On Volterra Integral Equations of the First Kind with a Bulge Function by Using Laplace Transform

Transcription

1 Applied Mahemaical Sciences, Vol. 9, 15, no., HIKARI Ld, hp://dx.doi.org/1.1988/ams On Volerra Inegral Equaions of he Firs Kind wih a Bulge Funcion by Using Laplace Transform P. Haarsa 1 and S. Poha 1 Deparmen of Mahemaics, Srinakharinwiro Bangkok 111, Thailand Wad Ban-Koh School, Bandara, Amphoe Pichai Uaradi 53, Thailand Copyrigh c 14 P. Haarsa and S. Poha. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac In his paper, we sudy he Volerra inegral equaions of he firs kind wih a bulge funcion. The Laplace ransform, inverse Laplace ransform and he convoluion heorem are used in his sudy o obain he exac soluion. The Simpson s quadraure rule is used o find he numerical soluions. Mahemaics Subjec Classificaion: 45B5, 44A5 Keywords: Laplace ransform, Volerra inegral equaions, Taylor series expansion, Convoluion heorem 1 Inroducion Problems in which inegral equaions [6] are faced include radiaive energy ransfer and he oscillaion of a sring, membrane, or axle. Oscillaion problems may also be solved as differenial equaions. Inegral equaions are significan in many applicaions. S. Xiang [5] focused on Laplace and inverse Laplace ransforms for approximaion of Volerra inegral equaions of he firs kind wih highly oscillaory Bessel kernels, where he explici formulae for he soluion of he firs kind inegral equaions are derived, from which he inegral equaions can also be efficienly calculaed by he Clenshaw-Curis-Filon-ype

2 5 P. Haarsa and S. Poha mehods. Furhermore, by applying he asympoics of he soluion, some simpler formulas for approximaing he soluion for large values of he parameers are deduced. F. Mirzaee [3] used he repeaed Simpson s quadraure rule o solve he linear Volerra inegral equaions of he firs kind, convered he linear Volerra inegral equaions of he firs kind o a recurrence relaion and shown ha he esimaes have a good degree of accuracy. In his paper, we sudy he Volerra inegral equaions of he firs kind wih a bulge funcion. The Laplace ransform, inverse Laplace ransform, he convoluion heorem and he Taylor series expansion are used in his sudy o derive he exac soluion. The Simpson s quadraure rule is used o find he numerical soluions. Preliminaries We began our sudy by giving ou he he Laplace ransform, he convoluion heorem and he Volerra inegral equaions of he firs kind which can be used in his sudy. Definiion.1. The Laplace Transform [1]. Given a funcion f() defined for all, he Laplace ransform of f is he funcion F defined as follow: F (s) = L {f()} = for all values of s for which he improper inegral converges e s f()d. (1) Theorem.. The Convoluion Theorem []. The convoluion of wo funcions f() and g() denoed f() g(), is given by f() g() = whenever he inegral is defined. f(τ)g( τ)dτ. () A linear Volerra equaion of he firs kind is of he form f() = K(, x)y(x)dx, x. (3) where f is a given funcion and x is an unknown funcion o be solved for. Simpson s quadraure rule [3, 4] can be used for he numerical soluions of he Volerra inegral equaion of he firs kind. If N is even, hen Simpson s

3 On Volerra inegral equaions 53 quadraure rule may be applied o each subinerval [x i, x i+1, x i+ ] ; i =, 1,..., N 1, individually yields he approximaion h 3 [f(x i) + 4f(x i+1 ) + f(x i+ )]. Summing N approximaion resuls in he composie version of Simpson s quadraure rule S(h) = h [f(a) + 4f(a + h) + f(a + h) + 4f(a + 3h) f(b h) + 4f(b h) + f(b)]. (4) for he enire inerval. The error of S(h) is he sum of all N ES(h) = b a f(x)dx S(h) = h5 9 N 1 individual error f (4) (ξ i ), ξ i = [x i, x i+ ]. (5) 3 The soluion of Volerra inegral equaion of he firs kind by using he Laplace ransform Lemma 3.1. The Laplace ransform of he bulge funcion e ( l) is expressed by { } L e ( l) = e l i= [ ] l + + l(s 3 + l ). (6) Proof. The Taylor series expansion e x is of he form e x x n = n! = 1 + x + x! + x3 3! + x (7) 4! n= Therefore, by subsiuing equaion (7) wih x = ( l), we obain ( e ( l) = e l + e l l + e l 1 ) + l + ( e l 1 ) + l3 3 + O( 4 ). (8) 6 By aking he Laplace ransform o equaion (8) and using he fac ha he Laplace ransform is linear, we derived { } [ ] L e ( l) = e l l + + l(s 3 + l ). (9)

4 54 P. Haarsa and S. Poha Lemma 3.. The soluion of he Volerra inegral equaion of he firs kind wih a bulge funcion can be expressed by y( η)e aη dη = e ( l). = e l [ 6 + 6a + 3( 3l + l 3 a + al ) + 6( 1 + l + la) 6 + l(6 + a 3 ( 3 + l )) ]. (1) Proof. By aking he Laplace ransform o he above equaion, we have { } L y( η)e aη dη = L e ( l). (11) Applying he convoluion heorem, i yields L { e a} [ ] = e l l + + l(s 3 + l ). (1) And again by applying he convoluion heorem and Lemma 3.1 o equaion (1), we obain L {} L { e a} [ ] = e l l + + l(s 3 + l ). (13) Or = L 1 {e l [ ] } l + + l(s 3 + l ) (s + a). (14) We can nex use he parial fracion mehod o equaion (14), we have { = L 1 e l (l + a) + e l } ( 1 + l + la) + e l ( 3l + l 3 a + l a) s s s 3 { } + L 1 e l ( 3 + l )la + e l s 4. (15) Then, he inverse Laplace ransform can be employed o equaion (15) o obain = e l [ 6 + 6a + 3( 3l + l 3 a + al ) + 6( 1 + l + la) 6 + l(6 + a 3 ( 3 + l )) ]. (16)

5 On Volerra inegral equaions 55 Example 3.3. we consider he Volerra inegral equaion of he firs kind wih a bulge funcion from lemma 3. which is y( η)e aη dη = e ( l). by fixing l =, a = 1 and h =.1,.5,.5,.1 in Simpson s quadraure rule, we compare he exac soluion from equaion (16) and he approximae soluion obained by he Simpson s quadraure rule as shown graphically A.8.7 B C.15 D Figure 1: Graphs of he comparison of exac soluions and numerical soluions of example. Graph A, we fix l =, a = 1 and h =.1. Graph B, we fix l =, a = 1 and h =.5. Graph C, we fix l =, a = 1 and h =.5. Graph D, we fix l =, a = 1 and h =.1.

6 56 P. Haarsa and S. Poha 4 Conclusion In his work, we sudied he Volerra inegral equaions of he firs kind wih a bulge funcion which is denoed by f() = e ( l) where l is a posiive consan. For his sudy, we fix he value of l = and a = 1 bu we decease he values of h as h =.1,.5,.5,.1. We applied he repeaed Simpson s quadraure rule for solving he numerical soluions. To reach he exac soluion, we applied he Laplace ransform, he inverse Laplace ransform, he Taylor series expansion and he convoluion heorem. Referring o our example, We can conclude ha he approximae soluions obained by he Simpson s quadraure rule are in good concord wih he exac soluions. References [1] C. Henry Edwards and David E. Penney, differenial equaions and boundary value problem, Pearson Educaion, Inc, USA, 4. [] D. Lomen and J. Mark, differenial equaions, Prenice-Hall Inernaional, Inc, USA, [3] F. Mirzaee, Numerical Soluion for Volerra Inegral Equaions of he Firs Kind via Quadraure Rule, AMS., 6()(1), [4] G. M. Phillips and P. J. Taylor, Theory and applicaions of numerical analysis, Academic Press, New York, [5] S. Xiang, Laplace ransforms for approximaion of highly oscillaory Volerra inegral equaions of he firs kind, Applied Mahemaics and Compuaion., 3(1)(14), hp://dx.doi.org/1.116/j.amc [6] Wikipedia: Inegral equaions in mahemaics. Received: November 9, 14; Published: December, 14