On the Integro-Differential Equation with a Bulge Function by Using Laplace Transform

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1 Applied Mahemaical Sciences, Vol. 9, 15, no. 5, 9-34 HIKARI Ld, hp://dx.doi.org/1.1988/ams On he Inegro-Differenial Equaion 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 su he inegro-differenial equaions wih a bulge funcion. The Laplace ransform, inverse Laplace ransform and he convoluion heorem are used in his su o obain he exac soluion. The rapezoidal rule is used o find he numerical soluion. Mahemaics Subjec Classificaion: 65R1, 65R, 65D3 Keywords: Laplace ransform, Inegro-differenial equaions 1 Inroducion Mahemaical modelling of real-life problems normally oucomes in funcional equaions, like ordinary or parial differenial equaions, inegral and inegrodifferenial equaions, sochasic equaions. Numerous mahemaical formulaion of physical phenomena be composed of inegro-differenial equaions, hese equaions originaes in many fields like fluid namics, biological models and chemical kineics. A. Bellour and M. Bousselsal [1] discovered he numerical soluion of delay inegro-differenial equaions. The main purpose of heir work is o provide a new numerical approach based on he use of coninuous collocaion Taylor polynomials for he numerical soluion of delay inegro-differenial

2 3 P. Haarsa and S. Poha equaions. In his paper, we su he inegro-differenial equaions wih a bulge funcion. The soluion is derived by using Laplace ransform, inverse Laplace ransform, he convoluion heorem and he Taylor series expansion. The numerical soluion is obained by he rapezoidal rule [4]. Preliminary Noes We began our su by giving ou he he Laplace ransform, he convoluion heorem and he inegro-differenial equaions which can be used in his su. Definiion.1. The Laplace Transform []. 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 [3]. 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. () Inegro-differenial equaion is an equaion ha involves boh inegrals and derivaives of an unknown funcion of he form d x dx y() + f(, y())d = g(x, y(x)), y(x ) = y, x. (3) x The Trapezoidal rule(see [4]) can be used for he numerical soluion of he inegro-differenial equaion as follows: and x k +h x k +h x x k x k f(x, y(x))dx = h [f(x k, y k ) + f(x k+1, y k + hy k)] + O(h 3 ). (4) x F (x, s, y(s))dsdx = h x k x F (x k, s, y(s)) + x k+1 x F (x k+1, s, y(s)) (5)

3 On he inegro- differenial equaion The soluion of he inegro-differenial equaion wih a bulge funcion by using Laplace ransform Lemma 3.1. The Laplace ransform of he bulge funcion e ( l) is expressed by { } L e ( l) = e l Proof. The Taylor series expansion e x is of he form e x = n= [ ] l + + l(s 3 + l ). (6) s s 3 s 4 x n n! = 1 + x + x! + x3 3! + x (7) 4! 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. (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) s s 3 s 4 Lemma 3.. The soluion of he inegro differenial equaion wih a bulge funcion can be expressed by d = ( l) e + y( η) cos ηdη, y() = ξ 1. y() = e l 7 [ 5 l l + 1l l l+ 6 4 l l ] + ξ 1( + ). (1)

4 3 P. Haarsa and S. Poha Proof. By aking he Laplace ransform o he above equaion, we have { } { } L = L e ( l) + L y( η) cos ηdη d. (11) Applying he convoluion heorem, i yields { } { } L = L e ( l) + L {y()} L {cos }. (1) d And again by applying he convoluion heorem and Lemma 3.1 o equaion (1), we obain [ ] sy (s) y() = e l l + + l(s 3 + l ) + s Y (s). (13) s s 3 s 4 s + 1 Or [ ] [ ] Y (s) = e l l + + l(s 3 + l ) s + 1 s s 3 s 4 s 3 [ ] s ξ 1. (14) s 3 We can nex use he parial fracion mehod o equaion (14), we have L {y()} = ξ 1 s + e l s + le l + ( 1 + l )e l s 6 s 3 + a + l e l s 4 + ( + l )le l s 5 + ( 3 + l )le l. (15) s 7 Then, he inverse Laplace ransform can be employed o equaion (14) o obain y() = e l 7 [ 5 l l + 1l l l+ 6 4 l l ] + ξ 1( + ). (16) Example 3.3. we consider he inegro-differenial equaion wih a bulge funcion from lemma 3. which is d = ( l) e + y( η) cos ηdη, y() = 1. by fixing l =, 6, ξ 1 = 1 and h =.1,.1 in he rapezoidal rule, we compare he exac soluion from equaion (16) and he approximae soluion obained by he rapezoidal rule [4] as shown graphically.

5 On he inegro- differenial equaion x 14 Exac soluion Approx. soluion 3.5 y() Figure 1: Exac soluion and numerical soluion of example for l = and h =.1. 5 Exac soluion Approx. soluion 15 y() Figure : Exac soluion and numerical soluion of example for l = 6 and h =.1. 4 Conclusion In his work, we sudied he inegro-differenial equaions wih a bulge funcion which is denoed by f() = e ( l) where l is a posiive consan. We applied he rapezoidal rule for solving he numerical soluions. To approach he exac soluion, we employed he Laplace ransform, he inverse Laplace ransform, he Taylor series expansion and he convoluion heorem. We can conclude, according o our examples, ha he approximae soluions obained by he rapezoidal rule [4] are in good agreemen wih he exac soluion.

6 34 P. Haarsa and S. Poha x 14 Exac soluion Approx. soluion 3.5 y() Figure 3: Exac soluion and numerical soluion of example for l = and h =.1. References [1] A. Bellour and M. Bousselsal, Numerical soluion of delay inegrodifferenial equaions by using Taylor collocaion mehod, Applied Mah.Sci., 37(1)(14), [] C. Henry Edwards and David E. Penney, differenial equaions and boundary value problem, Pearson Educaion, Inc, USA, 4. [3] D. Lomen and J. Mark, differenial equaions, Prenice-Hall Inernaional, Inc, USA, [4] J. Thomas Day, Noe on he numerical soluion of inegro-differenial equaions, Universiy of Wisconsin-Milwaukee., Received: November 3, 14; Published: January 1, 15