ON THE LAPLACE GENERALIZED CONVOLUTION TRANSFORM

Transcription

1 Annale Univ. Si. Budapet., Set. Comp. 43 4) ON THE LAPLACE GENERALIZED CONVOLUTION TRANSFORM Le Xuan Huy and Nguyen Xuan Thao Hanoi, Vietnam) Reeived February 5, 3; aepted Augut, 4) Abtrat. Several lae of integral tranform related to Laplae generalized onvolution are tudied. Neeary and uffiient ondition to enure that the tranformation i unitary are obtained, and a formula for the invere tranformation i derived in thi ae. In the appliation, we obtain olution of everal lae of integro-differential equation in loed form.. Introdution Laplae tranform i an integral tranform method whih i of the form Lf)y) = fx)e yx dx, y >. Here, the integral onverge for funtion f of the exponential order α >, i.e., there exit a poitive M and x uh that fx) Me αx, x x. Thi tranform find wide appliation in variou term of eletrial engineering, opti, ignal proeing, partial different equation, integral equation, invere problem and o on ee [3, 5, 6, 9,,,, 5). Key word and phrae: onvolution, Fourier ine, Fourier oine, Laplae, tranform, integrodifferential equation. Mathemati Subjet Claifiation: 33C, 44A35, 45E, 45J5, 47A3, 47B5. Thi reearh i funded by Vietnam National Foundation for Siene and Tehnology Development NAFOSTED) under grant.-4.8.

2 34 Le Xuan Huy and Nguyen Xuan Thao Although, o far there have been many artile about onvolution tranform ee [,, 4, 7,, 3, 4, 7). But generalized onvolution tranform related to the Laplae tranform ha not been tudied. In thi paper, we introdue everal generalized onvolution related to Laplae tranform. Then, we tudy lae of integral tranform related to thee generalized onvolution, whih are Laplae generalized onvolution tranform. Thi paper onit of four etion. The firt etion, reflet the ontent of the paper. The eond etion, we reall everal fundamental notation ued in thi paper. The third etion, we introdue everal new generalized onvolution for the Fourier ine, Fourier oine and the Laplae tranform; a well a together with the related integral tranform. The Waton type theorem whih give the neeary and uffiient ondition to enure that the above integral tranform are unitary. Additionally, the Planherel type theorem i alo obtained. In the final etion, a appliation, we obtain olution of everal lae of integro-differential equation in loed form. The reearh i intereted in L R + ) pae and the pae of funtion of exponential order α >.. Preliminarie The Fourier oine and Fourier ine tranform are defined a follow ee [, ).).) F f ) y) = π F f ) y) = π fx) o xydx = fx) in xydx = d π dy π d dy fx) fx) in xy dx, y >, x o xy dx, y >, x for f L R + ). The eond integral in.) and.) are alo well-defined for f L R + ). The Fourier ine and oine generalized onvolution of f and k i defined a in [ by.3) f k)x) = fy)[kx + y) + ignx y)k x y )dy, x >,

3 On the Laplae generalized onvolution tranform 35 whih atifie the following fatorization and Pareval type identitie ee [).4).5) F f k ) y) = F f ) y) F k ) y), y >, f, k L R + ), f k ) [ x) = F F f)f k) x), x >, f, k L R + ). The Fourier oine and ine generalized onvolution of f and k i defined a in [8 by.6) f k)x) = fy)[kx + y) ignx y)k x y )dy, x >, whih atifie the following fatorization and Pareval type identitie ee [).7).8) F f k ) y) = F f ) y) F k ) y), y >, f, k L R + ), f k ) [ x) = F F f)f k) x), x >, f, k L R + ). by The Fourier oine onvolution of two funtion f and k i defined a in [ f k)x) = fy)[kx + y) + k x y )dy, x >, F whih atifie the following fatorization identity.9) F f F k ) y) = F f ) y) F k ) y), y >, f, k L R + ). The onvolution of two funtion f and k for the Laplae tranform ee [5, ) f L k)x) = x fx y)ky)dy, x >, thi onvolution atifie the fatorization identity.) L f L k ) y) = Lf)y)Lk)y), y >. Thi fatorization identity hold for all funtion f and k of exponential order α >. Moreover, f L k)x) of exponential order α > ee Theorem.39, p.9 in [).

4 36 Le Xuan Huy and Nguyen Xuan Thao 3. The Laplae generalized onvolution tranform In thi etion, we introdue everal new generalized onvolution related to the Laplae tranform and tudy lae related to integral tranform. Definition 3.. The generalized onvolution with a weight funtion γy) = in y of two funtion f, k for the Fourier ine, Fourier oine and Laplae tranform are defined by γ f k ){ } x) = 3.) { [ v v + x u) ± v v + x + u) [ v v + x + u) ± v } v + x + + u) fu)kv)dudv, where x >. Theorem 3.. Suppoe that fx) L R + ) and kx) i a funtion of exponential order α >. Then, the generalized onvolution f γ k ) { } L R + ) atify the Pareval type identitie 3.) γ ) f k { } x) = F [ { } ± in yf{ } f)lk) x), x >, and the following fatorization identitie hold 3.3) γ ) F { } f k { } y) = ± in yf { f)y)lk)y), y >. } Proof. From 3.) and by uing the formula.3.5) in [5 e αx o xydx = α α + y, α >,

5 On the Laplae generalized onvolution tranform 37 we get γ f k ){ } x) = fu)kv)e vy{ [ ox u)y ± ox + u)y [ ox + u)y ± ox + + u)y } dudvdy = ± π = ± π = ± π [ fu)kv)e vy { in xy. in y. o uy o xy. in y. in uy { o uy fu) in uy } du. F { } f)y)lk)y) in y { in xy o xy } dudvdy { } in xy kv)e vy dv in y dy o xy } dy. Therefore Pareval type identitie 3.) hold. On the other hand, Lk)y) vanihe at infinity and f L R + ) therefore in yf { } f)y)lk)y) L R + ). Combining with 3.) we have f γ k ) { } L R + ) and fatorization identitie 3.3) hold. In the next part of thi etion, we reflet everal propertie of the integral tranform related to onvolution.3),.6) and generalized onvolution 3.), namely, tranformation of the form 3.4) fx) gx) = T k,k f ) ) { x) = d dx f γ } k ) { x) + f } { } k )x), x >. Theorem 3. Waton type theorem). Suppoe that k x) i a funtion of exponential order α > and k x) L R + ), then neeary and uffiient ondition to enure that the tranform 3.4) are unitary on L R + ) are that 3.5) ± in ylk )y) + F k )y) = + y. Moreover, the invere tranform have the form ) { fx) = d dx g γ } 3.6) k ) { x) + g } { } k )x).

6 38 Le Xuan Huy and Nguyen Xuan Thao Proof. Neeity. Aume that k and k atify ondition 3.5). We known that hy), yhy), y hy) L R) if and only if F h)x), d d dx F h)x), dx F h)x) L R) Theorem 68, pp.9, [). Moreover, d d F h)x) = dx dx + hy)e ixy dy = F [ iy) hy) x). Speially, if h i an even or odd funtion uh that hy), y hy) L R + ), then the following equalitie hold 3.7) d dx ) F{ } h) [ x) = F { + y } )hy) x). From ondition 3.5), therefore ± in ylk )y) + F k )y) are bounded, and hene + y ) ± in ylk )y) + F k )y) ) F { } f)y) L R + ). Sine 3.4), by uing Pareval type propertie.5),.8), 3.), and formula 3.7), we have d gx) = =F { } dx [ ) [ F { } ± in yf { } f)y)lk )y) + F { } f)y)f k )y) x) + y ) ± in ylk )y) + F k )y) ) F { } f)y) Therefore, the Pareval identitie f L R + ) = F { } f L R + ) and ondition 3.5) give x). g LR +) = + y ) ± in ylk )y) + F k )y) ) F { } f)y) L R +) = F { } f)y) L R + ) = f L R + ). It how that the tranform 3.4) are iometri. On the other hand, ine we have + y ) ± in ylk )y) + F k )y) ) F { } f)y) L R + ), F { } g)y) = + y ) ± in ylk )y) + F k )y) ) F { } f)y). Uing ondition 3.5), we have F { } f)y) = + y ) ± in ylk )y) + F k )y) ) F { } g)y).

7 On the Laplae generalized onvolution tranform 39 Again, ondition 3.5) how that + y ) ± in ylk )y) + F k )y) ) F { } g)y) L R + ). By uing.5),.8), 3.), and formula 3.7), we have [ fx) =F { } + y ) ± in ylk )y) + F k )y) ) F { } g)y) ) [ = d dx F { } ± in yf { } g)y)lk )y) + F { } g)y)f k )y) ) { = d dx g γ } k ) { x) + g } { } k )x). Thu, the tranform 3.4) are unitary on L R + ) and the invere tranform have the form 3.6). Suffiieny. Aume that, the tranform 3.4) are unitary on L R + ). Then the Pareval identitie for Fourier oine and ine tranform yield g LR +) = + y ) ± in ylk )y) + F k )y) ) F { } f)y) L R +) = F { } f)y) L R + ) = f L R + ). Therefore, the operator M θ [fy) = θy)fy), here θy) = + y ) ± in ylk )y) + F k )y) ) are unitary on L R + ), or equivalent, the ondition 3.5) hold. Theorem 3.3 Planherel type theorem). We et [ θ { } x, u, v) = v v + x u) ± [ v v + x + u) ± v v + x + u) v v + x + + u), and uppoe that k x) i a funtion of exponential order α > and two time ontinuouly differentiable, k x) L R + ) and two time ontinuouly differentiable, atifying ondition 3.5) and ) ) Θ { } x, u, v) = d dx θ { } x, u, v), K x) = d dx k x)

8 3 Le Xuan Huy and Nguyen Xuan Thao are bounded funtion. Let f L R + ) and for eah poitive integer N, et g N x) = N Θ { } x, u, v)fu)k v)dudv + N fu) [ K x + u) ± ignx u)k x u ) du. Then: ) We have g N L R + ), and if N then g N onverge in L R + ) norm to a funtion g L R + ) with g L R + ) = f L R + ). ) Set g N = g.χ, N), then f N x) = Θ { } x, u, v)gn u)k v)dudv + g N u) [ K x + u) ± ignx y)k x u ) du, alo belong to L R + ), and if N then f N onverge in norm to f. Proof. Beaue of the definition of f N and g N, thee integral are over finite interval and therefore onverge. Set f N = f.χ, N), we have g N x) = N Θ { } x, u, v)fu)k v)dudv + N fu) [ K x + u) ± ignx u)k x u ) du ) = { d dx θ { } x, u, v)f N u)k v)dudv + f N u) [ k x + u) ± ignx u)k x u ) } du. In view of Waton type theorem, we onlude that g N L R + ). Let g be the tranform of f under tranformation 3.4), we have g L R + ) = f L R + ),

9 On the Laplae generalized onvolution tranform 3 and the reiproal formula 3.6) hold. We have ) g g N )x) = { d dx θ { } x, u, v)f f N )u)k v)dudv + f f N )u) [ k x + u) ± ignx u)k x u ) } du. By uing Waton type theorem, we get g g N )x) L R + ) and g g N L R + ) = f f N L R + ). Sine g g N L R + ) a N then g N onverge in L R + ) norm to g L R + ). Similarly, one an obtain the eond part of the theorem. The following example how the exitene k and k whih atify ondition 3.5). Example. We hooe k x) = i in x i a funtion of exponential order α >, by uing 3..9) in [5, we have 3.8) Lk )y) = i + y. We hooe the funtion k x) L R + ) uh that 3.9) By uing..4) in [5, we have k x) =F [ o y + y = F k )y) = o y + y. in yx + ) + in yx ) + y dy = [ e x+) Eix + ) e x+) Ei x ) + e x ) Eix ) e x ) Ei x + ) L R + ), here, for real nonzero value of x, the exponential integral Eix) i defined a ee [5) Eix) = x e t t dt.

10 3 Le Xuan Huy and Nguyen Xuan Thao When, ine 3.8), 3.9), therefore ondition 3.5) are atified, i.e., ± in ylk )y) + F k )y) = + y. 4. A la of integro-differential equation Not many integro-differential equation an be olved in loed form. In thi etion, we onider the following integro-differential equation related to the tranform 3.4) 4.) fx) + d Tφ,ψ f ) x) = gx), x >. dx Here, φx) = φ L φ ) x), φ i given funtion of exponential order α >, φ x) = in t L in t ) x) and ψx) = eh t ψ ) x), ψ x) L R + ). gx) i given funtion in L R + ), and fx) i unknown funtion. Theorem 4.. Suppoe the following ondition hold 4.) + y + y 3 ) [ in ylφ)y) ± F ψ)y), y >. Then equation 4.) have unique olution in L R + ). Moreover, the olution an be preented in loed form a follow fx) = gx) q where qx) L R + ) i defined by Proof. 4.3) { F } g) x), F q)y) = y + y3 ) [ in ylφ)y) ± F ψ)y) + y + y 3 ) [ in ylφ)y) ± F ψ)y). The equation 4.) an be rewritten in the form ) d fx) + dx d3 [f γ φ){ dx 3 } x) + f { } ψ) x) = gx). By uing Pareval type identitie 3.),.5) and.8), we have 4.4) ) d dx d3 f γ d φ){ dx 3 } x) = dx d3 = F { } ) [ dx 3 F { } ± in yf{ } f) Lφ ) x) [ y + y 3 ) in yf { } f) Lφ ) x),

11 On the Laplae generalized onvolution tranform 33 and 4.5) ) d dx d3 f d dx 3 { } ψ) x) = dx d3 = ±F { } ) [ dx 3 F { } F{ } f) F ψ ) x) [ y + y 3 )F { } f) F ψ ) x). From 4.3), 4.4) and 4.5), we get [ fx) + F { y + y } 3 ) in yf { } f) Lφ ) [ x) ± F { y + y } 3 )F { } f) F ψ ) x) Therefore, = gx). F { } f)y) + y + y3 ) [ in y F { } f) y)lφ)y) ± F { } f)y)f ψ)y) or equivalent, 4.6) = F { } g)y), F { } f)y)[ + y + y 3 ) in ylφ)y) ± F ψ)y) ) = F { } g)y). From ondition 4.) and 4.6), we have 4.7) F { } f)y) = F { } g)y) [ y + y3 ) [ in ylφ)y) ± F ψ)y) + y + y 3 ) [ in ylφ)y) ± F ψ)y) On the other hand, by uing.7) in [ and fatorization identity.), we have 4.8) Lφ)y) = Lφ )y)lφ )y) Moreover, from formula.9.) in [3 = Lφ )y)l in t)y)l in t)y) = + y ) Lφ )y). F eh t)y) = π πy eh, and formula.9.4) in [3 for n = 4 + y ) eh πy = F eh 3 t)y),.

12 34 Le Xuan Huy and Nguyen Xuan Thao ombining with fatorization identity.4), we have 4.9) F ψ)y) = F eh t)y)f ψ )y) = + y F eh 3 t)y)f ψ )y). From 4.8) and 4.9), we have y + y 3 ) [ in ylφ)y) ± F ψ)y) y = in y + y Lφ )y) ± yf eh 3 t)f ψ )y). Then, ine by the formula.3.6) in [5 F e t )y) = y + y, and by uing partial integration, we eaily prove the following formula yf eh 3 t ) y) = 3F inh t eh 4 t ) y), ombining with fatorization identitie 3.3),.7), we have 4.) y + y 3 ) [ in ylφ)y) ± F ψ)y) π = in yf e t )y)lφ )y) 6F inh t eh 4 t)f ψ )y) π = F e t γ φ ) y) 6F inh t eh 4 ) t) ψ y) = F [ π e t γ φ ) 6inh t eh 4 t) ψ y) L R + ). From 4.), ondition 4.) and Wiener-Levy theorem in [6, there exit a funtion qx) L R + ) uh that F q)y) = y + y3 ) [ in ylφ)y) ± F ψ)y) 4.) + y + y 3 ) [ in ylφ)y) ± F ψ)y). From 4.7), 4.) and uing fatorization identitie.4),.9), we have Therefore, F { } f)y) =F { } g)y) F { } g)y)f q)y) =F { } g)y) F { q } { F } g) y). The proof i omplete. fx) = gx) q { F } g) x), fx) L R + ).

13 On the Laplae generalized onvolution tranform 35 Referene [ Al-Muallam, F. and V.K. Tuan, A la of onvolution tranformation, Frational Calulu and Applied Analyi, 3 3) ), [ Al-Muallam, F. and V.K. Tuan, Integral tranform related to a generalized onvolution, Reult Math., ) ), [3 Beteman, H. and A. Erdelyi, Table of Integral Tranform, MGraw- Hill Book Company, In., New York - Toronto, 954. [4 Britvina, L.E., A la of integral tranform related to the Fourier oine onvolution, Int. Tran. and Spe. Fun., 6 5-6) 5), [5 Debnath, L. and D. Bhatta, Integral Tranform and Their Appliation, Chapman and Hall/CRC, Boa Raton, 7. [6 Fujiwara, T. Matuura T., S. Saitoh and Y. Sawano, Numerial real inverion of the Laplae tranform by uing a high-auray numerial method, Further Progre in Analyi, World Si. Publ., Hakenak, NJ, 9, [7 Hong, N.T., T. Tuan and N.X. Thao, On the Fourier oine- Kontorovih-Lebedev generalized onvolution tranform, Appliation of Mathemati, 58 4) 3), [8 Kakihev, V.A., N.X. Thao and V.K. Tuan, On the generalized onvolution for Fourier oine and ine tranform, Eat-Wet Journal of Mathemati, ) 998), [9 Saitoh, S., V.K. Tuan and M. Yamamoto, Revere onvolution inequalitie and appliation to invere heat oure problem, J. of Ineq. in Pure and App. Math., 3 5) ), -. [ Shiff, J.L., The Laplae Tranform: Theory and Appliation, Springer- Verlag New York, In., 999. [ Sneddon, I.N., Fourier Tranform, MGray-Hill, New York, 95. [ Tithmarh, E.C., Introdution to the Theory of Fourier Integral, Third edition. Chelea Publihing Co., New York, 986. [3 Thao, N.X., V.K. Tuan and N.T. Hong, A Fourier generalized onvolution tranform and appliation to integral equation, Frational Calulu and Applied Analyi, 5 3) ), [4 Tuan, V.K., Integral tranform of Fourier oine onvolution type, J. Math. Anal. Appl., 9 999), [5 Tuan, V.K. and T. Tuan, A real-variable invere formula for the Laplae tranform, Int. Tran. and Spe. Fun., 3 8) ),

14 36 Le Xuan Huy and Nguyen Xuan Thao [6 Voloivet, S.S., On weighted analog of Wiener and Levy theorem for FourierVilenkin erie, Izv. Saratov. Univ. Mat. Mekh. Inform., 3) ) ), 3-7. [7 Yakubovih, S.B., Integral tranform of the Kontorovih-Lebedev onvolution type, Collet. Math., 54 ) 3), 99-. [8 Yakubovih, S.B. and A.I. Moinki, Integral-equation and onvolution for tranform of Kontorovih-Lebedev type, Diff. Uravnenia, 9 7) 993), in Ruian) Le Xuan Huy Faulty of Bai Siene Univerity of Eonomi and Tehnial Indutrie 456 Minh Khai Hanoi, Vietnam lxhuy@uneti.edu.vn Nguyen Xuan Thao Shool of Applied Mathemati and Informati Hanoi Univerity of Siene and Tehnology Dai Co Viet Hanoi, Vietnam thaonxbmai@yahoo.om