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Generalization of the Chebishev polynomials and their application for solving systems of three-point scalar and vector equations

Author: tamari mkheidze

ABSTRACT As is known boundary and initial-boundary value problems for partial derivatives differential equations may be considered as operator differential equations on corresponding function spaces. For such kind of operations one of the universal methods for construction numerical algorithms is a finite difference method (a linear method). Approximate solution of boundary value problems for differential equation by semi-discrete method is reduced to the solution of the systems of differential-difference equation of special structure which can be imagined as operator difference schemes in corresponding function spaces. For such schemes the obtaining of such a priori estimates from which follows stability and convergence of approximate solution to accurate solution in natural classes have essential practical and theoretical importance. The use of a linear method for elliptic equations gives three-point operator-coefficient vector equations system. By using the method of associated polynomials it is easy to prove correct solution of such systems both in linear and non-linear case. The master’s work deals with the systems of relevantly constant and three-dimensional scalar equations with variable coefficients and operator-coefficient (generally non-linear) three-point vector equations systems. The questions of these systems correct solution using classical Chebyshev polynomials and generalized polynomials have been studied. The system of three-point scalar equations is formed in the result of three-point approximation of the boundary-value problem for ordinary differential equations of the second order. It is proved that the system of three-point constant coefficient scalar equations in certain conditions has the only solution and this solution is constructed by means of two-variable Chebyshev polynomials. The solution is estimated by means of corresponding characteristic equation roots from which a conclusion about the state of the system is made. The system of three-point constant coefficient scalar equations has been considered. The solution of this system is constructed by means of generalized polynomials and the questions of existence the solution, relation and stability of the system is examined. The algorithm of persistent change of solution the system of three-point variable coefficient scalar equations which is based on explicit representation of the solution by means of generalized polynomials. The work also deals with the system of three-point operator-coefficient vector equations. As is known systems of such type are formed for elliptic boundary-value problems in the result of using semi-discrete, finite difference, variation (in case of a proper choice of coordinate functions) and finite elements methods. Two cases are regarded when the coefficients are linear and non-linear operators. Accordingly, in case of fulfillment of certain natural conditions on the basis of two-variable Chebyshev polynomials properties, it is shown (in both cases) correct solution of the system. It should be noted that the proof of the non-linear system leads to the use of iteration process for exciting system. The obtained results are used for investigation of the questions of stability and convergence of discrete problem corresponding to some elliptic boundary-value problems.

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