Abstract: In the thesis, the problems of transverse impact to the flexible elastic membranes possessing 

a hole are investigated. In general the dynamics of the membrane is described by three sets of partial 

differential equations in three different regions with a priori unknown boundaries. The impact velocity is 

assumed to be high, so that one of the regions degenerates into a line of a strong discontinuity across 

which some nonlinear relations, following from the equations of motion, hold. Two different type 

problems, appropriate for two different type of boundary conditions on the hole are studied. The exact 

solutions to the problems on the radial wavefront are obtained, which, in particular, show that for a rigid 

hole the radial wavefront is a strong discontinuity wave, while for a free hole the derivatives of unknown 

functions are continuous. The transverse wavefront is the strong discontinuity wave for both cases. 

The principle difference of the problems studied in the thesis from the already solved problems in 

dynamics of membranes is that the problems considered in the thesis do not possess a similarity 

solution like the problems studied previously and hence one has to deal with the nonlinear initial and 

boundary value problems for a set of partial differential equations in regions with a priori unknown 

boundaries.The algorithms for the numerical solution to the problems are worked out. The numerical 

algorithms are based on the characteristic method of solution of the hyperbolic type of systems by 

making use of the jump conditions on the unknown boundary of regions.The equations that appear in 

two different regions which are investigated in the thesis can also be viewed as a set of equations in one 

large region but in the latter case the coefficients become the impulsive type of functions. In the ransition 

through the strong discontinuity wave the coefficients of the equations change impulsively. In this sense 

the problems considered in the thesis are related to the study of initial and boundary value problems for 

impulsive partial differential equations. In the thesis the numerical solutions to the problems are obtained 

and the different graphs of the unknown quantities are brought. The convergence and stability of the 

numerical solutions have been established. 

 Key words: Transverse impact, flexible elastic membrane, impact velocity, strong discontinuity, radial 

wavefront, transverse wavefront, similarity solution, characteristic method, hyperbolic type of system.