Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 2. Decomposition-extension method for squared operators

Vassiliev, Nikolay; Parasidis, I. N.; Providas, E.

doi:10.31799/1684-8853-2019-2-2-9

Cited by 9 publications

(14 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Here, problem (1) is investigated and solved for the case ln, XY mainly by functional transformations and by employing the properties of linear operators. The results of this method coincide with the results of [14,15] for ln.…”

Section: Introductionsupporting

confidence: 83%

“…The last equation easily implies the unique solution of (12) defined by (15). The operator B 1 , by Theorem 1, is correct.…”

Section: R C D a mentioning

confidence: 77%

“…Then R(B 1 )Y, which since BuB 1 v gives R(B)Y. Since the operator 1 A  and the functionals F 1 , …, F n ,  1 , …,  l are bounded, the boundedness of B -1 follows from (15). Hence, the operator B is correct.…”

Section: R C D a mentioning

confidence: 90%

“…We solve problem (24) using the previous Mathematica script, by changing only the input parameters m, l, n, F, f(t), g(t), N, (u). The program checks if all the assumptions of Theorem 2 hold and the unique solution is determined explicitly in finitely many steps of computation, following (15)…”

Section: R C D a mentioning

confidence: 99%

“…Existence theorems for nonlocal multipoint boundary value problem and IDEs via analytical methods are given in [11][12][13]. Boundary value problems B: XX of the type of (1) for the specific case of ln Articles ТЕОРЕТИЧЕСКАЯ И ПРИКЛАДНАЯ МАТЕМАТИКА have been studied by Vassiliev, Parasidis, Providas in [14,15], using the extension method. The extension method is a generalization of the direct method, which is presented in [4].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

An exact solution method for Fredholm integro-differential equations

Tsilika

2019

ICS

View full text Add to dashboard Cite

Introduction: Linear boundary value problems for Fredholm ordinary integro-differential equations are seldom consideredwith integral boundary conditions in the literature. In our case, integro-differential equations are subject to multipoint or nonlocalintegral boundary conditions. It should be noted that finding exact solutions even for multipoint problems or problems with nonlocalintegral boundary conditions with a differential equation is a difficult task. Purpose: Finding the uniqueness and existencecriterion of solutions for Fredholm ordinary integro-differential equations with multipoint or nonlocal integral boundary conditionsand obtaining exact solutions in closed form of such problems. Results: Within the class of abstract operator equations, for thespecial case of Fredholm integro-differential equations with multipoint or nonlocal integral boundary conditions, a criterion for theexistence and uniqueness of an exact solution is proved and the analytical representation of the solution is given. A direct methodanalytically solving such problems is proposed, in which all calculations are reproducible in any program of symbolic calculations.If the user sets the input parameters and the initial conditions of the problem, the computer codes check the conditions of existenceand uniqueness and of solution generate the analytical solution. The stages of the solution method are illustrated by twoexamples. The article uses computer algebra system Mathematica to demonstrate the results.

show abstract

Section: Introductionsupporting

confidence: 83%

“…The last equation easily implies the unique solution of (12) defined by (15). The operator B 1 , by Theorem 1, is correct.…”

Section: R C D a mentioning

confidence: 77%

Section: R C D a mentioning

confidence: 90%

Section: R C D a mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An exact solution method for Fredholm integro-differential equations

Tsilika

2019

ICS

View full text Add to dashboard Cite

show abstract

Factorization and Solution of Linear and Nonlinear Second Order Differential Equations with Variable Coefficients and Mixed Conditions

Providas

2021

Nonlinear Analysis, Differential Equations, and Applications

View full text Add to dashboard Cite

Integro-Differential Equations Embodying Powers of a Differential Operator

Parasidis

Providas²

2019

Vestnik of Samara University. Natural Science Series

Self Cite

View full text Add to dashboard Cite

We establish solvability and correctness criteria for two Fredholm type linear integro-diﬀerential operators B2, B4 encompassing up to second and fourth powers, respectively, of a diﬀerential operator A� with aknown inverse I = A�−1. We also derive explicit solution formulae to corresponding initial and boundary value problems by using the inverse of the diﬀerential operator. The approach is based on the theory of the extensions of linear operators in Banach spaces. Three example problems for ordinary and partial integro-diﬀerential operators are solved.

show abstract

Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 2. Decomposition-extension method for squared operators

Cited by 9 publications

References 5 publications

An exact solution method for Fredholm integro-differential equations

An exact solution method for Fredholm integro-differential equations

Factorization and Solution of Linear and Nonlinear Second Order Differential Equations with Variable Coefficients and Mixed Conditions

Integro-Differential Equations Embodying Powers of a Differential Operator

Contact Info

Product

Resources

About