The Power Series Expansion on a Bulge Heaviside Step Function

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1 Applied Mathematical Science, Vol 9, 05, no 3, 5-9 HIKARI Ltd, wwwm-hikaricom The Power Serie Expanion on a Bulge Heaviide Step Function P Haara and S Pothat Department of Mathematic, Srinakharinwirot Bangkok 00, Thailand Wad Ban-Koh School, Bandara, Amphoe Pichai Uttaradit 530, Thailand Copyright c 04 P Haara and S Pothat Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited Abtract In thi paper, we introduce the Heaviide tep function on a bulge function and formulate the Laplace tranform of the Heaviide tep function of a bulge function by applying the Power erie expanion Mathematic Subject Claification: 44A0, 34A, 30B0 Keyword: Laplace tranform, Bulge function, Heaviide tep function Introduction Phyical problem and mathematical problem can be olved by uing the Laplace tranform In mathematic, the Laplace tranform can be applied to find a olution of the linear ordinary differential equation with contant coefficient and variable coefficient In addition, the Laplace tranform of derivative have been tudied in many way to olve the ODE Ig Cho and Hj Kim 3 howed that the Laplace tranform of derivative can be expreed by an infinite erie or Heaviide function T Lee and H Kim 4 found the repreentation of energy equation by Laplace tranform In thi paper, we introduce the Heaviide tep function on a bulge function and formulate the Laplace tranform of the Heaviide tep function of a bulge function by applying the Power erie expanion

2 P Haara and S Pothat Preliminarie We introduce the tudy by handing out the Laplace tranform, Heaviide tep function and the Power erie expanion which can be ued our tudy Definition The Laplace Tranform Given a function ft) defined for all t 0, the Laplace tranform of f i the function F defined a follow: F ) = L ft) = for all value of for which the improper integral converge 0 e t ft)dt ) The nonhomogeneou differential equation with contant coefficient An equation of the form d n y a n dx a d n y n n dx a d n y n n dx a dy n dx a 0y = fx) ) i called the higher order nonhomogeneou linear differential equation In thi paper, we tudy the nonhomogeneou econd order differential equation with a bulge function in the form y ω y = ft) = ; 0 < t < ξ The t ; t > ξ Laplace tranform of the firt and econd derivative are expreed repectively by L y = F ) y0) and L y = F ) y0) y 0) The Power erie expanion of i derived by = e l e l lt e l ) l t e l ) l3 t 3 3) Lemma The Laplace tranform of the bulge function i expreed by L = e l l l 3 l ) 4) Heaviide tep function of a bulge function of a piecewie continuou function ft) = ; 0 < t < ξ i expreed by t ; t > ξ where l, ξ are contant ft) = tut ξ ) ut ξ ) 5)

3 Power erie expanion 7 Lemma 3 The Laplace tranform of ut ξ ) i expreed by L ut ξ ) where Γ = e l, Γ = e ξ = Γ Γ le ξ ξ Γ Γ e ξ ξ 3 ξ Γ Γ 3 e ξ ) l, and Γ 3 = 4 ξ ) l 3 3 3ξ 3 ξ ) Proof The ditribution of Heaviide tep function and equation 3), we derive ut ξ ) = e l ut ξ ) e l ltut ξ ) ) e l l t ut ξ ) ) e l l 3 3 t 3 ut ξ ) 7) Therefore, by taking the Laplace tranform to equation 7), we obtain L ut ξ ) = Γ L ut ξ ) Γ ll tut ξ ) Γ BL t ut ξ ) Γ CL t 3 ut ξ ) e ξ = Γ Γ le ξ ξ Γ Γ e ξ ξ 3 ξ Γ Γ 3 e ξ ξ 4 3ξ ξ 3 = Ψ 8) ) ) where Γ = e l l, Γ = l, and Γ 3 = Main Reult Lemma 3 The Laplace tranform of Heaviide tep function of a bulge function of a piecewie continuou function ft) = ; 0 < t < ξ t ; t > ξ

4 8 P Haara and S Pothat can be expreed by Γ l )Γ 3 l )lγ e ξ ξ Ψ 9) where l, ξ are contant Proof By taking the Laplace tranform to equation 5) and lemma 3, we obtain L ft) = L L t ut ξ ) = L L tut ξ ) L ut ξ ) Γ = l )Γ 3 l )lγ e ξ ξ Φ Φ Φ 3 Φ 4 ) 0) e where Φ = Γ ξ, Φ = Γ le ξ ξ, Φ3 = Γ Γ e ξ ξ ξ 3 and Φ 4 = Γ Γ 3 e ξ 4 ξ 3ξ 3 ξ Lemma 3 The olution of the nonhomogeneou differential equation with contant coefficient y ω y = ft) = ; 0 < t < ξ where y0) = t ; t > ξ ω 0, y 0) = ω i expreed by yt) = ω 0 co ωt ω ω in ωt K K Γ t l )Γ ltγ lt3 l ) ) where ω, ω 0, ω are contant, K = L e ξ ξ and K = L Ψ Proof By taking the Laplace tranform to the nonhomogeneou differential equation with contant coefficient and the Heaviide tep function and by lemma 3, it yield L y t) ω 0 ω ω L yt) = L Heaviide tep function of ft) Or L yt) = ω e ξ Ψ ω 0 ω ω ξ Γ l )Γ 3 l )lγ )

5 Power erie expanion 9 By applying the invere Laplace tranform to equation ) to derive the olution of the nonhomogeneou differential equation with the Heaviide tep function a follow yt) = ω 0 co ωt ω ω in ωt K K Γ t l )Γ ltγ lt3 l ) where K = L e ξ ξ and K = L Ψ 4 Concluion 3) In thi paper, we introduce the Heaviide tep function on a bulge function and formulate the Laplace tranform of the Heaviide tep function of a bulge function which i denoted by ft) = where l i a poitive contant Ig Cho and Hj Kim 3 tudied the Laplace tranform of derivative expreed by Heaviide function In thi reearch, we dicovered the method to olve the nonhomogeneou econd order differential equation with a bulge function involved the Heaviide tep function The Laplace tranform, the invere Laplace tranform and the Power erie expanion were ued in thi method Reference C Henry Edward and David E Penney, Differential Equation and Boundary Value Problem, Pearon Education, Inc, USA, 004 D Lomen and J Mark, Differential Equation, Prentice-Hall International, Inc, USA, Ig Cho and Hj Kim, The Laplace Tranform of Derivative Expreed by Heaviide Function, AMS,790)03), T Lee and Hj Kim, The Repreentation of Energy Equation by Laplace Tranform, Int Journal of Math Analyi, 8)04), Received: December 5, 04; Publihed: February 5, 05

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