Singularly Perturbed Integro-Differential Equations with Diagonal Degeneration of the Kernel in Reverse Time

Бободжанов, А. А.; Safonov, V. F.

doi:10.1023/b:dieq.0000028721.81712.67

Cited by 9 publications

(8 citation statements)

References 1 publication

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“…Systems (11b) for j = 1, 2, ..., n, are solvable in the space C ∞ ([t 0 , T], C n ) if and only if the identities (H j (t), χ j (t)) ≡ 0, j = 1, n, ∀t ∈ [t 0 , T] hold. It is easy to see that these identities coincide with the identities (9). Thus, the conditions (9) are necessary and sufficient for the solvability of the system (8) in the space U.…”

Section: Iterative Problems and Their Solvability In The Space Umentioning

confidence: 77%

“…α k (t)ϕ k e τ k + 2 ∑ j=1 K(t, t)ϕ k λ j (t) α k (t)e τ j and subordinate the resulting sum to the orthogonality conditions (9). We will have…”

Section: Construction Of the Solution Of The First Iteration Problem mentioning

confidence: 99%

“…The presence of such forces creates serious problems for the numerical integration of the corresponding differential equations. Therefore, asymptotic methods are usually applied to such equations, the most famous of which are the Feshchenko-Shkil-Nikolenko splitting method [1,2] and the Lomov's regularization method [3][4][5][6][7][8][9][10][11][12][13][14]. The splitting method is especially effective when applied to equations with a rapidly oscillating inhomogeneity, and in the case of an inhomogeneity containing both rapidly and slow components, the Lomov's regularization method turned out to be the most effective.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Generalization of the Regularization Method to Singularly Perturbed Integro-Differential Systems of Equations with Rapidly Oscillating Inhomogeneity

Бободжанов

Kalimbetov

Safonov

2021

Axioms

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In this paper, we consider systems of singularly perturbed integro-differential equations with a rapidly oscillating right-hand side, including an integral operator with a slowly varying kernel. Differential equations of this type and integro-differential equations with slowly varying inhomogeneity and with a rapidly oscillating coefficient at an unknown function are studied. The main goal of this work is to generalize the Lomov’s regularization method and to reveal the influence of the rapidly oscillating right-hand side on the asymptotics of the solution to the original problem.

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Section: Iterative Problems and Their Solvability In The Space Umentioning

confidence: 77%

“…α k (t)ϕ k e τ k + 2 ∑ j=1 K(t, t)ϕ k λ j (t) α k (t)e τ j and subordinate the resulting sum to the orthogonality conditions (9). We will have…”

Section: Construction Of the Solution Of The First Iteration Problem mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Generalization of the Regularization Method to Singularly Perturbed Integro-Differential Systems of Equations with Rapidly Oscillating Inhomogeneity

Бободжанов

Kalimbetov

Safonov

2021

Axioms

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“…For a survey of early results in the theoretical analysis of singularly perturbed Volterra integro-differential equations (VIDEs) and in the numerical analysis and implementation of various techniques for these problems we refer to the book [17]. An analysis of approximate methods when applied to singularly perturbed VIDEs can also be found in [2,5,6,18,21,27,29,33].…”

Section: Introductionmentioning

confidence: 99%

A Fitted Approximate Method for a Volterra Delay-Integro-Differential Equation with Initial Layer

Amiraliyev¹,

Yapman²,

Kudu³

2018

HJMS

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This study is concerned with the finite-difference solution of singularly perturbed initial value problem for a linear first order Volterra integrodifferential equation with delay. The method is based on the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder terms in integral form. The emphasis is on the convergence of numerical method. It is shown that the method displays uniform convergence in respect to the perturbation parameter. Numerical results are also given.

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“…Various approximating aspects for singularly perturbed VIDE's have also been investigated in [2,5,6,18,20,26,28,31].…”

Section: Introductionmentioning

confidence: 99%

A finite-difference method for a singularly perturbed delay integro-differential equation

Kudu¹,

Amirali²,

Amiraliyev³

2016

Journal of Computational and Applied Mathematics

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We consider the singularly perturbed initial value problem for a linear first order Volterra integro-differential equation with delay. Our purpose is to construct and analyse a numerical method with uniform convergence in the perturbation parameter. The numerical solution of this problem is discretised using an implicit difference rules for differential part and the composite numerical quadrature rules for integral part. On a layer-adapted mesh error estimations for the approximate solution are established. Numerical examples supporting the theory are presented.

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Singularly Perturbed Integro-Differential Equations with Diagonal Degeneration of the Kernel in Reverse Time

Cited by 9 publications

References 1 publication

Generalization of the Regularization Method to Singularly Perturbed Integro-Differential Systems of Equations with Rapidly Oscillating Inhomogeneity

Generalization of the Regularization Method to Singularly Perturbed Integro-Differential Systems of Equations with Rapidly Oscillating Inhomogeneity

A Fitted Approximate Method for a Volterra Delay-Integro-Differential Equation with Initial Layer

A finite-difference method for a singularly perturbed delay integro-differential equation

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