# br Ki br K i br

Ki

K
i

K
i

K

h

h

k

k

Stage 2

Ki
H2

h

h
k

k

P

h

h

Ste3

P

Boundary condition of III kind

2
i
w
r

i
w
r

h

h

w

k

k

Ste2
P

h

h
k
k

P

Fig. 7. Moving layer thickness 2 in stage2.

Fig. 8. Moving layer thickness 1 in stage3.

The problem in all the three Stages with different boundary condi-

tion converted into generalized system of Sylvester equations

h

h

117) whose
solution is obtained by applying
Bartels-

(117)
Stewart algorithm (1972). And applying these results in interface con-

k

k

dition, the position oayer thickness in Stage 2 and Stage 3 with gen-

eralized boundary condition determined. For numerical computation,

Matlab Software is used.

Ste3

The exact solutions of equation (47- 56) are available for semi-

Fig. 9. Moving layer thickness 2 in stage3.

infinite one-dimension case. We have described the structure of one-dimensional problem in cryosurgery as a semi-infinite moving lte do-main, which is initially in liquid phase at temperature T0. The surface at x = 0 cooled by a cryoprobe at temperature Tc. The freezing starts and the domain is divided into three regions. In Stage 1 only unfrozen re-gion is formed, in Stage 2 mushy and unfrozen regions are formed while in Stage 3 all the three regions: frozen, mushy and unfrozen are formed. We used Boundary fixation method to find the exact solution in Stage 3. Exact solution for this 2-NBDG problem is as follows
f = ( C + 4) i 2erfc ( e1 z ) + Ci 2erfc ( e 1z)

where

erf

e1z
Ste3

erf

erf

erf

and

Ste3

L

X

x

Legendre wavelet Galerkin Method is presented in Section 5 of each Stage. The temperature was calculated by this method in all the three region. And with the help of temperature, we find the interface posi-tion. Figs. 2 and 3 show the temperature distribution and interface position calculated by Modified Legendre wavelet Galerkin Method, and their comparison with exact solutions. These Figs. 2 and 3 clearly

demonstrate that numerical Modified Legendre wavelet Galerkin Method agrees completely with exact solution.

6. Result and discussion

In this paper, a two-dimensional mathematical model for the freezing of tumor tissue in lung is developed. The procedure of freezing is accomplished in three stages by imposing on Organelles the boundary condi-tion of I kind or II kind or III kind. For easy understanding of the model, we established the non-dimensional parameters described in section 3. Consequently, the dimensionless form of the model is described in Eqs. 26–42. We applied Modified Legendre wavelet Galerkin method (Kumar et al., 2018) in each phase of boundary condition I, II and III kind for the solution of a dimensionless model. In stage 2 and 3, the model is a moving boundary problem of partial differential equations. Eq. (39) shows the interface condition in stage 2 and Eq.(41,42) re-present the interface condition in stage 3.We used interface condition of stage 2 and 3 to determine the moving layer thickness and obtain the values of 2 ( Fo2) in stage 2 and 1 (Fo3), 2 ( Fo3) in stage 3. In both stages, the effect of Stefan number on Moving layer thickness is ob-served. Moving layer thickness increases as the Stefan number de-creases. Also, the effect of Kirchoff number on moving layer thickness has been seen in Stage 3 of boundary condition II kind and the effect of Biot number on moving layer thickness has been seen in stage3 of boundary condition III kind. Moving layer thickness increases as the Kirchoff number increases and moving layer thickness decreases as the Biot number increases.Moving layer thickness and temperature dis-tribution are two important factors during the cryosurgical treatment of lung tumor tissue for the prophecy of extreme damage to diseased tissue and the least damage to healthy lung tissue (Kumar et al., 2018). Consequently, we have analysed the temperature distribution in all stages with generalized boundary condition and moving layer thickness in the mushy and frozen region of boundary condition I, II and III kind.