YU.N. DOBROVOLSKY

In this article test of Fuchs - Kovalevskaya - Painleve for the solution of a model task on a pressure filtration of gas mix in the solid environment is offered. Need of application of this test is explained by that is established by numerous researches that many known integrated nonlinear equations of mathematical physics possess property Fuchs - Kovalevskaya – Painleve. Also new equations with such property were finded. When checking more difficult equations and systems of the equations on Fuchs test - resonances with high numbers can appear Kovalevskaya – Painleve. Thus difficulties of the analytical solution quickly increase. However in a type of a high algoritmichnost the test allows successful use of methods of symbolical calculations. For example, by means of Maple V system it was succeeded to carry out full classification of integrated cases of the equations of small water with dissipation and dispersion of the lowest orders. The idea of a method consists in the following. By analogy to the ordinary differential equations of the solution of the equations to private derivatives it is possible to look for in the form of the decomposition containing feature like a mobile pole. The position of a pole is set by means of any function. Originally the solution is looked for in neighborhood of singular diversity x-x0(t) =0 in the form of the expansion justified in article, where an index α – the integral positive number that provides polar nature of mobile feature of the solution. The x0 (t) function is considered arbitrary. The received decision will be the general if this expansion includes arbitrary functions, and number of these functions equally to an order of the considered equation. Further the solution of a partial equation look for in neighborhood of singular diversity ε (x, t) =0 in the look set to article - the generalized expansion of the symmetric on independent variables. It is shown how test of Fuchs – Kovalevskaya – Painleve for the Burgers equation well works. It is set that there is one resonance of n=2 for which the compatibility ratio therefore its solution has a required arbitrariness in two functions is satisfied. When testing the first equation of the considered system by method of Fuchs - Kovalevskaya – Painleve, set the following: The first. As both items of the right member of equation have square nonlinearity, there is no feature like a mobile pole, and there is an algebraic point of branching. Therefore the decision was looked for in the form of a power series of a general view. Some solution completely matching the solutions found in the author the early are thus received. The second. As from the function point of sight all equations of system have the identical nature, this property the remaining equations of system possess also. The third. When taking rather large number of items in one or other expansion, difficulty of the analytical solution quickly rise therefore the coordinated breakaway of members of a row since some k was executed. Otherwise, it is possible to use system of character computation, for example, Maple V or Maple VII packet. The fourth. Usefulness of the received results is that selecting the initial solution thus, on them it is possible to quit a priori on the result necessary to the researcher. The fifth. In the following article it will be shown how to apply the received expansions to this system or to find conversion (if it exists), linearizing this system.

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