A note on solutions of wave, Laplace’s and heat equations with convolution terms by using a double Laplace transform

Classifications and duality relations for several integral transforms

“…In this process, the applications of linear property and differential property of ITs can simplify the solving process. In addition, Eltayeb and Kılıçman in literature [7], had emphasized the importance of convolution properties of ITs in solving the non-homogeneous wave with non-constant coefficients in detail. In this section, we will derive and prove these properties from the point of view of GLT and GST.…”

“…For instance, Kang et al [11] discovered that the using of LCT was more convenient to analyze the problem of American strangle options than LT and the numerical inversion of the LCT was more dominant than other ITs [6,15]. As well as, the ST has advantages in analyzing the non-constant coefficients wave equations [7,13]. In addition, LT and ST play the important roles in solving the fractional PDEs analytically or numerically [1,10,14,26].…”

“…Specially, the diffusion equations and wave equations, as fundamental equations in applied mathematics and engineering, have been widely studied [7,13,24,25]. According to the research shown in literatures [6,9,16,[21][22][23], several different ITs, such as Laplace transform (LT), Sumudu transform (ST), ELzaki transform (ET), Laplace-Carson transform (LCT), as well as two new integral transforms (NIT1 and NIT2), were equivalent in obtaining the analytic solutions of the diffusion or wave equations.…”

A note on double Laplace decomposition method and nonlinear partial differential equations

“…Also the double Laplace transform and some of its application are used to solve general linear telegraph equation and wave equation with initial and boundary conditions see [3]. Also double Laplace transform applied by Eltayeb and Kilicman [4] and [5] to solved non-homogeneous wave equation with variable coefficients. In this work we use the double Laplace decomposition methods to solve nonlinear partial differential equation.…”

Distributed Control of Heat Conduction in Thermal Inductive Materials with 2D Geometrical Isomorphism

“…The systems & control literature documents several kinds of 2D integral transform, as listed in Table 2. Therein, the Double Laplace transform, for example in [36,37], and Double Fourier transform, for example in [15,17,19], were always applied for signal processing and feedback control in semi-infinite and infinite space regions, respectively. The Laplace-Galerkin transform [1,[38][39][40] or Fourier-Galerkin transform [41,42] is justified to model the non-Fourier heat conduction and its controllers for bounded space regions that are of real concern in heat conduction practice.…”