Applied Mathematical Sciences, Vol. 9, 2015, no. 23, 1107 - 1112 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121006

On Heaviside Step Function with a Bulge Function by Using Laplace Transform

P. Haarsa1 and S. Pothat2

1Department of Mathematics, Srinakharinwirot Bangkok 10110, Thailand 2Wad Ban-Koh School, Bandara, Amphoe Pichai Uttaradit 53220, Thailand

Copyright c 2014 P. Haarsa and S. Pothat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribu- tion, and reproduction in any medium, provided the original work is properly cited.

Abstract In this paper, we study the nonhomogeneous second order differential equation of the Heaviside step function with a bulge function. The Laplace transform, inverse Laplace transform and Power series expan- sion are employed to obtain the solution.

Mathematics Subject Classification: 44A10, 34A12, 30B10

Keywords: Laplace transform, Heaviside step function, Bulge function

1 Introduction

The Laplace transform can be employed in not only solving the linear ordinary differential equations with constant coefficient but also can be used with ordi- nary differential equations with variable coefficients. In addition, the Laplace transforms of derivatives have been studied in numerous approaches to solve the ODEs. Ig. Cho and Hj. Kim [3] showed that the Laplace transform of derivative can be expressed by an infinite series or Heaviside function. T. Lee and H. Kim [4] found the representation of energy equation by Laplace trans- form. In this paper, we applied the Laplace transform to the nonhomogeneous second order differential equation with a bulge function involved the Heaviside step function. 1108 P. Haarsa and S. Pothat

2 Preliminary Notes

We begin our study by contributing out the Laplace transform, Heaviside step function and the Power series expansion which can be used in this study. Definition 2.1. The Laplace Transform [1]. Given a function f(t) defined for all t ≥ 0, the Laplace transform of f is the function F defined as follow: ∞ Z F (s) = L {f(t)} = e−stf(t)dt. (1)

0 for all values of s for which the improper integral converges The nonhomogeneous differential equation with constant coefficients [2]. An equation of the form dny dn−1y dn−2y dy a + a + a + ... + a + a y = f(x). (2) n dxn n−1 dxn−1 n−2 dxn−2 1 dx 0 is called the higher order nonhomogeneous linear differential equations. In this paper, we study the nonhomogeneous second order differential equation with 2 00 2 − (t−l) a bulge function in the form y + ω y = e 2 . The Laplace transform of the first and second derivatives are expressed respectively by L {y0} = sF (s)−y(0) and L {y00} = s2F (s) − sy(0) − y0(0).

2 − (t−l) Lemma 2.2. The Laplace transform of the bulge function e 2 is expressed by

 2  2  2 2 2  − (t−l) − l 1 −1 + l l(s − 3 + l ) L e 2 = e 2 + + . (3) s s3 s4 Proof. The Power series expansion ex is of the form ∞ X xn x2 x3 x4 ex = = 1 + x + + + + ... (4) n! 2! 3! 4! n=0

(t−l)2 Hence, by substituting equation (4) with x = − 2 , we derive

2 2 2 2  2  2  3  − (t−l) − l − l − l 1 l 2 − l 1 l 3 e 2 = e 2 + e 2 lt + e 2 − + t + e 2 − + t . (5) 2 2 2 6 By taking the Laplace transform to equation (5) and using the fact that the Laplace transform is linear, we obtain

 2  2  2 2 2  − (t−l) − l 1 −1 + l l(s − 3 + l ) L e 2 = e 2 + + . (6) s s3 s4 On Heaviside step function with a bulge function 1109

Heaviside step function of a bulge function of a piecewise continuous ( 2 − (t−l) function f(t) = e 2 ; 0 < t < ξ1 is expressed by a ; t > ξ1

2 2 − (t−l) − (t−l) f(t) = e 2 + au(t − ξ1) − e 2 u(t − ξ1). (7) where a, ξ1 are constants.

2 − (t−l) Lemma 2.3. The Laplace transform of e 2 u(t − ξ1) is expressed by

 2  −ξ1s   − (t−l) e −ξ s 1 ξ1 L e 2 u(t − ξ ) = A + Ale 1 + 1 s s2 s  2  2 2ξ1 ξ1 +ABe−ξ1s + + . s3 s2 s  2  6 6ξ1 3ξ1 ξ1 +ACe−ξ1s + + + . (8) s4 s3 s2 s

2 l  1 l2   1 l3  − 2 where A = e , B = − 2 + 2 , and C = − 2 + 6 .

Proof. From equation (4) and the Heaviside step function, we have

2 2 2 − (t−l) − l − l e 2 u(t − ξ1) = e 2 u(t − ξ1) + e 2 ltu(t − ξ1) 2  2  − l 1 l 2 +e 2 − + t u(t − ξ ) 2 2 1

2  3  − l 1 l 3 +e 2 − + t u(t − ξ ). (9) 2 6 1

Therefore, by taking the Laplace transform to equation (9), we obtain

 2  − (t−l) L e 2 u(t − ξ1) = AL {u(t − ξ1)} + AlL {tu(t − ξ1)}

 2  3 +ABL t u(t − ξ1) + ACL t u(t − ξ1)

−ξ1s   e 1 ξ1 = A + Ale−ξ1s + s s2 s  2  2 2ξ1 ξ1 +ABe−ξ1s + + s3 s2 s  2  6 6ξ1 3ξ1 ξ1 +ACe−ξ1s + + + (= K). (10) s4 s3 s2 s 1110 P. Haarsa and S. Pothat

3 Main Result

Lemma 3.1. The Laplace transform of Heaviside step function of a bulge ( 2 − (t−l) function of a piecewise continuous function f(t) = e 2 ; 0 < t < ξ1 a ; t > ξ1 can be expressed by

ae−ξ1s M + + K. (11) s where a, ξ1 are constants. Proof. By taking the Laplace transform to equation (7) and lemma 2.2 and 2.3, we have

 2   2   − (t−l) − (t−l) L {f(t)} = L e 2 + L a − e 2 u(t − ξ1)

 2   2  − (t−l) − (t−l) = L e 2 + L {au(t − ξ1)} − L e 2 u(t − ξ1)

2  2 2 2  −ξ1s − l 1 −1 + l l(s − 3 + l ) ae = e 2 + + + s s3 s4 s

−ξ1s    2  e 1 ξ1 2 2ξ1 ξ1 −A + Ale−ξ1s + − ABe−ξ1s + + s s2 s s3 s2 s  2  6 6ξ1 3ξ1 ξ1 −ACe−ξ1s + + + s4 s3 s2 s ae−ξ1s = M + + K (12) s

2 l h 1 −1+l2 l(s2−3+l2) i e−ξ1s 1 ξ − 2 −ξ1s  1  where M = e s + s3 + s4 and K = A s + Ale s2 + s + h 2 i h 2 i −ξ1s 2 2ξ1 ξ1 −ξ1s 6 6ξ1 3ξ1 ξ1 ABe s3 + s2 + s + ACe s4 + s3 + s2 + s . Lemma 3.2. The solution of the nonhomogeneous differential equation with ( 2 − (t−l) constant coefficients y00 + ω2y = f(t) = e 2 ; 0 < t < ξ1 where y(0) = a ; t > ξ1 0 ω0 , y (0) = ω1 is expressed by

 2 2 2 ω 2 t t l y(t) = ω cos ωt + 1 sin ωt + e−l /2 1 − + 0 ω 2 2 lt3 l3t3  +lt − + + a × Heaviside(t − ξ ) + L−1 {K} . (13) 3 6 1 where a, l, ω, ω0 and ω1 are constants. On Heaviside step function with a bulge function 1111

Proof. By taking the Laplace transform to the nonhomogeneous differential equation with constant coefficients and the Heaviside step function and lemma 3.1, we obtain

2 00 2 s L {y (t)} − sω0 − ω1 + ω L {y(t)} = L {Heaviside step function of f(t)} . Or

sω ω ae−ξ1s L {y(t)} = 0 + 1 + M + + K. (14) s2 + ω2 s2 + ω2 s The inverse Laplace transform can be used to equation (14) to obtain the solution of the nonhomogeneous differential equation with the Heaviside step function as  2 2 2 ω 2 t t l y(t) = ω cos ωt + 1 sin ωt + e−l /2 1 − + 0 ω 2 2 lt3 l3t3  +lt − + + a × Heaviside(t − ξ ) + L−1 {K} . (15) 3 6 1

4 Conclusion

In this paper, we study the nonhomogeneous second order differential equation of the Heaviside step function with a bulge function which is denoted by f(t) = 2 − (t−l) e 2 where l is a positive constant. Ig. Cho and Hj. Kim [3] studied the Laplace transform of derivative expressed by Heaviside function. For this research, we found the method to solve the nonhomogeneous second order differential equation with a bulge function involved the Heaviside step function. The Laplace transform, the inverse Laplace transform and the Power series expansion were employed in this method.

References

[1] C. Henry Edwards and David E. Penney, Differential Equations and Boundary Value Problem, Pearson Education, Inc, USA, 2004. [2] D. Lomen and J. Mark, Differential Equations, Prentice-Hall Interna- tional, Inc, USA, 1988. [3] Ig. Cho and Hj. Kim, The Laplace Transform of Derivative Ex- pressed by Heaviside Function, AMS.,7(90)(2013), 4455 - 4460. http://dx.doi.org/10.12988/ams.2013.36301 1112 P. Haarsa and S. Pothat

[4] T. Lee and Hj. Kim, The Representation of Energy Equation by Laplace Transform, Int. Journal of Math. Analysis, 8(22)(2014), 1093 - 1097. http://dx.doi.org/10.12988/ijma.2014.44120

Received: December 14, 2014; Published: February 5, 2015