The Optimization of the Approximate Method of the Solution of Linear Boundary Value Problems for Volterra Integrodifferential Equations with Functional Delays
This article studies the possibility of approximate solution of resolving equations for boundary value problems of Volterra linear integer differential equations with functional delays. These resolving equations are obtained using a new form of function of flexible structure deduced by means of boundary conditions and initial functions. Using this form, it was shown that all the linear boundary value problems of Volterra integer differential equations of delay type are converted to integral equations of Volterra- Fredholm mixed type with the common argument. The boundary value problems of certain types of equations of neutral and advanced types are also transformed to the resolving equations of the same type.
Further on the issue arises to solve the obtained resolving equations. Since firstly uncertain parameters of solutions of the boundary value problem include these formulas of functions and cores of resolving integral equations, then due to their optimal choice, the exact solution can be found, or if it is difficult or impossible, the approximate solution. The approximate solution of the resolving integral equations of Volterra-Fredholm mixed type with the common argument in this work is obtained by the method of successive approximations. In its implementation, as well as in the use of other methods, due to the optimal choice of parameters, the amount of calculations can be reduced and the process of convergence of the method can be accelerated. The formulas for calculating the error are obtained for the approximate solutions of resolving equations, and using them also the error calculation formulas for initially set boundary value problems. The possible variant of the solution is also considered for the case when due to the choice of parameters all the cores for integrals with constant limits of integration can be identically equal to zero. The above example is subject to this variant of solution. The formulas of error estimation for resolving equations and initially set boundary value problems are also obtained for this variant of solution.
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