Extension method for a class of loaded differential equations with nonlocal integral boundary conditions

In this paper we investigate a class of loaded ordinary differential equations with nonlocal integral boundary conditions in terms of an abstruse operator equation Bu = A 2 u - q Ψ( u ) = f, f ∈ Y, (1) D ( B ) = { u ∈ D ( A 2) : Φ( u ) = NF ( Au ) , Φ( Au ) = PF ( Au )} . A loaded part and nonlocal integral boundary conditions of these equations are described using functional vectors Ψ( u ) and F ( Au ) , respectively. Such equations follow from Extension Theory of linear operators. The necessary and sufficient solvability conditions of these equations are given by the determinant of some matrix. In the case when this determinant is nonzero, a direct method for exact solution of this class of loaded differential equations is proposed. If some problem can be reduced to the type of equation under consideration, then it can be easily solved using the extension method. This method, for q = 0 -> , also gives the necessary and sufficient solvability conditions and the exact solution of a class of ordinary differential equations with nonlocal integral boundary conditions in terms of an abstruse operator equation Bu = A 2 u = f, D ( B ) = { u ∈ D ( A 2) : Φ( u ) = NF ( Au ) , Φ( Au ) = PF ( Au )} , f ∈ Y

Factorization method for solving nonlocal boundary value problems in Banach space

This article deals with the factorization and solution of nonlocal boundary value problems in a Banach space of the abstract form B1 u = A u - S Φ( u )- G Ψ(A0 u) = f, u ∈ D (B1) , where A, A0 are linear abstract operators, S, G are vectors of functions, Φ, Ψ are vectors of linear bounded functionals, and u, f are functions. It is shown that the operator B1 under certain conditions can be factorized into a product of two simpler lower order operators as B 1 = BB0. Then the solvability and the unique solution of the equation B1 u = f easily follow from the solvability conditions and the unique solutions of the equations Bv = f and B0 u = v. The universal technique proposed here is essentially different from other factorization methods in the respect that it involves decomposition of both the equation and boundary conditions and delivers the solution in closed form. The method is implemented to solve ordinary and partial Fredholm integro-differential equations

Factorization method for solving nonlocal boundary value problems in Banach space

This article deals with the factorization and solution of nonlocal boundary value problems in a Banach space of the abstract form B1u = Au − SΦ(u) − GΨ(A0u) = f, u ∈ D(B1),where A, A0 are linear abstract operators, S, G are vectors of functions, Φ, Ψ are vectors of linear bounded functionals, and u, f are functions. It is shown that the operator B1 under certain conditions can be factorized into a product of two simpler lower order operators as B1 = BB0. Then the solvability and the unique solution of the equation B1u = f easily follow from the solvability conditions and the unique solutions of the equations Bv = f and B0u = v. The universal technique proposed here is essentially different from other factorization methods in the respect that it involves decomposition of both the equation and boundary conditions and delivers the solution in closed form. The method is implemented to solve ordinary and partial Fredholm integro-differential equations

Extension and decomposition method for differential and integro-differential equations

A direct method for finding exact solutions of differential or Fredholm integro-differential equations with nonlocal boundary conditions is proposed. We investigate the abstract equations of the form Bu = Au-gF(Au) = f and B1u = A2u-qF(Au)-gF(A2u) = / with abstract nonlocal boundary conditions Φ(u) = N Ψ(Au) and Φ(u) = N Ψ(Au); Φ(Au) = DF(Au) + N Ψ(A2u); respectively, where q,g are vectors. D, N are matrices, F, Φ, Ψ are vector-functions. In this paper: 1. we investigate the correctness of the equation Bu = f and find its exact solution. 2. we investigate the correctness of the equation B1U = f and find its exact solution. 3. we find the conditions under which the operator B1 has the decomposition B1 =B2. i.e. B1 is a quadratic operator, anil then we investigate the correctness of the equation B2u = f1 and find its exact solntion. © 2019 L.N. Gumilyov Eurasian National University

Factorization Method for Solving Multipoint Problems for Second Order Difference Equations with Polynomial Coefficients

This paper is devoted to the study of second order linear difference equations with polynomial coefficients subject to multipoint boundary conditions. We provide necessary and sufficient conditions for the existence and uniqueness of solutions and find the unique solution in closed form by using factorization techniques. © 2020, Springer Nature Switzerland AG

A procedure for factoring and solving nonlocal boundary value problems for a type of linear integro-differential equations

The aim of this article is to present a procedure for the factorization and exact solution of boundary value problems for a class of n-th order linear Fredholm integro-differential equations with multipoint and integral boundary conditions. We use the theory of the extensions of linear operators in Banach spaces and establish conditions for the decomposition of the integro-differential operator into two lower-order integro-differential operators. We also create solvability criteria and derive the unique solution in closed form. Two example problems for an ordinary and a partial intergro-differential equation respectively are solved. © 2021 by the authors. Licensee MDPI, Basel, Switzerland

On the Solution of Boundary Value Problems for Loaded Ordinary Differential Equations

This chapter is devoted to the solution of the so-called loaded ordinary differential equations which arise in applications in sciences and engineering. We propose a direct operator method for examining existence and uniqueness and constructing the solution in closed form to a class of boundary value problems for loaded nth-order ordinary differential equations with multipoint and integral boundary conditions. © 2021, Springer Nature Switzerland AG

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On the solution of boundary value problems for ordinary differential equations of order n and 2n with general boundary conditions

We present a method for examining the existence and uniqueness and obtaining the exact solution to boundary value problems consisting of the differential equation Au = f, where A is a linear ordinary differential operator of order n, and multipoint and integral boundary conditions. We also derive a formula for computing the exact solution to even order boundary value problems encompassing the differential equation A2u = f subject to 2n general boundary conditions. The method is based on the correct extensions of operators in Banach spaces. © Springer Nature Switzerland AG 2020