The Homotopy Perturbation Method (HPM) is used to solve the Burgers-Huxley non-linear differential equations. Three case study problems of Burgers-Huxley are solved using the HPM and the exact solutions are obtained. The rapid convergence towards the exact solutions of HPM is numerically shown. Results show that the HPM is efficient method with acceptable accuracy to solve the Burgers-Huxley equation. Also, the results prove that the method is an efficient and powerful algorithm to construct the exact solution of non-linear differential equations.

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Ion-acoustic (IA) solitary wave propagation is investigated by solving the fractional Schamel equation (FSE) in a homogenous system of unmagnetized plasma. This plasma consists of the nonextensive trapped electrons and cold fluid ions. The effects of the nonextensive q-parameter, electron trapping, and fractional parameter have been studied. The FSE is derived by using the semi-inverse and Agrawal's methods. The analytical results show that an increase in the amount of electron trapping and nonextensive q-parameter increases the soliton ion-acoustic amplitude in agreement with the previously obtained results. However, it is vice-versa for the fractional parameter. This feature leads to the fact that the fractional parameter may be used to increase the IA soliton amplitude instead of increasing electron trapping and nonextensive parameters.

The time fractional modified Korteweg-de Vries (TFMKdV) equation is solved to study the nonlinear propagation of small but finite amplitude dust ion-acoustic (DIA) solitary waves in un-magnetized dusty plasma with trapped electrons. The plasma is composed of a cold ion fluid, stationary dust grains, and hot electrons obeying a trapped electron distribution. The TFMKdV equation is derived by using the semi-inverse and Agrawal's methods and then solved by the Laplace Adomian decomposition method. Our results show that the amplitude of the DIA solitary waves increases with the increase of time fractional order β, the wave velocity v0, and the population of the background free electrons λ. However, it is vice-versa for the deviation from isothermality parameter b, which is in agreement with the result obtained previously.

The equation of magneto-hydrodynamic Jeffery-Hamel flow of non-Newtonian Casson fluid in a stretching/shrinking convergent/divergent channel is derived and solved using a new modified Adomian decomposition method (ADM). So far in all problems where semi-analytical methods are used the boundary conditions are not satisfied completely. In the present research, a hybrid of the Fourier transform and the Adomian decomposition method (FTADM), is presented in order to incorporate all boundary conditions into our solution of magneto-hydrodynamic Jeffery-Hamel flow of non-Newtonian Casson fluid in a stretching/shrinking convergent/divergent channel flow. The effects of various emerging parameters such as channel angle, stretching/shrinking parameter, Casson fluid parameter, Reynolds number and Hartmann number on velocity profile are considered. The results using the FTADM are compared with the results of ADM and numerical Range-Kutta fourth-order method. The comparison reveals that, for the same number of components of the recursive sequences over a wide range of spatial domain, the relative errors associated with the new method, FTADM, are much less than the ADM. The results of the new method show that the method is an accurate and expedient approximate analytic method in solving the third-order nonlinear equation of Jeffery-Hamel flow of non-Newtonian Casson fluid.