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The method of Lie group transformation is used to obtain an approximate analytical solution to the system of first-order quasilinear partial differential equations that govern a one-dimensional unsteady planer, cylindrically symmetric and spherically symmetric motion in a non-ideal gas, involving strong shock waves. Invariance groups admitted by the governing system of partial differential equations, which are indeed continuous group of transformations under which the system of partial differential equations remains invariant, are determined, and the complete Lie algebra of infinitesimal symmetries is established. The infinitesimal generators are used to construct the similarity variables. These similarity variables are used to reduce the governing system of partial differential equations into a system of ordinary differential equations.

In the present article, similarity solutions of second kind to a problem of imploding cylindrical shock wave in non-ideal magnetogasdynamics are investigated. The equation of state of the medium is assumed to be in the form of the Mie-Gruneisen type. A numerical study of singular points of the differential equations leads to determination of the similarity exponent. Numerical description of the flow field has been presented in non-ideal magnetogasdynamics. The results obtained are compared with the numerical solution obtained by using CCW approximation method. Detailed studies are carried out for two different physically meaningful non-ideal medium in the presence of magnetic field. Also, the effect on magnetic field of flow variables such as density, velocity, pressure and magnetic pressure behind the wave front is illustrated through figures.

A scheme is developed to study numerical solution of the time-fractional shock wave equation and wave equation under initial conditions by the homotopy perturbation method (HPM). The fractional derivatives are taken in the Caputo sense. The solutions are given in the form of series with easily computable terms. Numerical results are illustrated through the graph.

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