**5. The First Approximation**

The Equations (41)–(44) can be written in the following forms:

$$\frac{\prod\_{r}^{(0)}}{\prod^{(0)}} = \left(F\_r^{(0)} + \frac{mF^{(0)}}{r}\right) / (r - F^{(0)}),\tag{52}$$

$$(F^{(0)} - r)\Pi^{(0)}F\_r^{(0)} + \frac{1}{\gamma}(G\_r^{(0)} + H\_r^{(0)}) = \frac{(m+1)F^{(0)}\Pi^{(0)}}{2},\tag{53}$$

$$\frac{G\_r^{(0)}}{G^{(0)}} = \left[ -\left( m + 1 \right) + \frac{\gamma}{\left( 1 - \bar{b}\Pi^{(0)} \right)} \left( F\_r^{(0)} + \frac{mF^{(0)}}{r} \right) \right] / \left( r - F^{(0)} \right),\tag{54}$$

$$\frac{H\_r^{(0)}}{H^{(0)}} = \left(2F\_r^{(0)} + \frac{2mF^{(0)}}{r} - m - 1\right) / (r - F^{(0)}).\tag{55}$$

We substitute the values *G*(0) *r* and *H*(0) *r* from Equations (54) and (55) in Equation (53) to ge<sup>t</sup> *F*(0) *r* in the following form

$$F\_r^{(0)} = \frac{\left[\frac{m+1}{\gamma} - \frac{mF^{(0)}}{(1-\tilde{b}\Pi^{(0)})r} + \left(\frac{m+1}{\gamma} - \frac{2mF^{(0)}}{\gamma r}\right)\frac{H^{(0)}}{G^{(0)}} + \frac{(m+1)F^{(0)}\Pi^{(0)}(r-F^{(0)})}{2G^{(0)}}\right]}{\left[\frac{2H^{(0)}}{\gamma G^{(0)}} + \frac{1}{(1-\tilde{b}\Pi^{(0)})} - \frac{\Pi^{(0)}(r-F^{(0)})}{G^{(0)}}\right]}.\tag{56}$$

Taylor [23] performed similarity transformation and presented the approximate solution for intense explosion. This approximation has been used by Sakurai [3] to produce analytical solution in gas dynamics. We, therefore, has assumed the first approximation *F*(0) as given in the work of Taylor [23]

$$F^{(0)}(r) = \frac{r}{\gamma} + Ar^n. \tag{57}$$

We ge<sup>t</sup> the value of A in the following form using Equations (49) and (57),

$$A = \frac{\gamma(1 - 2\alpha) - 1}{\gamma(\gamma + 1)}.\tag{58}$$

We use Equations (56)–(58) to ge<sup>t</sup> *n*. After determining the values of *A* and *n*, we integrate Equations (52)–(55) with boundary conditions (49) to obtain

$$\Pi^{(0)}(r) = \frac{\gamma + 1}{\gamma - 1} \left( 1 - \frac{2a}{\gamma - 1} \right) \left[ \frac{\gamma}{\gamma + 1 - r^{n-1}} \right]^{\left( \frac{(n+m)/\gamma(1-2a) - 1}{(n-1)(\gamma - 1)} + \frac{m+1}{(n-1)(\gamma - 1)} \right)} r^{\left( \frac{m+1}{\gamma - 1} \right)},\tag{59}$$

$$G^{(0)}(r) = \left\{ \frac{2\gamma}{\gamma+1} (1-a) - \left(\frac{\gamma+1}{\gamma-1}\right)^2 \frac{\gamma C\_0}{2} \left(1 - \frac{4a}{\gamma-1}\right) \right\} \left[\frac{\gamma}{\gamma+1-r^{n-1}}\right]^{(B)} r^{\frac{(\gamma+1)(n+1)\tilde{\gamma}\left(1-\frac{2\tilde{a}}{\gamma-1}\right)}{\gamma-1\left(1-\frac{2\tilde{a}}{\gamma-1}\right)\left(1-\frac{2\tilde{a}}{\gamma-1}\right)}},\tag{60}$$

where

$$B = \frac{\gamma (m+n)(\gamma(1-2a)-1) + \gamma (m+1)\delta(\gamma+1)(1-\frac{2a}{\gamma-1})}{(n-1)(\gamma-1)\{1-\tilde{b}(\frac{\gamma+1}{\gamma-1})(1-\frac{2a}{\gamma-1})\}}.$$

and

$$H^{(0)}(r) = \frac{\gamma C\_0}{2} \left(\frac{\gamma + 1}{\gamma - 1}\right)^2 \left(1 - \frac{4a}{\gamma - 1}\right) \left[\frac{\gamma}{\gamma + 1 - r^{n-1}}\right]^{\left(\frac{2(m+n)(\gamma(1-2a)-1) + (1+m)(2-\gamma)}{(n-1)(\gamma-1)}\right)} r^{\frac{(m+1)(2-\gamma)}{(n-1)}}.\tag{61}$$

Dimensionless quantities *<sup>F</sup>*(0), Π(0) and *G*(0) are computed in Tables 1 and 2 for (*m* = 0, 1) in ideal gas (without magnetic field i.e., *C*0 = 0). A comparison of the obtained results is presented with the published work of Sakurai [4] in gas dynamics. Schematic of dimensionless flow variables are depicted in Figures 1–4.


**Table 1.** *<sup>F</sup>*(0), Π(0) and *G*(0) for *m* = 0, *γ* = 1.4 and *C*0 = 0.

**Figure 1.** Schematic of non-dimensional velocity (**a**) *<sup>F</sup>*(0), (**b**) pressure *<sup>G</sup>*(0), (**c**) density Π(0) and (**d**) magnetic pressure *H*(0) for *γ* = 1.4, *C*0 = 0, *m* = 0 and *α* = 0.0, 0.015, 0.025, 0.05.


**Table 2.** *<sup>F</sup>*(0), Π(0) and *G*(0) for *m* = 1, *γ* = 1.4 and *C*0 = 0.

**Figure 2.** Schematic of non-dimensional (**a**) velocity *<sup>F</sup>*(0), (**b**) pressure *<sup>G</sup>*(0), (**c**) density Π(0) and (**d**) magnetic pressure *H*(0) for *γ* = 1.4, *α* = 0, *m* = 0 and *C*0 = 0.00, 0.02, 0.04.

**Figure 3.** Schematic of non-dimensional (**a**) velocity *<sup>F</sup>*(0), (**b**) pressure *<sup>G</sup>*(0), (**c**) density Π(0) and (**d**) magnetic pressure *H*(0) for *γ* = 1.4, *C*0 = 0, *m* = 1 and *α* = 0.0, 0.015, 0.025, 0.05.

**Figure 4.** Schematic of non-dimensional (**a**) velocity *<sup>F</sup>*(0), (**b**) pressure *<sup>G</sup>*(0), (**c**) density Π(0) and (**d**) magnetic pressure *H*(0) for *γ* = 1.4, *α* = 0, *m* = 1 and *C*0 = 0.00, 0.02, 0.04.
