Series solution of nonlinear differential equations by a novel extension of the Laplace transform method

A technique for extending the Laplace transform method to solve nonlinear differential equations is presented. By developing several theorems, which incorporate the Adomian polynomials, the Laplace transformation of nonlinear expressions is made possible. A number of well-known nonlinear equations including the Riccati equation, Clairaut's equation, the Blasius equation and several other ones involving nonlinearities of various types such as exponential and sinusoidal are solved for illustration. The proposed approach is analytical, accurate, and free of integration.     

The accuracy of numerical inversion of the Laplace transform applied to the time-dependent reflection function from a semi-infinite atmosphere

In order to check the accuracy of the numerical inversion of the Laplace transform when it is applied to evaluate the time-dependent diffuse reflection function from a semi-infinite atmosphere, numerical experiments are done in virtue of comparing the numerical value with the exact one. It becomes clear that the numerical inversion method is sufficiently applicable if the time dependence of incident radiation is given by the Dirac δ function. A technique which reduces the error of the numerical inversion is proposed. It gives a remarkable improvement.     

MHD flow and heat transfer of a generalized Burgers’ fluid due to an exponential accelerating plate with the effect of radiation

This paper presents a research for the MHD flow and heat transfer of an incompressible generalized Burgers’ fluid due to an exponential accelerating plate with the effect of radiation. The fractional calculus approach is used to establish the constitutive relationship of the viscoelastic fluid. Exact analytic solutions are obtained for the velocity and temperature fields in integral and series form in terms of the G function by means of Fourier sine transform and Laplace transform. Moreover, the figures are plotted to show the effects of different parameters on the velocity and temperature fields.