On the Connections Between Laplace and ELzaki Transforms

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1 Adances in Theoretical and Applied Mathematics. ISSN Volume 6, Number (), pp. Research India Publications On the Connections Between Laplace and ELzaki Transforms Tarig. M. Elzaki and Salih M. Ezaki Math.Dept. Sudan Uniersity of Science and Technology ( Abstract This paper discusses some relationship between Laplace transform and the new transform Called ELzaki transform, we sole first and second order ordinary differential equations using both transforms, and showing The ELzaki transform is closely connected with The Laplace transform. Keywords: ELzaki transform. Laplace transforms. Differential Equation. Introduction The definitions, Properties and applications of the new integral transform (ELzaki transform) to ordinary differential equations are described in this paper. The ELzaki transform can be used to sole ordinary differential equation and Control engineering Problems. The ELzaki transform easier than the Laplace transform for beginners to understand and use, also this transform can still sere as an auxiliary method to the Laplace transform. A lot of work has been done on the theory and applications of transforms such as Laplace-Fourier-Melin- Hankel and Sumudu, to name a few, but ery little on the power series transformation or ELzaki transform, probably because it is little known,and not widely used, the ELzaki transform rials the Laplace transform in problem Soling, its main adantage is the rials that it many be used to sole problems without resorting to anew frequency domain because it presere scales and units properties, the ELzaki transform may be used to sole intricate problems in engineering, mathematics and applied science without resorting to a new frequency domain. The theory of ELzaki transform defined for functions of exponential order is adequate for many applications in mathematics, (ordinary and partial differential equations). The ELzaki transform has deeper connections with the Laplace transform.

2 Tarig. M. Elzaki and Salih M. Ezaki or We recall that the Laplace is defined by st () () L f t = e f t dt, Re s > While the ELzaki transform is defined by the following formulas. t () =Ε, =, [, ] T f t f t e dt k k t, () T =Ε f t = f t e dt k k > For any function, or by the conersion rule n+ T = n! a (3) For function f n= n t which can be expressed as a polynomial or as a conergent in finite series for t > applied it to the Solution of ordinary conergent equations and control engineering Problems. The sufficient conditions for the existence of the ELzaki transform are that of exponential order. This means that the ELzaki transform may or may not exist. Theorem (-): Let T ( ) is the ELzaki transform of f() t : f ( t τ), t τ ( f () t ) T Ε = and g( t) =, t < τ τ () Ε g t = e T t ( τ ) Ε g t = e f t dt τ Let t = λ+ τ we find that: λ+ τ ( λ) e f dλ = which is the desired result τ τ e λ e f d e T ( λ) λ =

3 On the Connections Between Laplace 3 The ELzaki transform can certainly treat all problems that are usually treated by the well- known and extensiely used Laplace transform. Indeed as the next theorem shows the ELzaki transform is closely connected with the Laplace transform F ( s ). Theorem (-): Let () f t M, k, k >, such that f () t A= With Laplace transform, T = F Let: f () t t / k j i f () t < Me, if t ( ) [, ) F s the ELzaki transform T ( ) of f t () = A. for k < < k T e f t dt t is gien by Let w= t then we hae: w w dw T = e f ( w) = e f ( w) dw = F. Also we hae that T () = F () so that both the ELzaki and Laplace transforms must coincide at = s =. In fact the connection of the ELzaki transform with the Laplace transform goes much deeper, therefore the rules of F and T in (4) can be interchanged by the following corollary. (4) Corollary (-3) Let f () t haing F and T for Laplace and ELzaki transforms respectiely, then: F( s) = st (5) s This relation can be obtained from ( 4 ) by taking = s The equations (4) and (5) form the duality relation goerning these two transforms and may sere as a mean to get one from the other when needed.

4 4 Tarig. M. Elzaki and Salih M. Ezaki ELzaki Transform of Deriaties and Integrals Being restatement of the relation ( 4 ) will sere as our working definition, since the Laplace transform of sin t is then iew of (4), its. ELzaki transform is + s 3 Ε [ sint ] = this exemplifies the duality between these two transforms. + Theorem (-) Let F ( s) and T () be the Laplace and ELzaki transforms of the deriatie of f () t. : T () () i T = f() (6) (7) ( n ) T ( n ) n+ k ( k ii T = f ) ( ), n n k = T and F s are the ELzaki and Laplace transforms of the nth n n Where ( n f ) ( t ) of the function f ( t ). deriatie : (i) Since the Laplace transform of the deriaties of f() t is F () s = sf() s f() T = F = F f = F f T = f (ii) By definition, the Laplace transform for f ( n ) () t is gien by Therefore n ( n ) n n ( k+ ) ( k ) F s S F s S f = k = F n f F = ( k ) ( ) n n k + k =

5 On the Connections Between Laplace 5 Now, since ( k ) =, n ( n ) T( ) n+ k ( k T = f ) ( n ) ( k ) T F for k m we hae k = Theorem (-) Let T ( ) and F ( s) denote the ELzaki and the Laplace transforms of the definite integral of f ( t ). t () = ( τ) h t f dτ. = = T E h t T = = By the definition of Laplace transform F ( s) L h( t) F ( s) s Hence T = F = F = F = T Theorem (-3) Let T ( ) is the ELzaki transform of the function dt ( ) ( i ) E t f ( t) = T ( ) d d d 4 ( ii ) E t f ( t ) = T ( ) From the definition of ELzaki transform we hae. f t then:

6 6 Tarig. M. Elzaki and Salih M. Ezaki dt d i = T = e f t dt= d d t () () t t t e f () t dt = e ( t f () t ) dt+ e f () t dt = E ( t f ( t) ) + E ( f ( t) ) dt = d Where that: E f ( t) = T ( ) The proof of (ii) is similar E tf t T Theorem (-4) If E f () t = T then: d T ( ) T ( ) ( i ) E t f ( t) = f f d d T ( ) T ( ) ( ii) E tf ( t) = f f f f d 4 d T ( iii) E t f ( t) = f f d (i) From theorem (-3) we hae d E t f ( t) = E ( f ( t) ) E f ( t) = d d T T f f d The proof of (ii) and (iii) are similar to that of (i) Theorem (-5) (shift) f t A Let with ELzaki transform T ( ).

7 On the Connections Between Laplace 7 at E e f () t = T a a From definition of ELzaki transform we hae: at ( a) t E e f () t = f ( t) e dt Let = ( ) = ( ) w a t dw a dt, w w f e dw = T a a a a Theorem (-6) (conolution) Let f () t and g() t be defined in A haing Laplace transforms F( s ) and Gs () and ELzaki transforms M ( ) and N () the ELzaki transform of the Conolution of f and g ( * ) = ( τ) f g t f t g t dτ Is gien by: E ( f * g) ( t) = M N The Laplace transform of ( f * ) L ( f * g) = F( s) G( s) g is gien by: By the duality relation ( 4 ) we hae: = and since E f * g ( t) L f * g ( t), M = F, N = G E ( f * g) ( t) = F. G M N. = M N

8 8 Tarig. M. Elzaki and Salih M. Ezaki Solution of Ordinary Differential Equations. Example (3-) Consider the first order Differential equation: dx px f () t, t dt + = > (8) With the initial Condition x = a Where panda are Constants and f ( t ) is an external input function so that its Laplace transform and ELzaki transform exist First Solution by Laplace transforms s x s x + px s = f s x s ( s) a f = = s + p s + p Where that x ( s) and f ( s) are Laplace transform of x ( t) and f ( t). pt = + ( τ) t p τ x t ae f t e dτ Thus the solution naturally splits into two terms the first one Corresponds to the response of the initial condition and the second one is entirely due to the external input function f ( t ). In particular if f () t = c c is the constant there the Solution of ( 8 ) becomes c c x t a e p p pt () = + Second Solution by ELzaki Transfrom 8 we hae: Using ELzaki transform of equation X x + PX = F (9) By applying the initial condition we hae F a X = + + P + P () Where that X ( ) and F ( ) are ELzki transform of x ( t) and f ( t ). By the inerse of ELzaki transform and Conolution theorem( 6) we find that:

9 On the Connections Between Laplace 9 t p x t ae f t e τ d pt () = + ( τ) τ () Which is the same solution The first term of this solution in (9) is independent of time t and is usually called the steady state solution, and the second term depends on time t and is called the transient solution.in the limit as t the transient solution decays to zero if p > the steady state solution is attained, on the other hand, when p <, the transient solution grows exponentially as t and the solution becomes unstable. Equation ( 8) describes the Law of natural growth or decay Process with an f t according as p > or P <. external forcing function In particular, if f () t = and P> The resulting equation ( 8) occurs ery frequently in chemical Kinetics. Such an equation describes the rate of chemical reactions. Example (3-) Consider the Ordinary differential equation with Variable Coefficients: ty ty + y =, y =, y = () Solution by Laplace Transform Using Laplace transform and apply the Conditions, we hae d Y ( s ) + Y ( s ) = ds s s ( ()) = Y( s) L y t This is a linear differential equation and the solution is c Y( s) =.[ c constant] s + s Take the inerse Laplace transform we find that: y t = + ct Solution by ELzaki Transform Take the ELzaki transform of equation d T ( ) we hae T y y y y d d T ( ) T ( ) y + y + T ( ) = d E ( y() t ) = T Now Appling the initial condition to obtain:

10 Tarig. M. Elzaki and Salih M. Ezaki Or T + 3 T = 3 T T = (3) Equation (3) is a linear differential equation, which has solution in the form 3 T = c + c constant [ ] Using inerse ELzaki transform to find y t = ct+, () Which is the same Solution Conclusions The origin of ELzaki transform is traced back to the classical Fourier integral. ELzaki transform is a conenient tool for soling differential equations in the time domain without the need for performing an inerse ELzaki transform. The connection of the ELzaki transform with the Laplace transform goes much deeper. Reference [] G.Kwatugala, Sumudu transform an integral transform to Sole differential equations Control engineering Problems.Internau J.Math, Ed. Sci. Tech.4,35-43,(993). [] G.Keatugala, Sumudu transforms- on integral transform to Sole differential equations Control engineering Problems.Math Engang. Ind 6,39-39(993). [3] Asairu, M.A Application of the Sumudu transform to discrete dynamic Systems. International Journal of Mathematical Education in Science and Technology, 34(6), PP , (3). [4] Eltayeb H. And A, Kilicman.On Some Application of A New Integral Transform, International Journal of Mathematical Analysis, 4(3), PP.3-3(). [5] Kilicman. A. and H. ELtayeb. A note on Integral Transforms and partial Differential Equations, Applied Mathematical Sciences, 4(3)pp.9-8() [6] Kilicman, A and H. E. Gadain. An Application of Double Laplace Transform and Double Sumudu transform, Lobacheskii Journal of Mathematics, Vol. 3, No.3, and PP.4-3. (9). [7] G.K.Watugala, Sumudu transform a new integral transform to sole differential equation and Control engineering Problems, Math. Engrg. Indust.6, No 4, 39-39(998). [8] Lokenath Debnath and Dambaru Bhatta, Integral Transforms And Their Applications (Second Edition),Chapman& Hall/CRC(6).

The Use of Kamal Transform for Solving Partial Differential Equations

Adances in Theoretical and Applied Mathematics ISSN 973-4554 Volume 12, Number 1 (217), pp. 7-13 Research India Publications http://www.ripublication.com The Use of Kamal Transform for Soling Partial Differential