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In this paper, the combination of Elzaki transform and Adomian polynomial is used to obtain the approximate analytical solutions of nonlinear Klein Gordon equations. The approximate analytical solutions of all these equations are calculated in series form. In total, four Klein-Gordon equations from mathematical physics were considered to show the performance and effectiveness of this method. A three dimensional graph of solutions of some problems considered were plotted to show the shape of the solutions obtained and compared with that given in the references and they were found to agree. By comparing this method with some other known methods, all the problems considered showed that the Elzaki transform method and Adomian polynomial are very powerful and effective integral transform methods in solving some nonlinear equations.

3 A new, accurate and general formula for evaluating the skin friction parameter in a Magneto-Hydrodynamics (MHD) Falkner-4 Skan flow over a permeable wall was obtained. The formula gives the value of the skin friction of the problem in an infinite 5 interval (0, ∞) for all various values of the parameters involved. Shooting method via Runge-Kutta method for solving two-point 6 boundary value problem in a truncated interval (0, L) is used to compare the results obtained. It was observed that the percentage 7 difference between the two sets of results is very small.8

In this study, the analytical solution of steady hydromagnetic double exothermic combustible reaction fluid flow in a porous medium with convective cooling wall is presented. The viscous heating reactive liquid is totaling exothermic without consumption of material. The combustion reaction of the fluid takes place in a Poiseuille device, and it is been propelled by pressure gradient and pre-exponential bimolecular kinetics. The device is exposed to convective cooling to keep the reactive hydromagnetic fluid from distortion. The weighted residual method (WRM) is analytically used to get the numerical values for the dimensionless nonlinear governing equations. The solution to temperature and velocity distribution is carried out and the result is graphically depicted. The Nusselt number and skin friction coefficient is also showed for some significant parameters engrained in the flow and the solution obtained is compared with numerical method. As obtained in the study, the second exothermic reaction term increases the combustion process; hence the term will assist in reducing toxic discharge from the engines that pollute the environment. The Frank-Kamenetskii parameter contributes highly to system thermo-fluid destruction; as such it must be monitored.