph eqution by the method of seprting vribles, Journl of Mthemticl Anlysis nd Applictions, vol. 338, no. 2, pp , V. A. Ostpenko, Boundry Problem without Initil Conditions for Telegrph Eqution, Dnepropetrovsk, V. A. Ostpenko, The First Initil-Boundry Vlue Problem for Region with Mobile Border, The Differentil Equtions nd Their Applictions in Physics, Dnepropetrovsk, Ukrine, V. A. Ostpenko, The second initil-boundry vlue problem for region with mobile border, The Bulletin of the Dnepropetrovsk University, Mthemtics, vol. 1, pp. 3 21, 1997 Russin. 9 V. A. Ostpenko, Dynmics of the wves in ropes of vrible length, The Bulletin of Poltv Ntionl Technicl University, vol. 16, pp , 25 Russin. 1 V. A. Ostpenko, Dynmic field of displcements in rods of vrible length, in Proceedings of the 8th Interntionl Conference on Dynmicl Systems Theory nd Applictions, pp , Lod, Polnd, V. A. Ostpenko, Exct solution of the problem for dynmic field of displcements in rods of vrible length, Archives of Applied Mechnics, vol. 77, no. 5, pp , V. A. Ostpenko, Initil-boundry vlue problem for rod of vrible length, perturbed from the mobile top end, The Bulletin of Dnepropetrovsk University, Mechnics, no. 2, pp , 26 Russin.

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