Boundary value problem for abstract first order differential equation with integral condition

Abstract Sufficient conditions of solvability and unique solvability of the boundary value problem u (m)(t) = f(t, u(τ 11(t)), . . . , u(τ 1k (t)), . . . , u (m–1)(τ m1(t)), . . . . . . , u (m–1)(τ mk (t))), u(t) = 0, for t ∉ [a, b], u (i–1)(a) = 0 (i = 1, . . . , m – 1), u (m–1)(b) = 0, are established, where τ ij : [a, b] → R (i = 1, . . . , m; j = 1, . . . , k) are measurable functions and the vector function f : ]a, b[×Rkmn → Rn is measurable in the first and continuous in the last kmn arguments; moreover, this function may have nonintegrable singularities with respect to the first argument.

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One method for the boundary value problem eigenvalues calcuating for a second-order differential equation with a fractional derivative

International Journal of Modeling Simulation and Scientific Computing ◽

10.1142/s1793962319410046 ◽

2019 ◽

Vol 10 (01) ◽

pp. 1941004

Author(s):

Temirkhan Aleroev ◽

Hedi Aleroeva ◽

Lyudmila Kirianova

Keyword(s):

Differential Equation ◽

Perturbation Theory ◽

Dirichlet Problem ◽

Boundary Value Problem ◽

Fractional Derivative ◽

Fractional Derivatives ◽

Order Differential Equation ◽

Boundary Value ◽

Second Order ◽

Linear Operators

In this paper, we give a formula for computing the eigenvalues of the Dirichlet problem for a differential equation of second-order with fractional derivatives in the lower terms. We obtained this formula using the perturbation theory for linear operators. Using this formula we can write out the system of eigenvalues for the problem under consideration.

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Numerical methods for solving the first-kind boundary value problem for a linear second-order differential equation with a deviating argument

Journal of Applied Analysis ◽

10.1515/jaa-2017-0018 ◽

2017 ◽

Vol 23 (2) ◽

Author(s):

Muhad H. Abregov ◽

Vladimir Z. Kanchukoev ◽

Maryana A. Shardanova

Keyword(s):

Differential Equation ◽

Numerical Methods ◽

Finite Difference ◽

Boundary Value Problem ◽

Order Differential Equation ◽

Boundary Value ◽

Sufficient Conditions ◽

Second Order ◽

Deviating Argument ◽

Second Order Differential Equation

AbstractThis work is devoted to the numerical methods for solving the first-kind boundary value problem for a linear second-order differential equation with a deviating argument in minor terms. The sufficient conditions of the one-valued solvability are established, and the a priori estimate of the solution is obtained. For the numerical solution, the problem studied is reduced to the equivalent boundary value problem for an ordinary linear differential equation of fourth order, for which the finite-difference scheme of second-order approximation was built. The convergence of this scheme to the exact solution is shown under certain conditions of the solvability of the initial problem. To solve the finite-difference problem, the method of five-point marching of schemes is used.

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