Existence and multiplicity of solutions of the problem x” = ‐q(t) |x|p sign x (i), x(0) = x(1) = 0 (ii) are investigated by reducing equation (i) to a quasi‐linear one so that both equations are equivalent in some domain O. If a solution of corresponding quasi‐linear problem is located in the domain of equivalence O, then this solution solves the original problem also. If this process of quasilinearization is possible for multiple essentially different linear parts, then multiple solutions to the problem (i), (ii) exist. Darbe nagrinejamas taip vadinamas Emdeno‐Faulerio kvazitiesines diferencialines lygties homogeninio kraštinio uždavinio sprendiniu egzistavimas ir daugialypumas. Parodyta, kad šio uždavinio sprendinio daugialypumas priklauso nuo tam tikru būdu gautos kvazilinearizuotos lygties tiesines dalies savybiu.

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Bioenergy is one of the widespread renewable energy sources. Energy from biomass thermal conversion can reduce greenhouse emissions and have a positive effect on climate change. Biomass conversion is generally carried out in reactors of cylindrical shape. From a modelling point of view many factors have to be taken into account in order to optimize thermal efficiency of the conversion process. One of the important methods for the analysis of complex fluid flows is hydrodynamic stability theory. Base flow solution in classical hydrodynamic stability problems is usually found as a simple analytical solution of the equations of motion. Biomass conversion problems lead to nonlinear boundary value problems, which can be either solved numerically or analyzed using the bifurcation theory. In the present paper we analyze a mathematical model of heat transfer in the presence of nonlinear heat sources. This model includes the study of positive solutions to a nonlinear boundary value problem with certain boundary conditions. The equations in a problem contain several parameters, which essentially affect the behaviour and the number of solutions. Bifurcation analysis of the problem, conducted with respect to the parameters, allows obtaining somewhat precise results on the number of positive solutions. Generally, two, one and zero positive solutions are possible, depending on the values of the parameters. The obtained solutions represent base flow for the hydrodynamic stability problem, which can be solved with the objective to identify the factors affecting the conversion process.

Consider two-point boundary value problems of resonance typeWe investigate existence and multiplicity of solutions to such problems by using the quasilinearization process.

An equation is considered together with the boundary conditions , . This problem under appropriate conditions can be reduced to quasilinear problem for two-dimensional differential system. The conditions for existence of multiple solutions to the original problem are obtained by multiply applying the quasilinearization technique.