A new integral transform on time scales and its applications

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1 Agwa et al. Advances in Difference Eqations 202, 202:60 RESEARCH Open Access A new integral transform on time scales and its applications Hassan Ahmed Agwa, Fatma Mohamed Ali and Adem Kılıçman 2* * Correspondence: akilicman@ptra.pm.ed.my 2 Department of Mathematics and Institte of Mathematical Research, Universiti Ptra Malaysia (UPM), UPM, Serdang, Selangor, Malaysia Fll list of athor information is available at the end of the article Abstract Integral transform methods are widely sed to solve the several dynamic eqations with initial vales or bondary conditions which are represented by integral eqations. With this prpose, the Smd transform is introdced in this article as a new integral transform on a time scale T to solve a system of dynamic eqations. The Smd transform on time scale T has not been presented before. The reslts in this article not only can be applied on ordinary differential eqations when T R, difference eqations when T N 0, bt also, can be applied for q-difference eqations when T q N 0, where q N 0 : {q t : t N 0 for q > } or T q Z : q Z {0} for q> (which has important applications in qantm theory) and on different types of time scales like T hn 0, T N 2 0 and T T n the space of the harmonic nmbers. Finally, we give some applications to illstrate or main reslts. 200 Mathematics Sbject Classification. 44A85; 35G5; 44A35. Keywords: Smd transform, time scales, dynamic eqation Introdction In the literatre there are several integral transforms that are widely sed in physics, astronomy as well as in engineering. Watgala [,2 introdced a new integral transform and named it the Smd transform that is defined by the formla F() S[f (t); 0 e t f (t)dt, ( τ, τ 2 ), (:) and applied it to find the soltion of ordinary differential eqations in control engineering problems. It appeared like the modification of the well known Laplace transform L[f(t);, where L[f (t); 0 e t f (t)dt. (:2) However in [3,4, some fndamental properties of the Smd transform were established. By looking at the properties of this transform one can notice that the Smd transform has very special and sefl properties and it can help with intricate applications in the sciences and engineering. For example, in [5, the Smd transform was extended to distribtions (generalied fnctions) and some of their properties were also stdied in [6,7. Recently Kılıçman et al. applied this transform to solve a system 202 Agwa et al; licensee Springer. This is an Open Access article distribted nder the terms of the Creative Commons Attribtion License ( which permits nrestricted se, distribtion, and reprodction in any medim, provided the original work is properly cited.

2 Agwa et al. Advances in Difference Eqations 202, 202:60 Page 2 of 4 of differential eqations, see [8. Frther in [9, a system of fractional linear differential eqations were solved analytically by sing a new method which was named fractional Smd transform. We note that by sing the Smd transform techniqe we can redce the comptational work as compared to the classical methods while still maintaining the high accracy of the nmerical reslt. We also note that an interesting fact abot the Smd transform is that the original fnction and its Smd transform have the same Taylor coefficients except for the factor n!. Ths, if f (t) a n t n,then F() n!a n t n ; see [0. Frthermore, Laplace and Smd transforms of the Dirac n0 delta fnction and the Heaviside fnction satisfy: and S 2 [H(x, t) 2 [δ(x, t), S 2 [δ(x, t) 2 [H(x, t) v. In this stdy, athors prpose is to introdce the Smd transform on a time scale and show the applicability of this interesting new transform and its efficiency in solving the linear system of dynamic eqations and integral eqations. Assme that T is a time scale sch that sp T and fix T. Let R (the set of regressive fnctions), then R and e (t, ) is well defined. The Laplace transform of the fnction f : T R was defined by L{f }() : f (t)e σ (t, ) t (:3) for Î Ω{f}, where Ω{f} consists of all complex nmbers R for which the improper integral exists, (see [-5). 2 Main reslts Definition 2. [5 The fnction f : T R is said to be of exponential type I if there exist constants M, c>0 sch that f(t) Me ct. Frthermore, f is said to be of exponential type II if there exist constants M, c>0 sch that f(t) Me c (t, ). Definition 2.2 Assme that f : T 0 R is a rd-continos fnction, then the Smd transform of f is S{f }() : f (t)e σ (t, ) t (2:) for Î D{f}, where D{f} consists of all complex nmbers R for which the improper integral exists. Theorem 2. (Linearity) Assme that S{f} ands{g} existfor Î D{f} and Î D{g}, respectively, where f and g are rd-continos fnctions on T and a and b are constants. Then S{αf + βg}() αs{f }() + βs{g}() (2:2) n0

3 Agwa et al. Advances in Difference Eqations 202, 202:60 Page 3 of 4 for Î D{f} D{g}. Proof. The proof follows directly from Definition 2.2. We will assme that T is a time scale with bonded graininess, that is, 0 <µ * µ(t) µ * for all t T. Let H denotes the Hilger circle, see [5, given by { H H t C :0< + μ(t) < }, μ(t) { H min C :0< + μ < } μ, { H max C :0< + < }. μ μ It is clear that H min H t H max. To give an appropriate domain for the transform, which of corse is tied to the region of convergence of the integral in (2.), for any c >0 define the set D { ( ) } C :Re μ > Re μ (c) forallt T { C : ( ) } Hc max and Re μ > Re μ (c) forallt T { C : H and Re μ ( ) } > Re μ (c) forallt T, where H c denotes the complement of the closre of largest Hilger circle corresponding to µ.. Note that if µ * 0 this set is a right half plane; see [5. max Lemma 2. If ( ) H and Re μ > Re μ (c) for all t T, then + μ(t) and ( ) c H. Proof. Since H then + < μ(t) μ(t) which implies ( ) since Re μ > Re μ (c) implies +μ(t) > +cμ(t), we see ( ) c + μ(t) +cμ(t) ( μ(t) + μ(t) ) < μ(t). + μ(t).also, Theorem 2.2 (Domain of the transform). The integral f (t)e σ (t, ) t converges absoltely for Î D if f(t) is of exponential type II with exponential constant c. Proof. If H, then + μ(t).

4 Agwa et al. Advances in Difference Eqations 202, 202:60 Page 4 of 4 Note that f (t)e σ (t, ) t M M e c(t, )e σ (t, ) t e c(t, ) t + μ(t) ( + μ )M γ γ where g (+µ. )M and α exp exp t t γ e αt dt +cμ log + μ. μ e c(t, ) t log ( + c) μ(s) s t μ(s) μ(s) log +cμ(s) s t ( + μ )M, α + μ(s) The same estimates sed in the proof of the proceeding theorem can be sed to ( ) show that if f(t) is of exponential type II with constant c and Re μ > Re μ (c), then lim e t (t, ) f (t) 0. In the following theorem, we state the relationship between Smd transform and Laplace transform: Theorem 2.3 Assme that f : T R is rd-continos fnction, then S{f }() L{f } ( ) (2:3) for Î D{f}, where D{f} consists of all complex nmbers R for which the improper integrals in (.3) and (2.) exist. Proof. By sing the definition of the transform we obtain L{f } ( ) f (t)e σ (t, ) t S{f }(). (2:4) Theorem 2.4 If T N 0, then S{f }() ( ) + Z{f } + (2:5)

5 Agwa et al. Advances in Difference Eqations 202, 202:60 Page 5 of 4 where Z{f} is the Z-transform of f, which is defined by Z{f }() t0 f (t) t (2:6) for those complex vales of for which this infinite sm converges. Proof. The proof follows directly from Theorem 2.3 and the relation ( +) L { f } () Z { f } ( +). Example 2. In this example, we find the Smd transform of f(t). From Definition 2.2 we have S{}() Therefore, we get e σ (t, ) t [ [ μ(t)e (t, )+e (t, t 0 ) t μ(t) e (t, )+e (t, t 0 ) t [ +μ(t) [ + μ(t)( ) +( )μ(t) [ ( ) +( )μ(t) ( ) e e (t, t 0 ) t e (t, t 0 ) t e (t, t 0 ) t (t, ) t e (t, ). S{} (). (2:7) Example 2.2 In particlar, S{e a (t, )}() is given by For, S{e α (t, )}() S{e α (t, )}() L{e α(t, )} From the above example, we have S{cosh α (t, )}() α. (2:8) α 2 2, ( ) ( ) α α.

6 Agwa et al. Advances in Difference Eqations 202, 202:60 Page 6 of 4 and α S{sinh α (t, )}() α 2 2, S{cos α (t, )}() S{sin α (t, )}() α 2 2, α α 2 2. Example 2.3 In the following, we make se of (2.5) and (2.6) to find S{a t }() where t T N 0. In fact, ( ) t S{α t α }() + t + +( α), α +. The following reslts are derived sing integration by parts. Theorem 2.5 Assme that f : T C is of exponential type II sch that f Δ is a rdcontinos fnction and lim e t (t, ) f (t) 0, then S{f }() S{f }() f (). (2:9) Proof. Integration by parts yields S{f }() Remark 2. Similarly, f (t)e σ (t, ) t [ f (t)e (t, t 0 ) + 2 S{f }() f (). f (t)e σ (t, ) t S{f }() 2 S{f }() 2 f () f ( ). More generally, we have S{f n }() n n S{f }() n k f k ( ). k0 This is becase from Theorem 2.5, we have S{f n + }() S{(f n ) }() S{f n }() f n ( ) [( ) n n S{f }() n k f k ( ) f n ( ) k0 n n+ S{f }() n k+ f k ( ). k0

7 Agwa et al. Advances in Difference Eqations 202, 202:60 Page 7 of 4 Theorem 2.6 Assme h : T C is a rd-continos fnction. If H(t) : t for t T, then h(τ) τ (2:0) S{H}() S{h}() (2:) for those regressive C {0} satisfying lim t {H(t)e (t, )} 0. (2:2) Proof. Integrating by parts we have S{H}() [ e σ (t, )H(t) t ( e H( )+ ) e σ (t, )H(t) t t (t, )H(t) + S{h} (). e σ (t, )h(t) t e σ (t, )H (t) t Remark 2.2 One can get the reslt in (2.) from (2.3) and the relation L{H(t)}() L{h(t)}(). Remark 2.3 From (2.) we have S {t} (), S{t 2 } 2 + S{σ (t)} (). Theorem 2.7 Assme f : T C is a rd-continos fnction and L{f(t)}() F L (). Then L{e σ α (t, )f (t)}() F L ( α). (2:3) Proof. Since e σ α (t, )e σ (t, )e σ ( α) (t, )

8 Agwa et al. Advances in Difference Eqations 202, 202:60 Page 8 of 4 and L{f(t)}() F L (), then L{e σ α (t, )f (t)}() e σ (t, )(e σ α (t, )f (t)) t e σ ( α) (t, )f (t) t F L ( α). From (2.3) and (2.3) we have the following reslt. Theorem 2.8 Assme f : T C is a rd-continos fnction and S{f(t)}() F S (). Then S{e σ α (t, )f (t)}() ( ) ( ) F L α ( α) F S ( α). (2:4) In [5, the convoltion of two fnctions f, g is defined by (f g)(t) t where ˆf is the shift of f and ˆf (t, σ (s))g(s) s for t T0, L { f g } () L{f } () L{g} (). Remark 2.4 When T R, then (f g)(t) t f (t s)g(s)ds which coincides with the classical definition. In the following, we present the relation between the Smd transform of the convoltion of two fnctions on a time scale T and the prodct of Smd transform of f and g. Theorem 2.9 Assme that f, g are reglated fnctions on T, then S { f g } () S { f } () S { g } (). Proof. Since S{f g}() ( ) L{f g} ( L{f } ( ( )) ( L{f } L{g} S { f } () S { g } (). ) L{g} ( )) ( ) 3 Applications I. To find the soltion of a homogeneos dynamic eqation with constant coefficients in the form

9 Agwa et al. Advances in Difference Eqations 202, 202:60 Page 9 of 4 n a k y k 0. k0 Applying Smd transform we get n [ a 0 S{y}()+ a k k S{y}() k y(0) k y (0) y k (0) 0, k or, ( n k0 ) a k k S{y}() n k k a k i0 y i (0) k i ψϕ(0), (3:) where a a 2 a n ψ ( a 2 a 3 a n 0.. ) a 3 a n a n 0 a n 0 0 ( ) T ϕ(0) y(0)y (0)...y n (0) Example 3. Consider the following dynamic eqation y (t) y (t) 6y(t) 0, y(0),y (0) 7, (3:2) then a 0 6, a, a 2. Conseqently from (3.) we have ( 6 + ) 2 S{y}() ( ) ( )( ) ; hence, S{y}() Therefore, y (t) e 3 (t,0) +2e 2 (t,0). (3:3) Remarks () When T R the Eqation (3.2) becomes y (t) y (t) 6y(t) 0, y(0), y (0) 7 and then from (3.3) its soltion is y (t) e 3t +2e 2t. (2) When T Z the Eqation (3.2) becomes as a difference eqation y (t) y (t) 6y (t) 0, y (0), y (0) 7

10 Agwa et al. Advances in Difference Eqations 202, 202:60 Page 0 of 4 and from (3.3) it has a soltion y (t) (4) t +2( ) t, t Z. (3) When T q N 0 {0} the Eqation (3.2) becomes y q q (t) y q (t) 6y(t) 0, y(0), y q (0) 7 where y y(qt) y(t) q(t), t Then from (3.3), we have (q )t y() 6, y(t) [ + 3(q )s+2 [ 2(q )s, t >. s [,t) s [,t) II. To find the soltion of a system of dynamic eqations with constant coefficients in the form P (D) Y F, where, P (D)...P n (D) P(D) P n (D)...P nn (D) P ij (D) n i a ik D k, D k y y k. k0 Analog to (3.), we have y, Y y n f, F f n Since S{P(D)Y}() (S{P ij (D)y j }()) S{F}(). S{P ij (D)y j }() a i0 S{y j }()+ n i [ a ik k S{y j}() k y j(0) k y j (0) y k j (0) k ( ni n i k0 ) a ik k S{y j }() k k a ik l0 y l j (0) k l and n i k k a ik l0 y l j (0) k l 2 n i T ij j (0), a i a i2 a ini a i2 a i3 a ini 0 a i3 a i4 0 a ini 0 a ini 0 0 y j (0) y j (0) y n i j (0) j (0) ( y j (0)y j (0)...y n j (0)) T,

11 Agwa et al. Advances in Difference Eqations 202, 202:60 Page of 4 then, [( ni ) [ a ik ni k k S{y j }() a ik i.e., k0 k l0 y l j (0) k l + S{F}() ( ( )) P (S{Y}()) n n m ( (0)) m +(S{F}()) n (3:4) n n where, Ψ is n m matrix, m S{Y} n j d j, d j max i n {deg P ij(d)}. Conseqently, [ ( ) [ P (0) + S{F}(). (3:5) Example 3.2 To solve the following system of two dynamic eqations we have, Since x +3y +2x 2e 3 (t, 0),x(0), x (0) 5, P y 4x +3y 9cos 2 (t, 0), y(0) 2, y (0) 3, ( ) , S{Y} S 3+ 2 x(0) (0) x (0) y(0) 5 2, y (0) 3 [ [ S{F} S 2 [ [ [ [ [ {[ 2e3 (t,0) 9cos 2 (t,0) [ 2 } S{Y} P ()[ (0) + S{F}(). [ {[ } x, y 22 [ Then [ S{Y} ( 3)(+4 2 )

12 Agwa et al. Advances in Difference Eqations 202, 202:60 Page 2 of 4 Conseqently, [ x y S {[ } [ e3 (t,0)+sin 2 (t,0). e 3 (t,0)+cos 2 (t,0) i.e., x(t) e 3 (t, 0) + sin 2 (t, 0) and y(t) e 3 (t, 0) + cos 2 (t, 0). II. To find the soltion dynamic integral eqation we provide the following example. Example 3.3 Consider the integral eqation t h(t) e 2 (t, 0)+4 h(τ) τ. 0 Applying the Smd transform we get S{h}() Then we get S{h}() +4S{h} (). 2 ( 2)( 4) and conseqently from (2.8), we have h(t) e 2 (t, 0) +2e 4 (t, 0). Example 3.4 For solving the following eqation x(t) cos 2 (t, 0) +3(x(t) sin 3 (t, 0)). Applying Smd transform we get [ 9 Then S{x} x(t) 4 S {[ } (9 5cos 2 (t, 0)), III. Assming that T has constant graininess µ(t) µ, wecanfinds(e σ α (t, 0)), S(e β (t, 0)e σ α (t, 0)), S(e β(t, 0)e σ α (t, 0)), S(eσ α (t, 0)cos β(t, 0)) and S(e σ α (t, 0)sin β(t, 0)). From (2.4), we have S(e σ α (t, 0)) S(te σ α (t, 0))() ( α), (3:6) ( α) 2, (3:7) S(e β (t, 0)e σ α (t, 0))() ( α) β, (3:8)

13 Agwa et al. Advances in Difference Eqations 202, 202:60 Page 3 of 4 S(e σ α (t, 0)cos β(t, 0)) ( ( α) α) 2, (3:9) + β 2 and S ( e σ α (t, 0)sin β(t, 0) ) β ( α)2 + β 2. When T R (as a special case) the reslts in (3.6), (3.7), (3.8), and (3.9) become S{e αt }() S{te t }() S{e βt e αt }() (+α), (+α) 2, +(α β), S{e αt ( + α) cos βt}() (+α) 2 + β 2 2 which coincide with previos reslts in [3. Ths it seems that the present reslts are more general. 4 Conclsion In this article, the Smd transform is introdced as a new integral transform on a time scale T in order to solve system of dynamic eqations. Frther, Smd transform on time scale T is not presented before. The reslts in this article not only can be applied to ordinary differential eqations when T R, difference eqations when T N 0, bt also, can be applied for q-difference eqations when T q N 0,where q N 0 : {q t : t N 0 for q > } or T q Z : q Z {0} for q> which has several important applications in qantm theory and on different types of time scales like T hn 0, T N 2 0,andT T n the space of the harmonic nmbers. Regarding the comparison between Smd and Laplace transform, see for example, [4-7,6. For example when T R, Maxwell s eqations were solved for transient electromagnetic waves propagating in lossy condcting media, see [6 where the Smd transform of Maxwell s differential eqations yields a soltion directly in the time domain, which netralies the need to perform the inverse Smd transform. Acknowledgement The athors express their sincere thanks to the referee(s) for the carefl and details reading of the manscript and very helpfl sggestions that improved the manscript sbstantially. Athor details Department of Mathematics, Ain Shams University, Faclty of Edcation, Roxy, Cairo, Egypt 2 Department of Mathematics and Institte of Mathematical Research, Universiti Ptra Malaysia (UPM), UPM, Serdang, Selangor, Malaysia Athors contribtions All athors contribted eqally to the manscript and read and approved the final draft.

14 Agwa et al. Advances in Difference Eqations 202, 202:60 Page 4 of 4 Competing interests The athors declare that they have no competing interests. Received: December 20 Accepted: 0 May 202 Pblished: 0 May 202 References. Watgala, GK: Smd transform: a new integral transform to solve differential eqations and control engineering problems. Int J Math Ed Sci Technol. 24(), (993). doi:0.080/ Watgala, GK: Smd transform: a new integral transform to solve differential eqations and control engineering problems. Math Eng Indstry. 6(4), (998) 3. Belgacem, FBM, Karaballi, AA: Smd transform fndamental properties investigations and applications. Jornal of Applied Mathematics and Stochastic Analysis2006, 23. Article ID 9083, doi:0.55/jamsa/2006/ Kılıçman, A, Eltayeb, H: On a new integral transform and differential eqations. J Math Probl Eng 200, 3 (200). Article ID , doi:0.55/200/ Eltayeb, H, Kılıçman, A, Fisher, B: A new integral transform and associated distribtions. Int Trans Spec Fnc. 2(5), (200). doi:0.080/ Kılıiçman, A, Eltayeb, H: A note on integral transforms and partial differential eqations. Appl Math Sci. 4(3), 09 8 (200) 7. Kılıçman, A, Eltayeb, H: On the applications of Laplace and Smd transforms. J Franklin Institte. 347(5), (200). doi:0.06/j.jfranklin Kılıçman, A, Eltayeb, H, Ravi Agarwal, P: On Smd transform and system of differential eqations. Abstract and Applied Analysis 200, (200). Article ID , doi:0.55/200/ Kadem, A, Kılıçman, A: Note on transport eqation and fractional Smd transform. Compt Math Appl. 62(8), (20). doi:0.06/j.camwa Zhang, J: A Smd based algorithm for solving differential eqations. Compt Sci J Moldova. 5(3), (2007). Ahrendt, C: The Laplace transform on time scales. Pan Am Math J. 9(4), 36 (2009) 2. Bohner, M, Gseinov, GSh: The convoltion on time scales. Abstr Appl Anal 2007, 24 (2007). Article ID Bohner, M, Peterson, A: Dynamic Eqations on Time Scales: An Introdction with Application. Birkhäser, Boston, MA (200) 4. Bohner, M, Peterson, A: Advances in Dynamic Eqations on Time Scales. Birkhäser, Boston, MA (2003) 5. Davis, JM, Gravagne, IA, Jackson, BJ, Marks, RJ, Ramos, AA: The Laplace transform on time scales revisited. J Math Anal Appl. 332, (2007). doi:0.06/j.jmaa Hssain, MGM, Belgacem, FBM: Transient Soltions of Maxwell Eqations based on Smd transform. Progress Electromag Res PIER. 74, (2007) doi:0.86/ Cite this article as: Agwa et al.: A new integral transform on time scales and its applications. Advances in Difference Eqations :60. Sbmit yor manscript to a jornal and benefit from: 7 Convenient online sbmission 7 Rigoros peer review 7 Immediate pblication on acceptance 7 Open access: articles freely available online 7 High visibility within the field 7 Retaining the copyright to yor article Sbmit yor next manscript at 7 springeropen.com

A Note on the Shifting Theorems for the Elzaki Transform

Int. Jornal of Mh. Analysis, Vol. 8, 214, no. 1, 481-488 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.214.4248 A Note on the Shifting Theorems for the Elzaki Transform Hwajoon Kim Kyngdong