Laplace Transform Analytical Restructure

In this paper, the Laplace transform definition is implemented without resorting to Adomian decomposition nor Homotopy perturbation methods. We show that the said transform can be simply calculated by differentiation of the original function. Various analytic consequent results are given. The simplicity and efficacy of the method are illustrated through many examples with shown Maple graphs, and transform tables are provided. Finally, a new infinite series representation related to Laplace transforms of trigonometric functions is proposed.


Introduction
Integral transforms methods have been used to a great advantage in solving differential equations.Limitations of Fourier series technique, were overcome by the extensive coverage of the Fourier transform to functions   f t , which need not be periodic [1].The complex variable in the Fourier transform is substituted by a single variable s to obtain the well known Laplace transform [1][2][3][4][5][6][7][8], a favorite tool in solving initial value problems (IVPs).The integral equation defined by Léonard Euler was first named as Laplace by Spitzer in 1878.However the very first Laplace transform applications were established by Bateman in 1910 to solve Rutherford's radioactive decay, and Bernstein in 1920 with theta functions.
For a real function   f t with variable the Laplace transform, designated by the operator, , giving rise to a function in

 
F s , in the right half complex plane, is defined by, While we completely focus on the Laplace transform, in this paper, many of the ideas herein stem from recent work on the Sumudu transform, and studies and observations connecting the Laplace transform with the Sumudu transform through the Laplace-Sumudu Duality (LSD) for and the Bilateral Laplace Sumudu Duality (BLSD) for t [9][10][11][12][13][14][15][16].Indeed, considering the   , , 0, e , if 1 0, While Analyses about the properties of the Sumudu transform, transform tables, and many of its physical applications can be found in [9][10][11][12]14,16], investigations, applications, and transform tables stemming from the Natural transform can be found in [17][18][19][20][21].This new integral transform combines both (one sided) Sumudu and (one sided) Laplace transforms by, Obviously, taking, , in the Natural transform, leads to the Laplace transform, and taking, , results in one sided Sumudu transform.We note that while the Natural can be bilateral like both Sumudu and Bilateral Laplace, when the variable is chosen positive in the definition, both Re s and the variable, , must be positive as well, as as this ought to correct a related one sided Natural Transform defintion misprint appearing in our papers [17,18].
The gist and essence of this work is solving the Laplace integral equation once by differention, and by integration by parts.Divided into two major sections, this paper in Section 2 explains the various multiple shift properties connected with the Laplace transform by just differentiating the original function.The new infinite series representation of trigonometric functions related with the Laplace transform is proved in Section 3. In consequence of our formulations and derivations, three tables are provided at the end of the Section 3 ended with concluding remarks and directions for some future work.The tables are respectively covering derivatives periods for the function E sin t  (in Table 1), 21 trignometric series expansions entries (in Table 2), and 16 main Laplace transform properties, as generated by Proposition 3 (in Table 3).Examples 1, 2, and 3 in the body of the text of Section 2, as well as Example 6 and Entry 17 of transform Table 2 in Section 3, are respectively afforded Maple graphs (see Figures 1-5), showing both the time function invoked in the corresponding example, and its resulting Laplace trasform.

Laplace Transforms by Function Differentiation
As stated earlier in the introduction, an ultimate goal of ours, among others, is calculating the Laplace transform of   f t by simple differentiation rather than usual integration.We show that we can do this, without resorting to the Adomian nor homotopy methods, ADM, and HPM, as was done in [22,23].Along with the Laplace series definition below, some elementary properties are proved.
Definition The Laplace transform (henceforth designated as F(s)) of the exponential order and sectionwise continuous function , is defined by, Remark We observe that, from the traditional Laplace transform, taking , so that , , and in Bernoulli's integration by parts, for solving the Laplace integral equation by parts?The detailed answer with analysis is given in Section 3.For simplicity,

Theorem 1 The Laplace transform of derivative of
, the proof is completed.  ; 0 The Laplace transform of antiderivative of f(t), in the domain performing the usual computations, yields the RHS of the equation above and proves out theorem.Theorem 3 For , the Laplace transform of the function From the theory of Laplace transform is given by Equation (1),  , when 2    .
Finally for the non-negative integer , after simplifim .
Theorem 4 The Laplace transform of the function and after the usual computations, Theorem 4 follows.

Example 1 As an application of Theorem 3, the Laplace transform of
, where   , denotes the first kind order one Bessel function, is calculated as follows, (graph shown in Figure 1), Now substituting the above derivatives in Equation (3), and after applying both the limits, and when the derivative of the function , with respect to is shifted by , then the Laplace transform is given by, ( Proof.The proof is simple, we have  is given by Theorem 1.
Theorem 6 For non-negative integers i and , when the derivative of the function Proof.The LHS of above equation is by Theorem 1, and after proper calculations, the proof is calculated.
Theorem 7 For , the Laplace transform of the antiderivative of the function ifted with m t , given by, , t sh is Proof.Computing the summation in the RHS of the Equation in Theorem 2 with respect to s in the domain

 
, s  , times, yields the proof.m We now establish the following results, Theorem 9 For , the Laplace transform of the derivative of . Theorem Proof.Substituting Theorem 4 in yields the expected derivatives, Therefore (Figure 3), Theorem 11 For non-negative integers and th i m , e Laplace transform of the - i th antiderivative with respect to t in the domain   From Theorem 5 through Theorem 12, there is no restriction on positive integers and , which means bot  L m i h can be same (or) different and either of the integer can less than (or) greater than to one another.
The Theorem 5 and the Theorem 9 varies only in the coefficients, that is the order of the derivative, the same holds for Theorem 6 and Theorem 10, again the Theorem 7 and Theorem 11 varies only in the coefficients, that is the order of the anti-derivative, similarly for Theorem 8 and 12. Hence we have the following propositions, respectively.

Proposition 1 If the function  
f t and its   1 i  derivative with respect to t go to zero as

Proposition 2
Copyright © 2013 SciRes.AM The following initial and final value, convo function periodicity related theorems can verified through conventional Laplace transform theory.
Theorem 13 Let the function, lution, and be easily  , Theorem 14 The Laplace transform of the convolution of two functions   f t , and,   g t , is given by,


  e equation so in the second infinite series of the abov that the limits     pletes the and after rea uating com Example 5 The full sine-wave rectifier is given by the rranging and eval function, Using Theorem 15, the Laplace transform of the full sine-wave rectifier is calculated by using the entries of column 5 of Table 1,

Laplace Transforms by Integration by Parts
The Laplace transform of is calculated by substituting Su ect to some constraints we then generally ave, osition 3 The Laplace transform of a or's Proof.Now , so that . Furthermore, in view of Proposition 3, when applying the upper and lower limits in the antiderivatives above, we get.
, whence we get, (function and its Laplace transform in Figure 4)  e agree that co roposition 3, since the continuous integration of constant and polynomials with respect to does not converge anywhere when and

New Infinite Series Representation for Trig Functions
In The inverse Laplace transform of Proposition 3 would be same as inverse Laplace transform of the ab ation, and hence it is enough to find the inverse Laplace transform of , 0 the inverse Laplace transform of be which is Dirac delta function [2] since, .
Again when  [2], and references therein for more details).
We prove the inverse Laplace transform of singular functions that satisfy the Tauberian (initial value) theorem in the following proposition where the trigonometric io ns are represen where nction, [24].
Proposition e necessary condition for the existence of Proposition 3 (and hence the above equation) is that, the Taylor serie is the first kind Bessel's function of order zero.And this particular function will play the major role in the exponential kerneled transforms (see Equations ( 30) through (35) in [20]).And the Laplace transform is taken with respect to , since v and ndependent, the perm rchange of of integration is consid ur.
Finally from Equations ( 19) and (20), ide of Equation ( 21) is zero, Since the second part of right hand s Substituting Equation (22) in Equation (15) for   . Applying Proposition 4, a tends to zero, a the s t ll and since From Equation ( 22   Finally, with in the entry 4 of Table 3, It is important to note that with respect to the entries 5 and 9 (and entries 6 and 10) of Table 3, the proposition 1 Equation (9) (and Equation ( 10)) holds true.Similarlly with respect to the entries 7 and 11 (and entries 8 and 12) of Table 3, the proposition 2 Equation (11) (and Equation (12)) remains the same.

Concluding Remarks and Future Work
As far as the Section 2 is concerned, when the function is Laplace transformed by differentiation, then the inverse Laplace is automatically an integration process.Having worked with various examples, our proposed methods lead to exact solutions.A remaining open query is that of defining the inverse for the Laplace transform by using similar tools and processes as in Proposition 3.But in vi ess of differentiating.For example consider the function , its Laplace trans-2 s s  ew of the concept of Section 2 above, Laplace and inverse Laplace transform are the respective reciprocal processes of differentiation and integration of the function.
If so, then with the Proposition 3, the inverse Laplace transform will be the proc cos t form by the Proposition 3 is given by  

Figure 5 .
Figure 5. Graph of entry 17 of Table 2 and its Laplace transform.(a)   f t t t 5 5 sin cos  ; (b)

3 .
be Taylor's seriezable trigonometric function is acceptable only on a theoritical point of view, from the evaluation of Laplace tra sform of trigonometric functions vice-versa of De on n finiti of Section 2 and Proposition 3. On the other hand, we show under what condition the Proposition 3 inve ansform of Proposition 3. ositio exists?Definitely the answer would be by finding the rse Laplace tr For simplicity's sake, re-writing the Prop n 3, is akin to evaluating the limits and representing,     The Laplace transform of Taylor's seriezable trigonometric function   f t is simply defined by,


In[14]  the Bilateral Laplace Sumudu Duality ( SD) was established.the inverse Laplace transform of is gi BL 1 ven by (see Equation (5.10) in [14]), . Hence for finding the original function, when equating the coefficients of identical powers of s with Proposition 3, we get    As t e sub-scripts denote the order

Table 3 . Laplace transform properties with respect to the Proposition 3.
Theorem 2 in LHS yields the RHS of the Equation above.
10The Laplace transform of the derivative with respect t From the property of Laplace transform, the  Laplace transform of s is given by the first derivative of Dirac delta function with respect to t ,    