Revisiting the 1D and 2D Laplace Transforms

Abstract

The paper reviews the unilateral and bilateral, one- and two-dimensional Laplace transforms. The unilateral and bilateral Laplace transforms are compared in the one-dimensional case, leading to the formulation of the initial-condition theorem. This problem is solved with all generality in the one- and two-dimensional cases with the bilateral Laplace transform. The case of fractional-order systems is also included. General two-dimensional linear systems are introduced and the corresponding transfer function is defined.

An Entropy Paradox Free Fractional Diffusion Equation

“…Assuming that Ψ α θ is constant, we can interpret (21) as the LT of a Mittag-Leffler function, as we did above (42). Therefore, we can write:…”

“…Let U(x, s) = L[u(x, t)] be the Laplace transform of u(x, t) relatively to t and U(κt) = F [u(x, t)] the Fourier transform relatively to x. The 2-D Laplace-Fourier transform (LT-FT) of u(x, t) is denoted by Ū(κ, s) = LF [u(x, t)] [41,42]. Assume also that we want to compute the output for t > 0 and that there exists an initial-condition (IC) u(x, 0) = v 0 (x) with V 0 (κ) = F v 0 (x).…”

“…where * * [42] denotes the 2-D convolution, and v(x, t) is any input function. However, in agreement with (1), we shall be interested in the free therm only that, if β > 0, it is given by the solution of…”

Substantial, tempered, and shifted fractional derivatives: Three faces of a tetrahedron

“…If the function at hand is defined on any sub‐interval in , then we can extend its definition to the whole real line with null values. We use the two‐sided Laplace transform (LT): where f ( t ) is any function defined on and F ( s ) is its transform, provided that it has a nonempty region of convergence (ROC). Sufficient conditions for the existence of the LT can be found in Roberts, 15 Zemanian, 16 and Ortigueira and Machado 17 The inverse LT is given by the Bromwich integral where is inside the region of convergence of the LT and . The Fourier transform (FT), , is obtained from the LT through the substitution , with . The functions and distributions have Laplace and/or Fourier transforms. Current properties of the Dirac delta distribution, δ (·), and its derivatives, δ ′ (·), δ ′′ (·)…, will be used. The standard convolution operation (denoted by the symbol ∗) will be adopted and written as The order of any fractional derivative, denoted by α , is any real number.…”

On Behavior Laplace Integral Operators with Generalized Bessel Matrix Polynomials and Related Functions

“…A number of studies on the generalizations of Laplace transform associated with special polynomials have been contributed by Ortigueira and Machado [14], Jarad and Abdeljawad [15], Ganie and Jain [16], and Saifa et al [17].…”