A Numerical Approach Technique for Solving Generalized Delay Integro-Differential Equations with Functional Bounds by Means of Dickson Polynomials

Kürkçü, Ömür Kıvanç; Aslan, Ersin; Sezer, Mehmet; İlhan, Özgül

doi:10.1142/s0219876218500391

Cited by 15 publications

(8 citation statements)

References 39 publications

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“…Kürkçü et al [34,35,37] recently applied residual error analysis to ordinary integro-differential-(difference) equations and they established the residual function-based convergence analysis for model problems [36]. Our aim is to improve the obtained approximate solutions by employing the residual error analysis based on the fractional derivative.…”

Section: Residual Error Analysis Based On Fractional Derivativementioning

confidence: 99%

A novel graph-operational matrix method for solving multidelay fractional differential equations with variable coefficients and a numerical comparative survey of fractional derivative types

Kürkçü¹,

Aslan²,

Sezer³

2019

Turk J Math

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In this study, we introduce multidelay fractional differential equations with variable coefficients in a unique formula. A novel graph-operational matrix method based on the fractional Caputo, Riemann-Liouville, Caputo-Fabrizio, and Jumarie derivative types is developed to efficiently solve them. We also make use of the collocation points and matrix relations of the matching polynomial of the complete graph in the method. We determine which of the fractional derivative types is more appropriate for the method. The solutions of model problems are improved via a new residual error analysis technique. We design a general computer program module. Thus, we can explicitly monitor the usefulness of the method. All results are scrutinized in tables and figures. Finally, an illustrative algorithm is presented.

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Section: Residual Error Analysis Based On Fractional Derivativementioning

confidence: 99%

A novel graph-operational matrix method for solving multidelay fractional differential equations with variable coefficients and a numerical comparative survey of fractional derivative types

Kürkçü¹,

Aslan²,

Sezer³

2019

Turk J Math

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“…Proof In view of Equation , we can obtain the fractional‐based residual function

R_{α_{k}, N} false(t false)

on L 1 [ a , b ] as

R_{α_{k}, N} (t) = \sum_{k = 0}^{m} P_{k} (t) y_{N}^{(α_{k})} (t) + \sum_{r = 2}^{5} Q_{r} (t) y_{N}^{r} (t) - g (t) .

We also know from Kürkçü et al and Oğuz and Sezer that

R_{α_{k}, N} false(t false)

can be stated with respect to α k as

(||, true \int_{a}^{b} R_{α_{k}, N} false(t false) d t) \leq true \int_{a}^{b} (||, R_{α_{k}, N} false(t false)) d t,

and using the mean value theorem for integrals, we can write

(||, true \int_{a}^{b} R_{α_{k}, N} false(t false) d t) = ((), b - a) (||, R_{α_{k}, N} false(t_{0} false)), t_{0} \in false(a, b false) .

Then inserting Equation into the inequality , it holds that

|R_{}|

…”

Section: Error Analysis By Means Of Fractional‐based Residual Functionmentioning

confidence: 99%

“…This analysis is basically made up of a fractional‐based residual function

R_{α_{k}, N} false(t false)

, the mean value theorem along with fractional derivative α k . Formerly, some authors employed an error analysis based on residual function for integer order integro‐differential equations . For the first time with this study, we construct fractional‐based error analysis for the present method.…”

Section: Error Analysis By Means Of Fractional‐based Residual Functionmentioning

confidence: 99%

“…It is well known that the polynomial‐based methods, such as the matching, Dickson, Legendre‐Laguerre, Lucas, Boubaker, Bernoulli, and Chelyshkov have provided a simple procedure for collocation and wavelet operational matrix systems. On the other hand, specific studies, which especially seek the numerical solutions of highly nonlinear FDEs, are scarcely found in the literature.…”

Section: Introductionmentioning

confidence: 99%

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An integrated numerical method with error analysis for solving fractional differential equations of quintic nonlinear type arising in applied sciences

Kürkçü

Aslan

Sezer

2019

Math Methods in App Sciences

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In this study, fractional differential equations having quintic nonlinearity are considered by proposing an accurate numerical method based on the matching polynomial and matrix-collocation system. This method provides an integration between matrix and fractional derivative, which makes it fast and efficient. A hybrid computer program is designed by making use of the fast algorithmic structure of the method. An error analysis technique consisting of the fractional-based residual function is constructed to scrutinize the precision of the method. Some error tests are also performed. Figures and tables present the consistency of the approximate solutions of highly stiff model problems. All results point out that the method is effective, simple, and eligible. KEYWORDS error analysis, fractional differential equations, matrix-collocation method, quintic nonlinearity, residual function MSC CLASSIFICATION34A08; 65L60

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“…The kernel function K r v (t, s) is of the form (Kürkçü et al 2016(Kürkçü et al , 2017(Kürkçü et al , 2018:…”

Section: Y N T Andmentioning

confidence: 99%

An inventive numerical method for solving the most general form of integro-differential equations with functional delays and characteristic behavior of orthoexponential residual function

2019

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In this study, we constitute the most general form of functional integro-differential equations with functional delays. An inventive method based on Dickson polynomials with the parameter-α along with collocation points is employed to solve them. The stability of the solutions is simulated according to an interval of the parameter-α. A useful computer program is developed to obtain the precise values from the method. The residual error analysis is used to improve the obtained solutions. The characteristic behavior of the residual function is established with the aid of the orthoexponential polynomials. We compare the present numerical results of the method with those obtained by the existing methods in tables. Keywords Collocation points • Dickson and orthoexponential polynomials • Error analysis • Matrix method Mathematical Subject Classification 34K06 • 45J05 • 65L60 • 65R20 Communicated by Hui Liang.

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A Numerical Approach Technique for Solving Generalized Delay Integro-Differential Equations with Functional Bounds by Means of Dickson Polynomials

Cited by 15 publications

References 39 publications

A novel graph-operational matrix method for solving multidelay fractional differential equations with variable coefficients and a numerical comparative survey of fractional derivative types

A novel graph-operational matrix method for solving multidelay fractional differential equations with variable coefficients and a numerical comparative survey of fractional derivative types

An integrated numerical method with error analysis for solving fractional differential equations of quintic nonlinear type arising in applied sciences

An inventive numerical method for solving the most general form of integro-differential equations with functional delays and characteristic behavior of orthoexponential residual function

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