Laplace Transform Associated with the Weierstrass Transform

International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 618 ISSN 2229-5518

Laplace transform associated with the Weierstrass transform

Dr. P.A. Gulhane and S.S. Mathurkar

Dept. Of Mathematics, Govt. College of Engineering, Amravati, (M.S), India [email protected] , [email protected]

Abstract: An elegant expression is obtained for the Laplace-Weierstrass transform LW of the function in terms of their existence. It is also shown , how the main result can be extended to hold for the LW transform of several functions.

Keywords: Transform,Laplace Transform,Weierstrass Transform, Laplace-Weierstrass transform, Generalized Function, Existence theorem,Multiplication Theorem.

1. Introduction: ∞ 2 In this paper we have introduced the 1 −−()xy /4 F() x= ∫ f() y e dy (1.2) concept of LW transform which has use in 4π −∞ several fields. The basic idea behind any transform is that the given problem can be where ()yf is a suitably restricted conventional solved more readilyIJSER in the transform domain. function on y ∞<<∞− and x is a complex The method is especially attractive in the linear variable. mathematical models for physical systems such Our purpose in this work is to define as a spring/mass system or a series electrical and study the Laplace transform associated with circuit which involve discontinuous functions. the Weierstrass transform. The distributional Laplace transform is This transform is defined by defined as

−st ∞∞ ()− yx 2 Fs()()= ft, e (1.1) 1 st−− (, xsF )= LW (, ytf ){}= ∫∫ (), 4 dtdyeytf The conventional Weierstrass transform 4π 00 is defined by ( 1.3)

2. The Testing Function Space LWa,b:

Let a and b be fixed numbers in , let ⇒ f is R-integrable over any finite interval in 1 ≥ (), ytf variable in , and let ,ba (), yth denote푅 the range ()yt 0, . 1 ()− yx 2 the function: 푅 st −− ⇒ 4 (), ytfe is R-integrable over any , < < 0 −풂풚 , ( , ) = (2.1) ퟐ finite interval in the range ()yt ≥ 0, . , 0 < 풂 풃 풆−풃풚 −∞ 푦 풉 풕 풚 � ퟐ Now by definition of LW transform ,

풆 ≤ 푦 ∞ 2 LWa,b as the linear space of all complex valued ∞∞ ()xy− 1 −−st = 4 smooth functions φ (), yt on 0 t ∞<< , LW{} f() t, y ∫∫f() t, y e dy dt (3.1) 4π 00 y ∞<<∞− such that for each p, q = 0, 1, 2, -

2 2 - - ∞∞ y ()−yx 1 st−− ≤ ' at 4t0 4 dtdyeeeM π ∫∫ y2 00 at+ +qp 4 2 ,,, qpba φγ (), = sup ,ba ()()φ ,, ytDytheyt ∞< −x 0 t ∞<< 4 0 y ∞<< − 'eM ≤ (2.2) ()− xas π

The space LWa,b is complete and a −x2 Frechet space. This topology is generated by the eM 4 ≤ where- '= MM total families of countably multinorms space π ()− asx given by (2.2). −x 2 Me 4 But is finite quantity. π ()− asx 3. Existence TheoremIJSER for LW Transform: ∴ LW (, ytf ){} exists provided, s > a and x > 0. If (), ytf is a function which is piecewise continuous on every finite interval in 4. Theorem: If LW ( ){}= (,, xsFytf ) then the range ()yt ≥ 0, and satisfies n 2 n d y i LW n ()ytft ()−= (),1,) xsF 4t {} n () ≤ ', at eeMytf 0 . For all ()yt ≥ 0, and for ds and some constant a, t0 and M ' , then the LW n n nn d transform of (), ytf exists for all > as and ) LWii {()()− ytfyx } ()()−= (),21, xsF dxn x > 0 . Proof: It is given that the function (), ytf is Proof: Given that, LW ( ){}= (,, xsFytf ) piecewise continuous on every finite interval in i) By definition of LW transform the range ()yt ≥ 0, .

F s,, x= LW f t y r+1 ()(){} r+1 r+1 d 2 LW t f() t, y =() − 1 F() s, x (4.4) ∞∞ ()xy− {} r+1 1 −−st ds = f() t, y e4 dy dt (4.1) 4π ∫∫ 00 d ⇒ Theorem is true for n = r + 1. F' () sx,,= {} F() sx ds Hence by induction , the theorem is true for all 2 ∞∞ ()xy− d 1 −−st = f() t, y e4 dy dt positive integral values of n. ds π ∫∫ 4 00 ii) Given that, LW ( ){}= (,, xsFytf )

(4.2) ∞ ∞ ()− yx 2 st−− 1 4 ∞∞ ()−yx 2 (, xsF )= LW (, ytf ){}= (), dtdyeytf st−− ∫∫ −1 4 π = ∫∫ (), dtdyeytft 4 0 0 4π 00 Now,

∞ ∞ ()− yx 2 d d  1 st−−  (− ) (,'1 xsF )= LW (, ytft ){} (),' xsF = (), xsF =  ∫∫ (), 4 dtdyeytf  dx dx  4π 0 0  or d (4.5) LW{} t f() t, y=() − 1 F() s , x (4.3) ds −1 = LW {()()− , ytfyx } 2 i.e. the theorem is true for n = 1. Next, let this theorem be true for n = r, then we d have LW{()() x−=− y f t, y } () 1 (2)F() s , x (4.6) r dx r d LW {}r ()ytft ()−= [](),1, xsF IJSERds r i.e. the theorem is true for n = 1. ()()()− r r ,1 xsF = LW {}r , ytft

∞∞ ()− yx 2 Next, let this theorem be true for n = r, then we 1 st−− = r (), 4 dtdyeytft π ∫∫ have 4 00 r On differentiating both sides w. r. t. s we get, r rr d − −= LW {()()fyx yt } ()() r (),21, xsF r r+1 dx ()()−1,F sx r 2 rr d r ∞∞ ()xy− − −= d 1 −−st ()()21 r [](), xsF LW {()(), ytfyx } = e4 tr f() t, y dy dt dx ds π ∫∫ 4 00 r+1 r+1 ∞ ∞ ()− yx 2 −− ()()−1,F sx 1 r st = ()()− , 4 dtdyeytfyx 2 ∫∫ ∞∞ ()xy− π 1 −−st 4 0 0 = r+1 4 ∫∫t f() t, y e dy dt 4π 00 Now differentiating both sides w. r. t. x we get,

r+1 rrd [5] Saitoh S : The Weierstrass Transform and an −  ()()12r+1 F() sx , dx Isometry in the Heat Equation Applicable 2 ∞∞ ()xy− d 1 r −−st Anal, (1983) = ()()x− y f t, y e4 dy dt dx π ∫∫ [6] Chelo Ferreira. Jose L. Lopez, 4 00 Approximation of the Poisson Transform for large and small values of −1 r+1 = LW {()()− , ytfyx } 2 the Transformation Parameter, Ramanujan J. LW {()()− r+1 ytfyx } ()()()−= rr ++ 11 r+1 ,21, xsF (2012) DOI 10,1007/11139-012-9438-y (4.7) ⇒ Theorem is true for n = r + 1. Hence by induction ,the theorem is true for all positive integral values of n.

5. Conclusion: In this paper we have introduced LW transform of a function (), ytf and seen how it is exists. Also we have proved another theorem which is useful to solve certain equation.

References : IJSER [1] Zemanian A H : Generalized Integral Transformation, Interscience Publishers, New York (1968). [2] Zemanian A H : Distribution Theory and Transform Analysis, McGraw-Hill, New York (1965). [3] Pathak R S : A Representation Theorem for a Class of Stieltjes Transformable Generalized Function (1974). [4] Srivastava H M : The Weierstrass-Laguerre Transform, National Academy of Sciences, Vol 68 , No 3, pp 554-556,(1971)