A Computational Method for n-Dimensional Laplace Transforms Involved with Fourier Cosine Transform

In 2007, the author published some results on n-dimensional Laplace transform involved with the Fourier sine transform. In this paper, we propose some new result in n-dimensional Laplace transforms involved with Fourier cosine transform; these results provide few algorithms for evaluating some n-dimensional Laplace transform pairs. In addition, some examples are also presented, which explain the useful applications of the obtained results. Therefore, one can produce some twoand threeas well as ndimensional Laplace transforms pairs.


Introduction and Preliminaries
Before a lunching into the main part of the paper, we define some notations and terminologies which will remain standard.The classification -dimensional Laplace transform under consideration for a function () is a function () through the relation where  = ( 1 ,  2 , . . .,   ),  = ( 1 ,  2 , . . .,   ),  ⋅  = ∑  =1     , and   () = ∏  =1   .The domain of definition of  is the set of all points  ∈ C  such that the integral in (1) is convergent.Instead of the -dimensional Laplace transform (1), sometimes we calculate the so-called -dimensional Carson-Laplace transform: Symbolically, we denote the pairs () and () by the following operational relation: In this notation, some of the formulas become more simple.We denote (3) in one-dimensional case by the following: Now, if the -dimensional Laplace transform is known, its inverse is given by the following: Herein, Br designates the appropriate Bromwich contour integral in the plane of integration.
For brevity, we will also use the following notation throughout this paper.
Let  ] = ( ] 1 ,  ] 2 , . . .,  ]  ) for any real exponent ], and let   () be the th symmetric polynomial in the component   of .Then we denote .The difficulties in obtaining multiple direct or inversion Laplace transforms (1) or (5) that appear in problems of physics and engineering lead to continuous efforts in expanding the transform tables for directs and designing algorithms generating new inverses transforms from known ones.While such tables are available, the actual evaluation of the direct and inversion integral is obviated and the solution of boundary value problems in several variables and some partial differential equations is reduced to a relatively routine procedure.For more details on this subject see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16].

The Main Results
In this section we state and give proof for our main theorems, which give some new -dimensional Laplace transforms pairs for arbitrary nonnegative integer  ≥ 2.
Theorem 1. Suppose that Also, let   () be the Fourier cosine transform of ( 2 ), and let provided the Laplace transform of functions (), () and  (−2)/2 (),  ≥ 2, exist and the integrals in the left side of (6) also exist in every variable.
Proof.By using the assumptions (i) and (ii) together, we get Now, interchanging the order of the integrals on the right side of (7) due to the Fubini's theorem [17] evaluating the inner integral and next by replacing  by ] 2 in the resulting equation, we have From ( 8) we can easily obtain Evaluating the inner integral in the rightside of ( 9), we get By the assumption, (10) can be rewritten as Next, we replace  by  1 ( 1/2 ) in ( 11) and multiply both sides of the resulting relation by   (), in order to obtain Now, we use the following operational relation which is given in [18], in ( 12) for  = 1, 2, . . ., , (12) reads Therefore, This completes the proof.

Theorem 2. Suppose all conditions given in Theorem 1 hold true but replace the condition (iii) by the following:
(iii)  L{ −1/2 (); } =  −1/2 (). Then Proof.The proof of Theorem 2 is similar to that of Theorem 1, and we therefore omit it.
The following examples will illustrate the applications of Theorems 1 and 2. We will consider the function  to be an elementary or some special function to construct certain functions with  variables, and we calculate their Laplace transforms, using Theorems 1 and 2. The first two examples are related to Theorem 1, and Examples 3 and 4 illustrate the application of Theorem 2.