Group Transformation Theory Applied to The Solution of Partial Differential Equations

The application of the group method in this thesis to different physical problems; two boundary layer problems and a wave problem, proved its ability in reducing nonlinear partial differential equations to ordinary ones. During the last steps of reduction the initial or boundary conditions were algebraically deduced. The most important results obtained are summarized in the following paragraphs. In the first application; 1- Two types of similarity transformations are obtained; A translational transformation η= y (α1x + β1)m and a helical one η= y erx. The boundary condition; c0(x) isanalytically evaluated and not assumed as for example Anderson et al [4], Birkhoff [94] and Gebhart [92] works. The obtained boundary conditions match the physical boundary conditions. 2- A relation between ‘m’ and ‘n’ was analytically deduced in chapter-3 in the form; where ‘m’; the transformation ratio relating ‘x’ to ‘y’ , n is the reaction order. This relation is new and physically means that the similarity transformation of y with respect x is bounded by the reaction order n. 3- The numerical solution proved that the greater the Schmidt number, the less the maximum velocity attained by the fluid and the thinner the concentration layer. This is due to the decrease in the molecular diffusivity of the fluid which results in the previous behaviour.In the second application; 1- The concentration of species next to the wall was analytically derived 2- The numerical results show that the greater the Schmidt number, the less the maximum velocity attained by the fluid and the thinner the concentration layer. This is due to the decrease in the molecular diffusivity of the fluid and the reduction of the buoyancy effect.3- The perturbation analysis about the singularity, n=1, gives an approximate analytical solution comparable with the numerical solution .In the third application; 1- Two similarity variables on the form; and are obtained. Both cases are new and differ from the usual assumption η= x-ct used in the exact solution of Hirota-Satsuma KdV problems 2- m =-3 is deduced analytically and implies that α2=3 α1 i.e. the amplitude time factor equal three times the x amplification factor. This result is compatible with transformations of KdV equations. 3- The initial conditions for ‘u’ and ‘v’ were derived analytically and found functions of (α1x.+β1)-2. This result corresponds to the group transformation of and described in (5.18). 6.2 Conclusions from the previous results, the group method proved that it’s an effective way to get similarity solutions for both boundary and initial value problems and to release new solutions that had never been obtained by other methods. Also, it can get comparable results with miscellaneous methods in the field.