A problem with parameter for the integro-differential equations

• Published in: Mathematical modelling and analysis Vol. 26; no. 1; pp. 34 - 54
• Main Authors: Bakirova, Elmira A., Assanova, Anar T., Kadirbayeva, Zhazira M.
• Format: Journal Article
• Language: English
• Published: Vilnius Vilnius Gediminas Technical University 01.01.2021
• Subjects: algorithm; Algorithms; Approximation; Boundary value problems; Differential equations; integro-differential equation; Linear algebra; numerical solution; Ordinary differential equations; Parameters; problem with parameter; solvability criteria; Vectors (mathematics); δm(θ) general solution; numerical solution; solvability criteria; integro-differential equation; δm(θ) general solution; problem with parameter; algorithm
• ISSN: 1392-6292; 1648-3510; 1648-3510
• DOI: 10.3846/mma.2021.11977

The article proposes a numerically approximate method for solving a boundary value problem for an integro-differential equation with a parameter and considers its convergence, stability, and accuracy. The integro-differential equation with a parameter is approximated by a loaded differential equation with a parameter. A new general solution to the loaded differential equation with a parameter is introduced and its properties are described. The solvability of the boundary value problem for the loaded differential equation with a parameter is reduced to the solvability of a system of linear algebraic equations with respect to arbitrary vectors of the introduced general solution. The coefficients and the right-hand sides of the system are compiled through solutions of the Cauchy problems for ordinary differential equations. Algorithms are proposed for solving the boundary value problem for the loaded differential equation with a parameter. The relationship between the qualitative properties of the initial and approximate problems is established, and estimates of the differences between their solutions are given.