AN APPLICATION OF THE DOUBLE SUMUDU TRANSFORM

Transcription

1 Applied Mathematical Sciences, Vol. 1, 27, no. 1, AN APPLICATION OF THE DOUBLE SUMUDU TRANSFORM Jean M. Tchuenche 1 and Nyimua S. Mbare Mathematics Department, Uniersity of Dar es Salaam P.O. Box 3562, Dar es Salaam, Tanzania jmt biomaths@yahoo.com Abstract The double Sumudu transform of functions expressible as polynomials or conergent infinite series are deried. The relationship of the former to the double Laplace transform is obtained, and it turns out that they are theoretical dual to each other. The applicability of this relatiely new transform is demonstrated using some special functions, which arise in the solution of eolution equations of population dynamics as well as partial differential equations. Keywords: Laplace transform, Sumudu transform, population dynamics, partial differential equations Mathematics Subject Classification: 44A1,44A35,45A5, 92D25 1. Introduction A lot of work has been done on the theory and applications of transforms such as Laplace, Fourier, Mellin, Hankel, to name a few, but ery little on the power series transformation or Sumudu transform, probably because it is little known, and not widely used [3]. An ingenious solution to isualizing the Sumudu transform was proposed originally by Watugala [6], and subsequently exploited by Weerakoon [8,9] and Asiru [1,2]. Neertheless, this new transform rials the Laplace transform in problem soling [3]. Its main adantage 1 Corresponding author

2 32 Jean M. Tchuenche and Nyimua S. Mbare is the fact that it may be used to sole problems without resorting to a new frequency domain, because it preseres scales and units properties. Indeed, The Sumudu transform, is still not widely known, nor used. Haing scale and unit-presering properties, the Sumudu transform may be used to sole intricate problems in engineering mathematics and applied sciences [1] without resorting to a new frequency domain. As shown in [6], the theory of Sumudu transform, defined for functions of exponential order, is adequate for many applications in mathematics (ordinary and partial differential equations), and control engineering problems. Belgacem et al., [3], showed that the Sumudu transform has deeper connections with the Laplace transform. Our approach is somewhat different as we consider its double transform counterpart. Watugala [7] extended the transform to functions of two ariables with emphasis on solutions to partial differential equations, but his approach is slightly different from ours. Thus, the aim of this paper is to derie, we beliee for the first time, the double Sumudu transform. Motiated by the study carried in [5], where the double Laplace transform is used to sole an age-physiology dependent population model, and taking cognisance of the fact that the Sumudu transform is the theoretical dual of the Laplace transform, we ask: - Will the double Sumudu transform also rials its double Laplace dual in common differential equations problem soling? Our interest also comes from the fact that this new transform can certainly treat most problems (may they be of integral, differential, or engineering control nature) that are usually treated by the well-known and extensiely used Laplace transform [3]. Since this new transform has ery special and useful properties, and can help with intricate applications in sciences and engineering, we do beliee its double transform will also be a natural choice in soling problems with scale and units presering requirements. Therefore, our goal is to apply the double transform to the age- and physiology-dependent population dynamic problem studied in [5]. We recall that the Laplace transform is defined by, L[f(t)] = e st f(t)dt, Re(s) >, (1.1) while the Sumudu transform is defined oer the set of functions A = {f(t) M,τ 1,τ 2 >, f(t) <Me t τ j,ift ( 1) j [, ), (1.2)

3 Double Sumudu Transform 33 by F (u) =S[f(t)] = f(ut)e t dt, u ( τ 1,τ 2 ). (1.3) In the sequel, and by analogy with the double Laplace transform, we shall denote the double Sumudu transform by S The Double Sumudu Transform The Sumudu transform offers a straightforward and relatiely uncluttered way of introducing the double Sumudu transform, proided the function has a power series transformation with respect to its ariables. The double Laplace transform of a function of two ariables defined in the positie quadrant of the xy plane [see 4] is gien by L 2 [f(x, y); (p, q)] = f(x, y)e (px+qy) dxdy, where p and q are the transform ariables for x and y, respectiely. Definition 2.1 Let f(t, x); t, x IR + be a function which can be expressed as a conergent infinite series, then, its double Sumudu transform is gien by F (u, ) =S 2 [f(t, x); (u, )] = S[S{f(t, x); t u}; x ] [{ 1 } ] = S e t u f(t, x)dt ; x u = 1 e ( u t + x ) f(t, x)dtdx. (2.1) We present the applications of the double Sumudu transform to some special functions, similar to those obtained in the solution of equations of population dynamics with age structure. Howeer, it is a triial exercise to show that the double Sumudu and Laplace transforms are also theoretical dual. That is; ( )] 1 F (u, ) =L 2 [f(x, y); u, 1, (2.2) where L 2 represents the operation of double Laplace transform. Theorem 2.2 Let f(x, y),x,y IR + be a real-alued function, then, S 2 [f(x + y); (u, )] = 1 {uf (u) F()}. (2.3)

4 34 Jean M. Tchuenche and Nyimua S. Mbare The case f(x y) is more interesting from the biological point of iew where such functions are frequently encountered in mathematical biology, with f representing the population density, x the age, and y the time, or ice-ersa. The proof for the case x y is simple and robust enough, but with a tedious manipulation. We limit ourseles to the first quadrant because negatie populations are biologically irreleant. Thus, geometrically, if the line separating the first quadrant into two equal parts represents the η-axis (the lower part being represented by Q 1, and the upper part Q 2 ), while that separating both the second and fourth quadrants represents the ζ-axis (arrow pointing upwards) and ξ- axis (arrow from origin into the fourth quadrant), respectiely, then the proof is as follows: Let f be an een function, then S 2 [f(x y); (u, )] = Q 1 u + y ) dxdy Q 2 u + y ) dxdy, (2.4) while for f( ) odd, we hae S 2 [f(x y); (u, )] = 1 u + y ) dxdy Q 1 1 u + y ) dxdy. (2.5) Q 2 Let x = 1 2 (ξ + η); y = 1 (ξ η), then, 2 u + y ) dxdy = 1 Q 1 2 = Similarly, Hence, for f een, f(ξ)dξ ξ e ξ f(ξ)dξ e 1 2( 1 u + 1 )ξ 1 2( 1 u 1 )η dη = u2 F (u). (2.6) Q 2 u + y ) dxdy = S 2 [f(x y); (u, )] = 2 u F (). (2.7) uf (u)+f(), (2.8)

5 Double Sumudu Transform 35 and for odd functions S 2 [f(x y); (u, )] = uf (u) F(). (2.9) u + From equations (2.3) and (2.9), it is obious that if f is an een function, then (u + )S 2 [f(x y)] = ()S 2 [f(x + y)]. (2.1) We state without proof the following Lemmas. Lemma 2.3 Let f and g be two real-alued functions satisfying (2.1), then (i) S 2 [f(ax)g(by); (u, )] = F (au)g(b), (ii) S 2 [f(ax, by); (u, )] = F (au, b), where a and b are positie constants (the double Sumudu transform like its single counterpart is scale presering). Lemma 2.4 Let f(x), x IR + and H( ) represents the Heaiside function then (i) S 2 [f(x)h(x y); (u, ) =F (u)+( w where w = u +, and 1)F (w), (ii) S 2 [f(x)h(y x); (u, ) = ( 1 w ) F (w). The proof is simple, for instance, by rewriting the left-hand side of the equation in (ii) as 1 f(x)e x u e y dxdy, x and performing the integrations, bearing in mind that f satisfies Fubini s Theorem, the result follows. Corollary 2.5 (i) S 2 [H(x y); (u, )] = (ii) S 2 [H(y x); (u, )] = u +. u u +.

6 36 Jean M. Tchuenche and Nyimua S. Mbare Consequently, S 2 [f(x)h(y x)] + S 2 [f(x)h(x y)] = F (u). (2.11) The application of the double Sumudu transform to partial deriaties is as follows: Let f(,a)=f (a), then [ ] f S 2 f(t, a); (u, ) = 1 e ( t u + a ) f(t, a)dtda t t = 1 { e a 1 } e t u f(t, a)dt da. u t The inner integral gies (see [6]), F (u, a) f(,a), (2.12) u [... f(t, a) S 2 t ] ;(u, ) Also, [ ] f(t, a) S 2 ;(u, ) a = 1 u { 1 e a 1 } F (u, a)da e a f (a)da = 1 u {F (u, ) F ()}. (2.13) = 1 { e a 1 } e t u f(t, a)dt da u a = 1 e a d F (u, a)da da = F (u, ), (2.14) alternatiely, [ ] f(t, a) S 2 ;(u, ) a = 1 ( e t 1 ) u e a f u a da dt = 1 e t 1 u [F (t, ) f(t, )]dt u = 1 (F (u, ) F (u)), (2.15) where F (u, ) = F (u) and F (,)=F (). It is obious from equations (2.14) and (2.15) that, F (u, ) = F (u, ) F (u).

7 Double Sumudu Transform 37 If u and are equal, we obtain a special case of the double Sumudu transform, herein referred to as the iterated Sumudu transform, by analogy to the iterated Laplace transform [3]. Thus, the iterated Sumudu transform of any gien function of two ariables f(x, y), say, is defined by J[f(x, y); (u, u)] = S 2 [f(x, y); (u, u)] = 1 x+y e ( u 2 u ) f(x, y)dxdy. (2.16) Therefore, the Sumudu transform of the general conolution function x f (2) (x) := f(x y, y)dy is related to the iterated Sumudu transform as follows: S[f (2) (x); u] = 1 e x u f (2) (x)dx = 1 e x u f(x y, y)dxdy. (2.17) u u By letting x = s + t and y = t in equation (2.17), we obtain S[f (2) (x)] = 1 s+t f(s, t)e ( u ) dsdt = us2 [f(s, t); (u, u)] = uj[f(s, t); (u, u)], u Ω (2.18) with the conolution integral taken in the classical sense, and x, y, s, t IR Applications In this section, we establish the alidity of the double Sumudu transform by applying it to an eolution equation of population dynamic, namely the famous Kermack-Mackendrick Von Foerster type model. Let f be the population density of indiiduals aged a at time t, λ the death modulus. Then population eoles according to the following system f t + f a + λ(a)f =, x f(,a)=f (a), (4.1) f(t, ) = B(t). Taking the double Sumudu transform of equation (4.1) with with u, as the transform ariables for t, a, respectiely, after some little arrangements, we obtain [ ] F ()+uf (u) [F (u, ); (t, a)] = (4.2) u + + λ In order to find the inerse double Sumudu transform of equation (4.2), which we assume it exists and satisfy conditions of existence of the double Laplace

8 38 Jean M. Tchuenche and Nyimua S. Mbare transform, we proceed as follows: Let the right-hand side of equation (4.2) be written as F ()+uf (u) = F ()+uf (u) then, taking the inerse u + + λ (u + )(1 + λ ), u+ double Sumudu transform of (4.2) using Corollary (2.5) and Lemma (1.3(i)), we hae: [ ] S2 1 F ()+uf (u) [F (u, ); (t, a)] = S 1 2 (u + )(1 + λ ) u+ = S2 1 [ u + F ()+ u u + F (u)]s2 1 λ (1 + u + ) 1 H(t a)b(t a)e ( λa) + H(a t)f (a t)e ( λt) (4.3) We obtain an approximate solution, but it is important to not here that the surial function exp( λ ) does not disappear as in [5]. Thus, in order to obtain for instance e ( λa), we assume without loss of reality that u = 1 in the expansion, which gies us an approximation, whence the inequality in (4.3). Comparing the aboe result with that obtain in [5], we conclude that the double Sumudu transform also rials his double Laplace counterpart. Acknowledgements. Thanks to M.A. Asiru of the Federal Polytechnic, Bida, Nigeria, who brought this transform to our attention. References 1. Asiru, M.A., (21) Sumudu Transform and the Solution of Integral Equations of Conolution type. Int. J. Math. Educ. Sci. Technol. 32(6), Asiru, M.A.,(22) Further Properties of the Sumudu Transform and its Applications. Int. J. Math. Educ. Sci. Technol. 33(2), Belgacem, F.B.M, Karaballi, A.A, Kalla, L.S (23) Analytical Inestigations of the Sumudu Transform and Applications to Integral Production Equations. Math. Probl. Engr. 3 (23), Sneddon, I.N. The Use of Integral Transforms (New York: McGraw-Hill) Tchuenche, J.M. (23) Solution of an Age-Physiology Dependent Population Dynamic Problem using Double Laplace Transform, Proc. African Mathematical Union Int. Conf. of Mathematical Scientists (AMU-ICMS), , Uniersity of Agriculture, Abeokuta, Nigeria, No , 23.

9 Double Sumudu Transform Watugala, G.K.,(1993) Sumudu Transform - a new integral transform to sole differential equations and control engineering problems. Int. J. Math. Educ. Sci. Technol. 24(1), 35-43; (also see Math. Engr. Indust. 6(4), (1998), ). 7. Watugala, G.K., (22) The Sumudu transform for functions of two ariables, Math. Engr. Indust. 8(4), 293? Weerakoon, S.,(1994) Applications of Sumudu Transform to Partial Differential Equations. Int. J. Math. Educ. Sci. Technol., 25(2), Weerakoon, S., (1998) Complex inersion formula for Sumudu transform, Int. J. Math. Edu. Sci. Techn. 29(4), 618? Belgacem, F.B.M, and Karaballi, (26) Sumudu transform fundamental properties inestigations and applications, J. Appl. Math. Stoch. Anal Receied: May 22, 26