Some Remarks on the Sumudu and Laplace Transforms and Applications to Differential Equations
We study the relationship between Sumudu and Laplace transforms and further make some comparison on the solutions. We provide some counterexamples where if the solution of differential equations exists by Laplace transform, the solution does not necessarily exist by using the Sumudu transform; however, the examples indicate that if the solution of differential equation by Sumudu transform exists then the solution necessarily exists by Laplace transform.
In order to solve the differential equations, the integral transform is extensively applied and thus there are several works on the theory and application of integral transforms. In the sequence of these transforms, Watugala introduced a new integral transform, named the Sumudu transform, and further applied it to the solution of ordinary differential equation in control engineering problems; see . For further details and properties of Sumudu transform see [2–7] and many others. The Sumudu transform is defined over the set of the functions by the following formula:
The existence and the uniqueness was discussed in ; for further properties of Sumudu transform and its derivatives, we refer to . In , some fundamental further properties of Sumudu transform were also established.
Similarly, this new transform was applied to the one-dimensional neutron transport equation in . In fact, one can easily show that there is strong relationship between Sumudu and other integral transforms. In particular, the relation between Sumudu transform and Laplace transforms was proved in .
Further in , the Sumudu transform was extended to the distributions (generalized functions) and some of their properties were also studied in . Recently, Kılıçman et al. applied this transform to solve the system of differential equations; see [7, 11].
Now, let us recall the following definition which is held in Estrin and Higgins; see ; the double Laplace Transform is defined by where and , are complex numbers. The double Sumudu transform of second partial derivative with respect to is given by Then, the interior integral is given by
By taking Sumudu transform with respect to for (1.5), we get double Sumudu transform in the form of
Similarly, double Sumudu transform of is given by And double Laplace transform defined the first-order partial derivative as
The double Laplace transform for second partial derivative with respect to is given by and double Laplace transform for second partial derivative with respect to similarly as above is given by
In a similar manner, the double Laplace transform of a mixed partial derivative can be deduced from single Laplace transform as
Our purpose here is to show the difference between Laplace transform and Sumudu transform by solving partial differential equations. In fact, the double Sumudu transform and double Laplace transform have a strong relationship that may be expressed either as where represents the operation of double Laplace transform. In particular, this relation is best illustrated by the fact that the double Sumudu and double Laplace transforms interchange the image of and . It turns out that
Further, the double Laplace and Sumudu transforms interchange the images of the Dirac function, and the Heaviside function, , since where the symbol means the tensor product thus the relation between the double Sumudu and double Laplace transform of convolution was given by (see ).
Note that since many practical engineering problems involve mechanical or electrical systems acted upon by discontinuous or impulsive forcing terms, then the Sumudu transform can be effectively used to solve ordinary differential equations as well as partial differential equations in engineering problems; see . In this paper, we study the relationship between Sumudu and Laplace transforms and further make some comparison on the solutions. Provide a counterexample where if the solution of differential equation by Laplace transform exists then it does not necessarily exist by using the Sumudu transform, however, if the solution of differential equation by Sumudu transform exist, then solution necessarily exists by Laplace transform. First of all we need the following concept related to the Sumudu transform of derivatives.
Proposition 1.1 (Sumudu Transform of Derivative). Let be differentiable on and let for . Suppose that . Then, , , and
More generally, if is differentiable on , for , and , then
Proof. For the proof of this proposition, see .
In general case, if is a differentiable function on with , and for or and , then, for all , Proposition 1.1 can be extended to higher derivatives, before extension, we introduce the following notation as in . Let be a polynomial in , where and . We define to be the matrix of polynomials given by the matrix product: For each complex number , the map defines a linear mapping of into in obvious way. We will write vectors in as row vectors or column vectors interchangeably, whichever, is convenient, although when is to be compute and the matrix representation by (1.19) of is used, then of course must be written as a column vector: for any . If , then defines a unique linear mapping of into (empty matrix). In general, if and is times differentiable on an interval , with , then we shall write If , we write for . If , then we define
Now, we need to consider the transform of higher derivatives as follows.
Proposition 1.2 (Sumudu transform of higher derivatives). Let be times differentiable on , and let for . Suppose that Then, for , and, for any polynomial of degree , for . In particular, (with here written as a column vector). For , one has
Proof. For the proof of this proposition, see .
In general, if is differentiable on with , and for or and then we have, for all ,
2. Solution of Differential Equations by Convolution Methods
In this section, we give the solution of the following equation: on . We prove an existence and uniqueness and provide a formula for the solution.
If we define and to be zero on , then (2.1) is equivalent to the equation where
First of all, we establish first an important result for homogeneous equation.
Theorem 2.1 (Properties of solution of the homogeneous equation). Let and let be complex constant such that . Let be differentiable on and zero and satisfy then one has the following.(1) is infinitely differentiable on .(2) For every integer , the limits exist.(3) If , then is given (except at 0) by the formula where
Proof. To avoid trivial statements, suppose that . Let . The function is times differentiable on , and , and are locally integrable and . By (1.26) and the relation , ; we have, for all Since , we obtain for large Now, is a proper rational function of ; there is a function , analytic on , such that, for sufficiently large, With defined analogously, we deduce from (2.8) and the shift rule that if , this gives thus is analytic on every open interval , with ; hence, is analytic on . However, on the interval , we have . We conclude that this formula must hold for all , which allows us to write for all ; from this formula, (1.1) and (1.2) follow immediately. Now, we write, for , The first term on the right is clearly Sumudu transformable. Similarly, the second term is also transformable since it is merely the translated function , thus is Sumudu transformable. To obtain the formula (2.5), we apply (1.23) and get Since is a polynomial of degree less than , by using Sumudu inverse transform, we obtain
Now, we extend the above theorem to the nonhomogeneous equation as follows.
Proposition 2.2. Let be as in the above theorem. Let be continuous on and zero on and let be locally integrable. Let be times differentiable on and zero on and satisfy then one has the following.(1) is continuous on and locally integrable on .(2) For , the limits exist.(3) If , then is given by the formula
Proof. The result is trivial if . Suppose that . Let . if is continuous on an open interval , then is times differentiable on . We have . Also, consider that is continuous for , and hence is locally integrable on and continuous on . Let . Then, . By using the above theorem, is locally integrable and exist for all , and we have . If we now write , then the properties (1) and (2) follow immediately. Equation (2.17) also follows because by using the statement if for , then for , where , and hence .
In the next theorem, we provide a complete solution of a non-homogeneous equation.
Theorem 2.3 (Existence and Uniqueness). Let be continuous function on , zero on , and locally integrable. Let . Then, there exists a unique (except on zero) function that is times differentiable on and zero on satisfying If , then is given (except on zero) by
Proof. Uniqueness is obvious by (2.17), thus is given by (2.19). To establish existence, we consider by (2.19). Since by taking Sumudu transform and using (1.23), we have This is a polynomial in which by virtue of the statement (if be Sumudu transformable and satisfy for . Then, ) must be identically zero. Thus, , then (2.20) is true. We have To verify that satisfies the initial conditions, we first observe that by (2.17) we must have . We deduce that Taking Sumudu transform and using the relation, if is a polynomial of degree less than , then , then it follows that for all sufficiently large (and hence for all real since the left side is a polynomial). From the relation and for all , we have . We conclude that . This establishes existence.
The last term in the right-hand side of (2.20) can be written in the form of for any vector . In the next section, we provide an example to make a comparison.
3. A Comparison on Solutions
Consider that the steady-state temperature distribution function in a long square bar with one face held at constant temperature and the other faces held at zero temperature is governed by the boundary-value problem under the boundary conditions
3.1. Solution with Laplace Transform
If we apply the multiple Laplace transform with respect to the variables and for (3.1), and single Laplace transform for the first and second boundary condition, we obtain If we look at (3.3) then we can easily notice that the right-hand side is transforms of the functions and which are not among the boundary condition of (3.1). Hence, and are taken to be the unknown functions and , respectively. Then, by using single Laplace transform, we have
On using single inverse Laplace transform with respect to for (3.3) and using the Laplace transform of convolution, we have Now, we take Laplace transform for third boundary condition and substitute in (3.6), using integral property and trigonometric manipulation, then (3.6) can be written in the form of
In order to obtain the single inverse Laplace transform with respect to , we use Cauchy’s residue theorem, so we have a simple pole at ; we use We compute the residues at , and adding together, then we obtain by using the last boundary condition and Fourier series to compute the unknown integral, then we obtain the solution of (3.1) as follows:
3.2. Solution with Sumudu Transform
Now, we apply multiple Sumudu transform for the same problem to check the solution whether equal or not equal or probably does not exist. By applying multiple Sumudu transform for (3.1) as follows: and by taking single Sumudu transform for first two boundary conditions of (3.2), we have where and are taken to be unknown functions and . Then, the Sumudu transform of unknown functions is given by By substituting (3.13) and (3.14) into (3.12) and rearranging, we have By taking inverse Sumudu transform with respect to for (3.15), and using convolution, we have Now by taking single Sumudu transform for third boundary condition, we have, and use , we have Substituting (3.17) into (3.16), we obtain By using trigonometric properties, we have By rearrangement the above equation, we have In order to obtain inverse Sumudu transform for (3.20), we use the Cauchy’s residue theorem, and then we have If we replace the variable by and divide the equation by , then we obtain
By using Cauchy’s residue theorem, we have simple poles at and thus it follows that the limit does not exist at , then the solution does not exist. That leads us to make a remark that if the solution of differential equation by Laplace transform exists then it does not necessarily exist by using the Sumudu transform, but if the solution of differential equation by Sumudu transform exists then it necessarily exists by Laplace transform.
The authors gratefully acknowledge that this research was partially supported by University Putra Malaysia under the Research University Grant Scheme 05-01-09-0720RU.
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