A Computational Method for -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 two- and three- as well as n-dimensional Laplace transforms pairs. 1. 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 , , , and . The domain of definition of is the set of all points 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, designates the appropriate Bromwich contour integral in the plane of integration. For brevity, we will also use the following notation throughout this paper. Let for any real exponent , and let be the th symmetric polynomial in the component of . Then we denote(i) , (ii) . 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 [1–16]. 2. 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 . Theorem 1. Suppose that(i) (ii) (iii) , .Also, let be the Fourier cosine transform of , and let belong to . Then provided the Laplace transform of functions , and , , exist and the integrals in the left