I. Influence Element Method to derive Green’s functions and Green’s matrices and respective Poisson’s type integral formulas (especially in closed form) for two and 3D boundary-value problems for canonical orthogonal domains (Cartesian, polar, cylindrical, spherical etc., and composite structures from them) for elliptic, parabolic and hyperbolic differential equations in the fields of:

II. Application of the Green’s functions and Green’s matrices to solve non homogeneous boundary value problems, arising from different branches of modern science, industry and technologies.

Mechanics: Integral Equations in Elasticity, Continuum Mechanics, Solid Mechanics, Structural Mechanics, Static, Dynamics, Theory of Elasticity, Thermo-elasticity, Poro-elasticity, Thermo-viscous-elasticity, Creeping Theory of Concrete, Polymers, Metals.

My plans in future are to develop this theory in general case for domains of curvilinear system of co-ordinate both in static and dynamics and to obtain new concrete results and analytical expressions for Green's functions and matrices. On this basis to publish a new Encyclopaedia on Green's Functions and Matrices which will be very useful for researchers, engineers, involved in solving of boundary value problems arising from different branch of industry and Engineering. This Encyclopaedia will play the seam role in Engineering as Table of Integrals and Sums in Mathematics. The Encyclopaedia will present an impact in development of computational methods and possibilities to solve Engineering Problems.

2. I elaborated a new method to solve 2-D and 3-D boundary-value problems in Solid Mechanics (so called Influence Elements Method). Influence Element Method (IEM) is based on the constructed in analytical form the Green's functions and matrices for elastic bodies with the simple canonical forms. IEM can be also use for the analyse of elastic and thermo elastic displacements and stresses fields for the more complicate in forms bodies, which represent a synthesis of simple canonical bodies. This method is expected to be more exactly and more efficient that classical method of finite elements, boundary element method and others. These results were summed in my book "Influence Elements Method" published in Romanian in Republic of Moldova.

My plans in future are to organize on the basis of Influence Elements Method and on the basis of constructed Green's matrices in analytical form the elaboration of the new computer software for analysis of stress-strain states of Civil and Mechanical Engineering Structures. On the basis of constructed Green's matrices and Influence Elements Method to compute stiffness matrices for Finite Element Method. To elaborate the compatibility between Influence Elements Method and Finite Element Method. On this basis to generate the new development of computational methods, including Finite Element Method. The established compatibility between Influence Elements Method and Finite Element Method will permit to solve boundary value problems not only for elastic bodies, which are composed from several bodies with the simple regular shape (for all regular domains can be used IEM), but also for the complicated bodies, which include additional the non regular domains (for these non regular domains can be used FEM).

3. For the first time I suggested a new formula for determination of Dynamical and Static Thermal Stresses in different Civil Engineering Elements and Structures. This formula represents a generalization of Green's and Maysel's formulas onto uncoupled thermo elasticity. These results were published in journals and conferences in USA, Japan and in my book, published by WIT press in UK and USA.

My plans in future are to develop this formula in the area of Coupled Thermo elasticity. On the basis of this formula to elaborate a new Encyclopaedia of Thermo elastic Static and Dynamical Influence Functions for Displacements.

4. Suggested methodology, applied for elaboration of the new theory of Green's matrices constructing for elastostatics boundary value problems is universally. It means that this methodology is valuable for elaboration of others new theories of Green's matrices constructing, not only for the elliptic Lame's set of system of equations, but also for the others systems of differential equations in partial derivatives for the bodies of numerous systems of orthogonal coordinates. This demonstrates that the methodology and the results of realization of this methodology will contribute essentially to appearance of the new theories of solving of boundary value problems. This will have a considerable influence on development of different areas of sciences, based on the different phenomena which are describe by with the systems o differential equations in partial derivatives. It is well known that at present for solving the vector boundary value problems are used directly the classical methods of mathematical physics, which were elaborated for solving of scalar boundary value problems. We have to mention that the main idea of the proposed methodology is to represent the solutions of the vector boundary value problems in a form of integrals, the kernels of which are the influence functions for the scalar boundary value problems, solvable in analytical form. As a final result we bring the process of solution of vector boundary value problems (are describe with the system of differential equations in partial derivatives) to the solution of some scalar boundary value problems (to the construction of some influence functions for the scalar differential equations in partial derivatives), solvable in the analytical form (also it is necessary to compute certain surface integrals or sometimes, in the more general cases of boundary conditions it is necessary to solve boundary integral equations). This is the main difference between the proposed in this project methodology and the existing methods to solve vector boundary value problems (or for Green's matrices constructing).

Decision nr 603 of Superior Attestation Commission of Republic of Moldova from 14.04.06