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A Tutorial to Approximately Invert the Sumudu Transform

Unlike the traditional Laplace transform, the Sumudu transform of a function, when approximated as a power series, may be readily inverted using factorial-based coefficient diminution. This technique offers straightforward computational advantages for approximate range-limited numerical solutions of certain ordinary, mixed, and partial linear differential and integro-differential equations. Furthermore, discrete convolution (the Cauchy product), may also be utilized to assist in this approximate inversion method of the Sumudu transform. Illustrative examples are provided which elucidate both the applicability and limitations of this method.


Introduction
Transform mathematics has traditionally been utilized for obtaining solutions of differential and integro-differential equations (DEs and IDEs) which arise in control theory, engineering, and related areas such as pharmacologic and mathematical modeling [1] [2] [3]. In general, transform mathematics allows for the conversion of differentiation and integration into algebraic processes which yield a preliminary solution that is expressed within the transform domain. Subsequent inversion of this transform function then produces the actual solution of the original DE or IDE which is represented in terms of moments.
The Laplace transform is classically utilized for this purpose [4] [5]. Typically, the inversion of a Laplace transform is accomplished using pre-existing tables which facilitate this process. Partial fraction expansion may be helpful, in sim-plifying an expression within the Laplace domain, but is not always reliable in generating a readily invertible result.
Therefore, when a tabular result is unavailable, inversion of the Laplace transform may be difficult; requiring integration within the complex plane. Approximate Laplace transform inversion techniques exist which are based upon numerical methods [6], Fourier analysis [7], or repetitive symbolic differentiation [8].
Relatively recently, alternative mathematical transforms have been developed. Specifically, the Sumudu transform is defined as [9]: It should be noted that the Sumudu transform of a real function yields results, within the u-domain, which are also real. Additionally, the Sumudu transform "preserves dimensions." Thus, whatever physical dimension ( ) g t has ( ) G u will also have [9] [10] [11]. This is particularly helpful when checking for algebraic accuracy.
For simplicity, the Sumudu transform process will be referred to using the S operator: Whereas the inversion process will be referred to as: Note that the traditional Laplace transform is defined using the L operator: Inspection of (1) and (4) demonstrates that the conversion, from a Laplace to a Sumudu transform, is obtained by using the substitution of 1 u for s with the subsequent multiplication of the Laplace transform by 1 u : A "duality" between the Laplace and Sumudu transforms therefore exists in terms of similar mathematical properties regarding linearity, convolution, differentiation, and integration. These topics have been explored and discussed previously [10] [11]. Furthermore, tables of Sumudu transforms are available which allow for straightforward conversions, between the u-domain and t-domains, for many commonly used functions [10] [11].
If t in the above equation has units of time, then s consequently has units of complex frequency, whereas u would also have the dimension of time. As previously stated, the Sumudu transform "preserves units." The purpose of this paper is to demonstrate an approximate inversion process of the Sumudu transform utilizing a geometric power series technique which uses non-negative integer values for n: Furthermore, this approximate inversion process may only apply within a narrow numerical range owing to the limitations of a truncated geometric power series as a means of representing a function.
Note that ( ) G u in (6) would also be continuously differentiable or "smooth." In addition, n a is a constant coefficient. Moreover, this geometric power series method can also be combined, utilizing superposition, with single or multiple known "pre-existing" Sumudu transform or transforms; such as those available from a table: Multiple geometric power series may also be combined in an additive or multiplicative manner.

The Sumudu Transform of a Power Series
The Sumudu transform of a power series in the t-domain is a "factorial-based" amplified power series in the u-domain (See Appendix A) [9]. Therefore: Thus, for integer values of 0 n ≥ : A summation of multiple power terms, in the time domain, could then be expressed as a power series: The corresponding Sumudu transform of the above equation would therefore be: Note that t has to be substituted for u on the RHS of the above equation.
Therefore, functions which can be approximated with a geometric power series, expressed with the form of (6) in the u-domain, may be readily inverted, back to the t-domain, using the aforementioned technique. Thus, approximate solutions of certain types of both linear differential equations, as well as linear integro-differential equations, can be generated. However, these approximate solutions may also be range-limited. In addition, a Sumudu transform is frequently a rational function expressed within the u-domain: However, ( ) G u may sometimes be expressed utilizing transcendental functions. Nonetheless, approximating ( ) G u as a geometric power series in the form of (6) can often be accomplished using commonly-known mathematical techniques and algorithms. Computer-based symbolic processors can also be utilized. As previously stated, geometric power series representations of functions may have limitations which restrict the acceptable accuracy of this approximation to that of a relatively narrow numerical range.
Lastly, a Sumudu transform may sometimes be expressed as a product of two or more rational functions: Each rational function would then be approximated as a unique geometric power series. Consequently, ( ) G u would be represented as the product of two or more infinite series: So that: where: coefficients. Furthermore, the above product of multiple series can be combined using discrete convolution (the Cauchy product) [12] [13]. However, each series must be truncated. As will be shown later, this technique can also be utilized as an approximation-based means to invert a partial fraction expansion from the u-domain back to the t-domain.

Methods
Numerical analysis and the conversion of rational functions into geometric power series were accomplished using Mathcad (PTC Corporation, MA, USA). Graphs were prepared using Excel (Microsoft Corporation, WA, USA).

Exponential Function
The Sumudu transform of the exponential function is [9]: Thus: Therefore, the Sumudu transform of the exponential function subsequently expressed as a power series is: Use of FBCD with substitution of t for u then yields the well-known Taylor's series of an exponential function. Thus, the above Sumudu transform is approximately inverted without the need for integration within the complex plane:

Sine Function
Using (1) the Sumudu transform of the sine function is [9]: Expressing the above as a power series: By using FBCD, the above Sumudu transform, expressed as a power series, is approximately inverted. This yields the Taylor series of a sine function. Note that u has been replaced by t: ( ) 3

Cosine Function
The Sumudu transform of the cosine function is found in a manner similar to that of the sine function [7]: Therefore: ( ) 2 2 4 4 6 6 8 8 The Taylor's series for the cosine function is then approximated using FBCD along with substitution of t for u within the above equation: So that:

Differentiation and Integration within the Sumudu Domain
Differentiation and integration, with the Sumudu transform, have an "inverse resemblance" to that of Laplace transforms [10] [11]. The first derivative is: Whereas the second derivative is: Higher-order derivatives can also be determined: Furthermore, integration within the Sumudu transform domain can be summarized as:

Sumudu Shift Theorem
The Sumudu transform of a function ( ) t g t ⋅ can be expressed using the Sumudu shift theorem and ( ) Making use of the product rule yields:

Example 1
Consider the fourth-order linear inhomogeneous differential equation with all initial conditions equal to zero: Using the Sumudu transform technique: Which is equivalent to: Thus: The approximate solution is readily obtained by means of FBCD and substitution of t for u: Thus: ( ) ( ) ( ) The numerical results of this are graphically illustrated in Figure 1.

Example 2
The FBCD technique can also be used to determine the approximate solution to this inhomogeneous first order integro-differential equation (IDE): Note that a straightforward ordinary differential equation (ODE) results: Assume that ( ) Representing the above using series notation: 6 .
Therefore:  This approximate solution is shown in Figure 2 whereas the "components" of the equation are illustrated in Figure 3.

Example 3
Consider the following integro-differential equation (IDE) which incorporates the sine integral (Si) function:   ( ) ( ) The above can be expressed as: Thus: Note that [14]: ( ) Use of the Cauchy product, or discrete convolution with two truncated series, results in a double or "nested" series: This allows for a single expression for u as a power function and ultimately a power series. Thus, the two series, which have undergone convolution, can also be algebraically combined: In addition, a double series which resulted from the discrete convolution of two series, is readily integrated and differentiated utilizing the same "term-byterm" methodology as a single series:

Example 4
Consider the following third-order differential equation in which all initial conditions are equal to zero:   ( ) It is expressed within the Sumudu transform domain as: Note that the following relationship: where: Applying discrete convolution (the Cauchy product) with two truncated series, each of m terms, yields: Utilizing further algebraic simplification: The use of FBCD and substitution of t for u yields the approximate solution: "Term-by-term" differentiation results in the following: The results of this are graphically illustrated in Figure 6 and

Example 5
The following double integral equation can be expressed as an ordinary differential equation (ODE) using a Sumudu transform and the shift theorem: Thus: Rearranging: Separating variables and simplifying:   Integrating both sides: This subsequently yields: Exponentiation results in: Allowing the constant of integration, C, to equal zero yields: The above equation can subsequently be expressed as an infinite series: is the Dirac delta function.
Secondly, combining the Sumudu transform ( ) for the remaining terms of (116) yields the following: Note that the variable t is the indefinite integral of the Heaviside step function. Furthermore, t is also the double indefinite integral of the Dirac delta function.
Lastly, the constants of integration for (119) and (120) are equal to zero. Figure  8 and Figure 9 illustrate the solution of this example.

Example 6
The aforementioned techniques can also be applied to certain partial differential equations (PDEs). As an example: The above equation can be readily converted to an ODE using the Sumudu transform:   An integrating factor is then utilized to take advantage of the product rule: This yields an expression such that the integrating factor now cancels from both sides of the resulting equation: ( ) 2 , e e e .  with the utilization of convolution. Further research and applications of this technique, particularly with non-linear and fractional differential equations, may be warranted.

Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.