Using a help system to make easier the CAS-user connection, we propose the problem of solving the equation of diffusion with variable coefficients. We transform the problem at the Laplace domain and we use the Bromwich integral and the residue theorem in order to do the inverse Laplace transform and then explicit solution is obtained.

This paper gives a generalization of the Pseudo-Laplace transform. In the special cases of semirings, the pseudo-exchange formula is proved. Also, for these semirings the forms of the Pseudo-Laplace transform and inverse operator are given. The results can be applied in dynamical programming for finding the maximum and minimum of the utility functions. AMS Mathematics Subject Classification (19...

We present a form of the discrete dipole approximation for electromagnetic scattering computations in time domain. We show that the introduction of complex frequencies, through the Laplace transform, significantly improves the computation time. We also show that the Laplace transform and its inverse can be combined to extract the field inside a scatterer at a real resonance frequency.

An automatic algorithm evaluating numerically an abscissa of convergence of the inverse Laplace transform is introduced.

The main idea behind the Laplace transformation is that we can solve an equation (or system of equations) containing differential and integral terms by transforming the equation in time (t) domain into Laplace (2) domain. For example, we can use Laplace transforms to turn an initial value problem into an algebraic problem which is easier to solve. After we solved the problem in Laplace domain w...

In this article using the inverse Laplace transform, we show analytical solutions for the generalized mass transfers with (and without) a chemical reaction. These transfers have been expressed as the Couette flow with the fractional derivative of the Caputo sense. Also, using the Hankel contour for the Bromwich's integral, the solutions are given in terms of the generalized Airy functions.

In this study, the second type of Green and Naghdi's thermoelasticity theory is applied to present the vibration of a nanobeam subjected to rectified sine wave heating based upon the nonlocal thermoelasticity theory. Both Young's modulus and thermal conductivity are considered to be linear functions of the temperature. The Laplace transform domain is adopted to solve the governing partial diffe...

In this article, the electro-osmotic flow of Oldroyd-B fluid in a circular micro-channel with slip boundary condition is considered. The corresponding fractional system is represented by using a newly defined time-fractional Caputo-Fabrizio derivative without singular kernel. Closed form solutions for the velocity field are acquired by means of Laplace and finite Hankel transforms. Additionally...

In this paper, inverse Laplace transform method is applied to analytical solution of the fractional Sturm-Liouville problems. The method introduces a powerful tool for solving the eigenvalues of the fractional Sturm-Liouville problems. The results  how that the simplicity and efficiency of this method.