NTMSCI 5, No. 1, (2017) 26

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1 NTMSCI 5, No. 1, - (17) New Treds i Mathematical Scieces The geeralized successive approximatio ad Padé approximats method for solvig a elasticity problem of based o the elastic groud with variable coefficiets Mustafa Bayram Departmet of Computer Egieerig, Faculty of Egieerig ad Architecture, Istabul Gelisim Uiversity, Avcilar, Istabul, Turkey Received: December 1, Accepted: 5 Jauary 17 Published olie: 5 Jauary 17. Abstract: I this study, we have applied a geeralized successive umerical techique to solve the elasticity problem of based o the elastic groud with variable coefficiet. I the first stage, we have calculated the geeralized successive approximatio of beig give BVP ad i the secod stage we have trasformed it ito Padé series. At the ed of study a test problem has bee give to clarify the method. Keywords: The geeralized successive approximatio method, Itegral Equatios, BVPs, Padé series. 1 Itroductio The solutio of BVPs has a lot of methods i literature. Oe of the most kow is the itegral equatios method. By usig the metioed method, we ca achieve a itegral equatio which is equivalet to the BVP. It is well kow that the solutio of the itegral equatio ca be defied as the solutio of the BVPs. This equatio is geerally kow as Fredholm equatio i the mathematical society. But i our paper we acquire a Fredholm-Volterra itegral equatio differet from the kow as so far. The elasticity problem based o the elastic groud with variable coefficiets has the followig form; d 4 x a(t)x= f(t),( t T) (1) dt4 d x() dt = A 1, d x() dt = B 1 () x(t)=a, dx(t) dt = B () where a(t) ad f(t) are previously give cotiuous fuctios o the iterval t T. At first, the successive approximatios method has bee applied to the problem ad the coverted to Padé series, 4. The equivalet itegral equatio The followig liear equatios K(t, s)x(s)ds (4) Correspodig author mbayram@gelisim.edu.tr

2 7 M. Bayram: The geeralized successive approximatio ad Padé Approximats method for solvig... K 1 (t,s)x(s)ds K(t, s)x(s)ds (5) K (t,s)x(s)ds () are kow i the literature as Volterra, Fredholm ad Volterra-Fredholm itegral equatios, respectively. We call the fuctio f(t) as free term of (4)-(5), K(t,s) ad K i (t,s) are kerels of equatios (4)-(5), ad x(t) is a ukow fuctio defied o the iterval t T. Let C,T be the space which cotais all of the cotiuous fuctios defied o the closed iterval,t. I this space the orm of x(t) C, T is a real fuctio ad give as follows, We ca defie F x ad V x as ad x = max t T x(t). F x V x K(t, s)x(s)ds K(t, s)x(s)ds o the C,T, ad the above operators are called as Fredholm ad Volterra operators. If F x C,T for x(t) C,T, the it is said that operator F x affects C,T. If the operator F x acts from C,T to R the we call the operator F x as liear fuctioal. We defie degeerated kerel fuctio as the followig K(t,s)= i=1 a i (t)b i (s) (7) If kerel fuctio is degeerated, the itegral equatios which has this sort of kerel are kow as itegral equatios with a degeerated kerel 5. Let equatio (4) has kerel (7) the the equatio (4) ca be arraged as Here, we search the solutio of the eq. (8) as follows To getc j, we ca arrage a system as follows C i = b i (s) f(s)ds i= j= a i (t) i= b i (s)x(s)ds. (8) a i (t)c i. a i (s)b i (s) f(s)dsc j (i=1,,)

3 NTMSCI 5, No. 1, - (17) / 8 Let be the determiat of above system. If, the we fid out C i = 1 T i j j=1 b j (s) f(s)ds where i j is kow as algebraic complemet of. The i j ca be costructed by deletig ith row ad jth colum of the. If equatio (4) has degeerated kerel, the solutio of the eq. (4) will be or i, j=1 a i (t) i j b j (s) f(s)ds a i (t)b j (s) i j i, j=1 f(s)ds. The gree fuctios ad solutio of BVPs We are cosiderig the followig BVPs x (t)b(t)x a(t)x= f(t), t T a x()β x ()=γ (9) a 1 x()β 1 x ()=γ 1. where a(t), b(t) ad f(t)( t T) are previously defied fuctios. Let α i, β i ad γ i (i =,1) are costats. We covert the equatio (9) ito its homogeeous form as followig x b(t)x a(t)x=, ( t T), (1) a x()β x ()=, (11) α 1 x()β 1 x ()=. (1) Defiitio 1. Let G(t,s) be fuctio which has the followig properties with its kow value s (,T) (I) If t s, the G(t,s) is solutio of the give problem (1). (II) If t = s, the G(t,s) is cotiuous fuctio with respect to t. Partial derivative of the G(t,s) with respect to t has first kid of discotiuity ad its jumpig umber 1. That is, G(s,s)=G(s,s), G t(s,s)g t(s,s)=1. Now, we are goig to costruct the Gree fuctio: Let we thik eq. (1) has two distict solutio such as x 1 (t), x (t) respectively ad satisfies the boudary coditios (11) ad (1), respectively. (1) Let cosider the followig fuctio G(x,s)= { ϕ(s)x1 (t), t s, ψ(s)x (t), s t T. (14) Now we choose the fuctios ϕ(t) ad ψ(t) which satisfy (.5). That is, ψ(s)x (s)=ϕ(s)x 1 (s),ψ(s)x (s)ϕ(s)x 1 (s)=1. If we solve the above system we ca get the fuctios ϕ(s) ad ψ(s). By substitutig ϕ(s) ad ψ(s) i (14) we get the fuctio G(x, s) as Gree fuctio of the problem (1)-(1).

4 9 M. Bayram: The geeralized successive approximatio ad Padé Approximats method for solvig... Theorem 1. Let G(x,s) be the Gree fuctios of the problem (1)-(1) ad let f(t) be a cotiuous fuctio, the followig fuctio x(t)= G(t,s) f(s)ds will be the solutio for ohomogeeous problem (9). 4 The equivalet Fredholm Volterra itegral equatios Let F(t) = f(t) a(t)x. Whe we cosider the boudary coditios () ad the equatio d 4 x dt 4 = F(t) has itegral order of four, o the iterval, t, the followig equatios ca be arrived where x (t)=x () t F(s)ds, x (t)=x ()x ()t t (t s)f(s)ds, x (t)=x ()x ()t x ()t t (ts) F(s)ds, x(t)=x()x ()t x ()t x ()t t (ts) F(s)ds x(t)=x()x ()t A 1t B 1t I additio to this, the boudary coditios (), () ad x(t), x (t) have bee used, A = x()x ()T A 1T T B = x ()A 1 T B 1T T (Ts) F(s)ds are gaied. The solutio of the above system yields the followig equatios, F(s)ds (15) (Ts) F(s)ds x()=a TB A 1T T x ()=B A 1 T B 1T T 1 (T s) (T s)f(s)ds, (Ts) F(s)ds. (1) If we put (1) i (15), i that case we acquire x(t)=a TB A 1T t(t s) F(s)ds A 1T (T s) (T s) F(s)ds F(s)ds (B A 1 T B 1T )

5 NTMSCI 5, No. 1, - (17) / or x(t)=a TB A 1T (T s) (T s) for this reaso, whe we thik F(t) = f(t) a(t)x, here we get x(t)=a TB A 1T (T s) (T s) (T s) (T s) h(t)=a TB A 1T x(t)=h(t) (T s) (T s) (T s) (T s) F 1 x T (B A 1 T B 1T )t A 1T F(s)ds F(s)ds t(t s) (B A 1 T B 1T )t A 1T f(s)ds f(s)ds t(t s) t(t s) a(s)x(s)ds a(s)x(s)ds, (B A 1 T B 1T )t A 1T f(s)ds f(s)ds t(t s) t(t s) (Ts) (Ts) (Ts) a(s)x(s)ds a(s)x(s)ds. (17) The equatio (17) is kow as liear Volterra Fredholm itegral equatio where Fredholm operator has degeerated kerel. Let we defie Vx t (ts) a(s)x(s)ds a(s)x(s)ds So, the equatio (17) ca be arraged F x T a(s)x(s)ds. x(t)=h(t)vxf 1 x tf x (18) by the reaso of F 1 x, F x Fredholm ad V x Volterra operators, respectively. Thereby, the problem (1)-() will be equivalet to the itegral equatio (18). 5 The geeralized successive approximatio method for elasticity problem To obtai the approximatio of Volterra-Fredholm itegral equatio (18), we ca use the followig formula x (t)=h(t)vx 1 F 1 x 1 tf x 1, (=1,, ) (19)

6 1 M. Bayram: The geeralized successive approximatio ad Padé Approximats method for solvig... here h(t)=x (t) is kow as discretioary ad cotiuous fuctio. By solvig the liear Volterra-Fredholm itegral equatio ca calculate the approximatio x (t), y(t)= h(t)f 1 y tf y () where () has a degeerated kerel ad has a solutio y(t)= h(t)c 1 tc (1) I additio to that we ca calculate the ukow terms C 1 ad C by solvig the followig liear equatio system (1F 1 1)C 1 (F 1 t)c = F 1 h (F 1)C 1 (1F t)c = F h () If we assume the determiat of the coefficiet matrix of () is ot zero, amely =(1F 1 1)(1F t)(f 1 t)(f 1) = (T s) (T s)a(s)ds 1 1 s(t s) (T s)a(s)ds s(t s) a(s)ds (T s) a(s)ds. We ca calculate C 1 ad C as follows C 1 = 1 (F1 h)(1f t)(f 1 t)(f h) C = 1 (1F1 1)(F h)(f 1 h)(f 1). If we put ito place C 1 ad C ito the equatio (1) we achieve the solutio of the equatio () as y(t)= h 1 1F t tf 1(F 1 h) 1 (1F 11)t F 1 t(f h). () To get the approximatio of x (t) we ca use (19) ad the equality h(t) = h(t) Vx 1 thus it yields the followig approximatio formula, here x (t)=h (t) 1F t tf 1 h (t)=h(t) 1F t tf 1 F 1 Vx 1 (1F 11)t F 1 t F V x 1 Vx 1 (4) F 1 h (1F 11)t F 1 t F h. (5) To guaratee that the approximatios of x (t) is coverget to the solutio of the problem (1-), the followig liear operator must satisfy the iequalities A(x)= 1F t tf 1 F 1 Vx (1F 11)t F 1 t F VxVx A(x) β x,

7 NTMSCI 5, No. 1, - (17) / β = 1F t T F 1 (T s) (T s) a(s) ds T 1F 11 F 1 t T (T s) a(s) ds1 Θ < 1. ( ) 1 T where Θ = (T s) a(s) ds. The covergece speed of the above approximatios satisfies the iequalities x x β x x or x x β 1β x 1 x. Padé series The Padé series is defied as follows. a a 1 xa x = p p 1 x p M x M 1q 1 x q L x L () After we multiply both sides of () by the deomiator of right side of () ad compare to the coefficiets of both sides i (). We obtai followig equatios a l a l M k=1 L k=1 a lk q k = p l,(l =,,M) (7) a lk q k =,(l = M 1,,M L). (8) By solvig the liear equatio i (8), we ca acquire the values of q k,(k= 1,,L). Furthermore by substitutig q k ito (), we obtai the values of p k,(l =,,M)1,. 7 Test problem Let thik the followig elasticity problem with homogeeous boudary coditios ad elasticity a(t) = 1. Accordig to these, the BVP ca be give as d 4 x x=t dt 4 x ()=, x ()= (9) x(1)=, x (1)=1 If we calculate the approximate solutio of (9) by usig The Geeralized Successive Approximatio give i (4) we arrive at a solutio as follows x (t)=.t 9.t 8.777t.74t 5.759t t.9 This is the approximate solutio of the problem (9) with the determiat value =

8 M. Bayram: The geeralized successive approximatio ad Padé Approximats method for solvig... Furthermore the solutio of x (t) ca be trasformed ito Padé series as follows ( t t.49419t ) 7/ =.79517t t t ( t t.14491t t t 5 ) Table 1: The compariso of the geeralized successive approximatio with the exact solutio o the iterval, 1. t i x(t i )Exact Sol. x (t i ) 7/ x(ti )x (t i ) 7/ Coclusio The aim of this paper is to costruct a approximate solutio of the equatio (9) which is elastic groud problem with variable coefficiets. I Table 1 the solutio of (9) is see i detailed. The umerical outputs i the Table 1 show us that the approximate solutio is very close to the exact solutios of (9). Competig iterests The authors declare that they have o competig iterests. Authors cotributios All authors have cotributed to all parts of the article. All authors read ad approved the fial mauscript. Refereces 1 E. Celik ad M. Bayram, O the Numerical Solutio of Differetial-Algebraic Equatio by Padé Series, Applied Mathematics ad Computatio, 17() E. Celik, E. Karaduma ad M. Bayram, A Numerical Method to Solve Chemical Differetial-Algebraic Equatios, Iteratioal Joural of Quatum Chemistry, 89() A. Aykut, E. Celik,ad M. Bayram, The Ordiary Successive Approximatios Method ad Padé Approximats for Solvig a Differetial Equatio with Variat Retarded Argumet, Applied Mathematics ad Computatio,, AMC E. Celik, A. Aykut ad M. Bayram, The Modified Two-Sided Approximatios Method ad Padé Approximats for Solvig a Differetial Equatio with Variat Retarded Argumet, Applied Mathematics ad Computatio,, AMC Fracis B. Hildebrad. Methods of applied Mathematics (secod editio), pretice-hall, ic., New Jersey E. Celik ad M. Bayram, The basic successive substitute approximatios Method ad Padé Approximatios to solve the elasticity problem of settled of the wrokler groud with variable coefficiets, Applied Mathematics ad Computatio, 154(4)