where ∆ is the Laplace operator. Here σ > 0 is a constant (the diffusion constant). It has dimensions of distance squared over time, so H0 has dimensions of inverse time. The operator exp(−tH0) for t > 0 is an self-adjoint integral operator, which gives the solution of the heat or diffusion equation. Here t is the time parameter. It is easy to solve for this operator by Fourier transforms. Sinc...

Fractional Dzherbashian-Nersesian operator is considered and three famous fractional order derivatives named after Riemann-Liouville, Caputo Hilfer are shown to be special cases of the earlier one. The expression for Laplace transform constructed. Inverse problems recovering space dependent time source terms a diffusion equation with involution involving considered. results on existence uniquen...

Exact expressions of velocity, temperature and mass concentration have been calculated for free convective flow of fractional MHD viscous fluid over an oscillating plate. Expressions of velocity have been obtained both for sine and cosine oscillations of plate. Corresponding fractional differential equations have been solved by using Laplace transform and inverse Laplace transform. The expressi...

A novel implementation of the Laplace transform method for the calculation of the density of states of molecules, for which the partition function can be explicitly given is described. It consists of doing the inverse Laplace transform numerically after multiplying the integrand by a smoothing factor. This evaluation is more accurate than the method of steepest descent, and the computation can ...

This paper deals with the solution of the spherically symmetric time-dependent Hartree-Fock approximation applied to nuclear giant monopole resonances in the small amplitude regime. The problem is spatially unbounded as the resonance state is in the continuum. The practical requirement to perform the calculation in a finite-sized spatial region yields an artificial boundary, which is not presen...

Fourier and hyperbolic models of heat transfer on a fin that is subjected to a periodic boundary condition are solved analytically. The differential equation in Fourier and non-Fourier models is solved by the Laplace transform method. The temperature distribution on the fin is obtained using the residual theorem in a complex plan for the inverse Laplace transform method. The thermal shock is ge...