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 f() y 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 ∞ ∞ ()x− y 2 Fs()()= ft, e (1.1) 1 −st − F( s, x )= LW {f ( t, y )}= ∫∫ f() t, y e4 dy dt The conventional Weierstrass transform 4π 0 0 is defined by ( 1.3) 2. The Testing Function Space LWa,b: IJSER © 2013 http://www.ijser.org International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 619 ISSN 2229-5518 Let a and b be fixed numbers in , let ⇒ f is R-integrable over any finite interval in 1 ≥ f() t, y variable in , and let ha, b () t, y 푅denote the range ()t, y 0 . 1 ()x− y 2 the function: 푅 −st − ⇒ e4 f() t, y is R-integrable over any , < < 0 −풂풚 , ( , ) = (2.1) ퟐ finite interval in the range ()t, y ≥ 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 ()t, y 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 ()x− y 1 −st − ≤ M' eat e4t0 e4 dy dt π ∫∫ y2 0 0 at+ p+ q 4 2 γa,,, b p q φ()t, y= sup e ha, b ()() t,, y Dφ t y < ∞ −x 0<t <∞ 4 0<y <∞ − M' e ≤ (2.2) ()s− a x π The space LWa,b is complete and a −x2 Frechet space. This topology is generated by the Me 4 ≤ where- MM'= total families of countably multinorms space xπ () s− a given by (2.2). −x 2 Me 4 But is finite quantity. xπ () s− a 3. Existence TheoremIJSER for LW Transform: ∴ LW{f ( t, y )} exists provided, s > a and x > 0. If f() t, y is a function which is piecewise continuous on every finite interval in 4. Theorem: If LW{f ( t,, y )}= F ( s x ) then the range ()t, y ≥ 0 and satisfies n 2 n d y i)LW tn f() t , y =() − 1F() s , x 4t {} n f() t,' y≤ M eat e 0 . For all ()t, y ≥ 0 and for ds and some constant a, t0 and M ' , then the LW n n n n d transform of f() t, y exists for all s> a and ) LWii {()()x− y f t, y }=()() − 1 2F() s , x dxn x > 0 . Proof: It is given that the function f() t, y is Proof: Given that, LW{f ( t,, y )}= F ( s x ) piecewise continuous on every finite interval in i) By definition of LW transform the range ()t, y ≥ 0 . IJSER © 2013 http://www.ijser.org International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 620 ISSN 2229-5518 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{f ( t,, y )}= F ( s x ) (4.2) ∞ ∞ ()x− y 2 −st − 1 4 ∞ ∞ ()x− y 2 F( s, x )= LW {f ( t, y )}= f() t, y e dy dt −st − ∫∫ −1 4 π = ∫∫ t f() t, y e dy dt 4 0 0 4π 0 0 Now, ∞ ∞ ()x− y 2 d d 1 −st − (−1 )F ' ( s , x )= LW {t f ( t, y )} F',() s x = F() s, x = ∫∫ f() t, y e4 dy dt dx dx 4π 0 0 or d (4.5) LW{} t f() t, y=() − 1 F() s , x (4.3) ds −1 = LW {()()x− y f t, y } 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 {}tr f() t, y =() − 1[]F() s , x IJSERds r i.e. the theorem is true for n = 1. ()()()−1r Fr s , x = LW {}tr f t, y ∞ ∞ ()x− y 2 Next, let this theorem be true for n = r, then we 1 −st − = ∫∫ tr f() t, y e4 dy dt 4π 0 0 have r On differentiating both sides w. r. t. s we get, r r r d − = − LW {()()x y f t, y } ()() 1 2r F() s , x r r+1 dx ()()−1,F sx r 2 r r d r ∞∞ ()xy− − = − d 1 −−st ()()1 2 r []F() s, x LW {()()x y f t, y } = e4 tr f() t, y dy dt dx ds π ∫∫ 4 00 r+1 r+1 ∞ ∞ ()x− y 2 − − ()()−1,F sx 1 r st = ()()x− y f t, y e4 dy dt 2 ∫∫ ∞∞ ()xy− π 1 −−st 4 0 0 = tr+1 f() t, y e4 dy dt ∫∫ 4π 00 Now differentiating both sides w. r. t. x we get, IJSER © 2013 http://www.ijser.org International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 621 ISSN 2229-5518 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) IJSER © 2013 http://www.ijser.org .