Pell–Lucas series approach for a class of Fredholm-type delay integro-differential equations with variable delays

Demir, Duygu Dönmez; Lukonde, Alpha Peter; Kürkçü, Ömür Kıvanç; Sezer, Mehmet

doi:10.1007/s40096-020-00370-5

Cited by 10 publications

(4 citation statements)

References 20 publications

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“…On the other hand, we shall directly determine the accuracy of the proposed method with the aid of the mean values of the residual functions with respect to N and M . Formerly, this mathematical procedure was used in (Dönmez Demir et al ., 2021; Kürkçü et al ., 2019) for integro-differential and fractional differential equations of non-system type. Now, we can adapt this procedure to the proposed method dealing with the system (1) as follows:which implies that the accuracy of the method is detected by the residual error bound trueRˆj,N,M.…”

Section: Residual Error Estimation: Correction Of Set Of Approximate ...mentioning

confidence: 99%

A generic numerical method for treating a system of Volterra integro-differential equations with multiple delays and variable bounds

Kürkçü,

Sezer

2024

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PurposeThis study aims to treat a novel system of Volterra integro-differential equations with multiple delays and variable bounds, constituting a generic numerical method based on the matrix equation and a combinatoric-parametric Charlier polynomials. The proposed method utilizes these polynomials for the matrix relations at the collocation points.Design/methodology/approachThanks to the combinatorial eligibility of the method, the functional terms can be transformed into the generic matrix relations with low dimensions, and their resulting matrix equation. The obtained solutions are tested with regard to the parametric behaviour of the polynomials with $\alpha$, taking into account the condition number of an outcome matrix of the method. Residual error estimation improves those solutions without using any external method. A calculation of the residual error bound is also fulfilled.FindingsAll computations are carried out by a special programming module. The accuracy and productivity of the method are scrutinized via numerical and graphical results. Based on the discussions, one can point out that the method is very proper to solve a system in question.Originality/valueThis paper introduces a generic computational numerical method containing the matrix expansions of the combinatoric Charlier polynomials, in order to treat the system of Volterra integro-differential equations with multiple delays and variable bounds. Thus, the method enables to evaluate stiff differential and integral parts of the system in question. That is, these parts generates two novel components in terms of unknown terms with both differentiated and delay arguments. A rigorous error analysis is deployed via the residual function. Four benchmark problems are solved and interpreted. Their graphical and numerical results validate accuracy and efficiency of the proposed method. In fact, a generic method is, thereby, provided into the literature.

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Section: Residual Error Estimation: Correction Of Set Of Approximate ...mentioning

confidence: 99%

A generic numerical method for treating a system of Volterra integro-differential equations with multiple delays and variable bounds

Kürkçü,

Sezer

2024

Self Cite

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“…Many researchers have recently utilized the collocation method to solve this differential equation. Taylor [24][25][26][27][28], Pell-Lucas [29], Morgan-Voyce [30][31][32][33][34], Dickson [35], ortho-exponential [36], and Bernoulli [37] matrix methods have been used to solve the linear differential, integro-differential equations, fractional differential equations [38], nonlinear differential equations, multi-pantograph [39], delay differential equations and difference and Fredholm integro-difference equation [40]. This method can significantly aid in reducing the complexity of solutions for large-scale epidemic models and can contribute to obtaining results that closely approximate the exact solutions of ordinary differential equation systems.…”

Section: Introductionmentioning

confidence: 99%

A numerical approach for an epidemic SIR model via Morgan-Voyce series

İlhan,

Şahin

2023

International Journal of Mathematics and Computer in Engineering

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This study presents the problem of spreading a disease that is not fatal in a population by using the Morgan-voyce collocation method. The main aim of this paper is to find the exact solutions of the SIR model with vaccination. The problem may be modeled with a nonlinear system of ordinary differential equations, mathematically. The presented method reduces the problem into a nonlinear algebraic system of equations by using unknown coefficients Morgan-Voyce polynomials and expanding approximate solutions. The Morgan-Voyce Polynomials are used. These unknown coefficients are calculated via the collocation method and matrix operations derivations. Two examples are given to show the feasibility of the method. To calculate the solutions, MATLAB R2021a is used. Additionally, comparing our method to Homotopy perturbation method (HPM) and Laplace Adomian decomposition method (LADM) proves the accuracy of the solution. The method studied can be seen as effective from these comparisons. So, it is essential to find solutions for the governing model. The study will contribute to the literature since we also discuss the vaccination situation. The results of this study are valuable for controlling an epidemic.

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“…Therefore, we need uniform and robust numerical techniques [6,8,10,11,15,23,25,28](see, also references therein).…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of a Singularly Perturbed Fredholm Integro Differential Equation with Robin Boundary Condition

2021

Turk J Math

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In this paper, we deal with singularly perturbed Fredholm integro differential equation (SPFIDE) with mixed boundary conditions. By using interpolating quadrature rules and exponential basis function, fitted second order difference scheme has constructed on a Shishkin mesh. Stability and convergence of the difference scheme have analyzed in the discrete maximum norm. Some numerical examples have solved and numerical outcomes are obtained.

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Pell–Lucas series approach for a class of Fredholm-type delay integro-differential equations with variable delays

Cited by 10 publications

References 20 publications

A generic numerical method for treating a system of Volterra integro-differential equations with multiple delays and variable bounds

A generic numerical method for treating a system of Volterra integro-differential equations with multiple delays and variable bounds

A numerical approach for an epidemic SIR model via Morgan-Voyce series

Numerical Solution of a Singularly Perturbed Fredholm Integro Differential Equation with Robin Boundary Condition

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