An efficient numerical method for a singularly perturbed Fredholm integro-differential equation with integral boundary condition

Durmaz, Muhammet Enes; Amirali, Ilhame; Amiraliyev, Gabil M.

doi:10.1007/s12190-022-01757-4

Cited by 20 publications

(10 citation statements)

References 39 publications

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“…To observe the behaviour of the considered problem, we plot the numerical solution profiles for various small values of the perturbation parameter in Figure 1 and Figure 2 . Furthermore, in Tables 2 and 4 , we compare the maximum absolute error of the proposed scheme with that of the scheme in [ 9 ] for both of the considered examples. The tables demonstrate that the proposed scheme outperforms the result in [ 9 ] in terms of the maximum absolute error.…”

Section: Numerical Examples and Discussionmentioning

confidence: 99%

“…Furthermore, in Tables 2 and 4 , we compare the maximum absolute error of the proposed scheme with that of the scheme in [ 9 ] for both of the considered examples. The tables demonstrate that the proposed scheme outperforms the result in [ 9 ] in terms of the maximum absolute error.…”

Section: Numerical Examples and Discussionmentioning

confidence: 99%

“…The uniqueness of the solution is guaranteed by this discrete maximum principle. The existence follows easily since, as for linear problems, the existence of the solution is implied by its uniqueness [ 9 ]. The discrete maximum principle enables us to prove the following lemma which provides the boundedness of the solution.…”

Section: The Fitted Operator Difference Schemementioning

confidence: 99%

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A fitted operator numerical method for singularly perturbed Fredholm integro-differential equation with integral initial condition

Oljira,

Woldaregay

2024

BMC Res Notes

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Objectives In this paper, a uniformly convergent numerical scheme is proposed for solving a singularly perturbed Fredholm integro-differential equation with an integral initial condition. The equation involves a left boundary layer which makes it difficult to solve it using the standard numerical methods. A fitted operator finite difference method is used to approximate the differential part of the equation and the composite Simpson $$\frac{1}{3}$$ 1 3 rule is used for the integral parts of the equation and the initial condition. Result The stability bound and error estimation of the approximated solution are performed, to show the uniform convergence of the scheme with order one in the maximum norm. Numerical test examples are provided to calculate the maximum absolute errors, thrgence, and the uniform error for a couple of examples to support theoretical analysis.

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Section: Numerical Examples and Discussionmentioning

confidence: 99%

Section: Numerical Examples and Discussionmentioning

confidence: 99%

Section: The Fitted Operator Difference Schemementioning

confidence: 99%

See 1 more Smart Citation

A fitted operator numerical method for singularly perturbed Fredholm integro-differential equation with integral initial condition

Oljira,

Woldaregay

2024

BMC Res Notes

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“…More specically, in [30,31], a difference scheme of the exponential type on a uniform mesh is considered. A fitted difference scheme on a piecewise uniform mesh is utilized in [32,33] to solve the problem. A difference scheme with an accuracy of O(N −2 lnN ) on a Shishkin mesh for SPFIDE with Robin boundary condition is given in [34].…”

Section: Introductionmentioning

confidence: 99%

First-order numerical method for the singularly perturbed nonlinear Fredholm integro-differential equation with integral boundary condition

Amirali

Durmaz

Acar

et al. 2023

J. Phys.: Conf. Ser.

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In this work, we consider first-order singularly perturbed quasilinear Fredholm integro-differential equation with integral boundary condition. Building a numerical strategy with uniform ε-parameter convergence is our goal. With the use of exponential basis functions, quadrature interpolation rules and the method of integral identities, a fitted difference scheme is constructed and examined. The weight and remainder term are both expressed in integral form. It is shown that the method exhibits uniform first-order convergence of the perturbation parameter. Error estimates for the approximation solution are established and a numerical example is given to validate the theoretical findings.

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“…As a result, in recent years, few works on the numerical solution of singularly perturbed Fredholm/Volterra integral equations have been recorded in the literature [ 11 , 12 ]. Durmaz et al [ 8 ] developed a fitted difference scheme on Shishkin mesh using interpolating quadrature rules and an exponential basis function for the numerical treatment of the singularly perturbed Fredholm integro-differential equation with mixed boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Solving singularly perturbed fredholm integro-differential equation using exact finite difference method

Badeye,

Woldaregay,

Dinka

2023

BMC Res Notes

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Objectives In this paper, a numerical scheme is designed for solving singularly perturbed Fredholm integro-differential equation. The scheme is constructed via the exact (non-standard) finite difference method to approximate the differential part and the composite Simpson’s 1/3 rule for the integral part of the equation. Result The stability and uniform convergence analysis are demonstrated using solution bound and the truncation error bound. For three model examples, the maximum absolute error and the rate of convergence for different values of the perturbation parameter and mesh size are tabulated. The computational result shows, the proposed method is second-order uniformly convergent which is in a right agreement with the theoretical result.

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An efficient numerical method for a singularly perturbed Fredholm integro-differential equation with integral boundary condition

Cited by 20 publications

References 39 publications

A fitted operator numerical method for singularly perturbed Fredholm integro-differential equation with integral initial condition

A fitted operator numerical method for singularly perturbed Fredholm integro-differential equation with integral initial condition

First-order numerical method for the singularly perturbed nonlinear Fredholm integro-differential equation with integral boundary condition

Solving singularly perturbed fredholm integro-differential equation using exact finite difference method

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