Abstract
The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found
Highlights
The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced
Solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations
Using the Green’s function, one can solve the problem of finding eigenvalues, which are very relevant in quantum field theory
Summary
The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. Solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Let’s note that using the Green’s function for a homogeneous problem, it is possible to calculate the solution of an inhomogeneous differential equation. Using the Green’s function, one can solve the problem of finding eigenvalues, which are very relevant in quantum field theory. Actual and important in mathematical research are the problems of integrating linear ordinary differential equations of the third order, as well as constructing on their basis the Green’s function of the boundary value problem for an ordinary differential equation of the third order
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