An Application of Generalized Laplace Transform in PDEs

Abstract

We would like to consider an application of generalized Laplace transform in partial differential equations(PDEs) by using the n-th partial derivatives. The tool of this research is the induction, and the proposed idea gives an easy solution to engineering problems by freely selecting the integer α in the definition.

Closed-form asymptotic solution for the transport of chlorine concentration in composite pipes

“…Here, we make consideration of the model expressed in equations (11)- (13). Thus, on applying the Laplace transform on the x variable in the model, one obtains from equation (11) the following…”

“…Besides, the literature is heavily rich with various analytical and semi-analytical methods, which indeed pave the way for the acquisition of exact and closed-form solutions for subsequent use in computational methods as benchmark solutions. To mention a few, we recall the following methods that are widely used to analytically tackle a wide class of differential equations, including the numerical Laplace transform method [11], a mixture of Laplace transform and binomial expansion method for the solution coupled dynamical models [12], the generalized Laplace transform [13], the decomposition method by Adomian [14], the application improved decomposition technique for complex-valued evolution methods [15], generalized Hankel transform method [16], the Laplace-Fourier-sine transform method [17], the Wiener-Hopf method [18], the eigen-value approach [19], and the Lie's symmetry method [20] to state but just a few. In the same vein, it will be soothing to recall some of the reliable numerical methods that play a part in fluid flow problems.…”

Using $G_{\alpha}$-transform to Study Higher-order Differential Equations with Polynomial Coefficients

“…Furthermore, the application of the n-th partial derivatives to the G α -transform in partial differential equations was presented by Kim Hj. et al [22]. Kim Hj.…”

Application of Double Sumudu-Generalized Laplace Decomposition Method for Solving 21-Pseudoparabolic Equation

“…Ahmeda et al [21] discussed the convergence of the double Sumudu transformation and used it to obtain the solution of the Volterra integropartial differential equation. The generalized Laplace transforms were used to study the solution of partial differential equations (PDEs) and also, the suggested ideas gave an easy solution to engineering problems by freely selecting integer α in the definition [22].…”