A numerical method for solving some model problems arising in science and convergence analysis based on residual function

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

doi:10.1016/j.apnum.2017.06.015

Cited by 32 publications

(35 citation statements)

References 32 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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“…Eventually, in order to find the Lucas polynomial solution of the problem (1)-(2), by replacing m row matrices (16) into any m rows of the form (15). Thus, we have the augmented matrix…”

Section: Lucas Matrix-collocation Techniquementioning

confidence: 99%

“…In recent years, the residual error analysis has been applied by some authors [5,11,13,14,16,17,19,22]. Furthermore, the reader can refer to [15,27,28] for converge analysis based on residual function; residual correction and its theory. Let us now construct the residual error analysis for the Lucas polynomials.…”

Section: Residual Error Analysismentioning

confidence: 99%

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A numerical technique based on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points for solving functional integro-differential equations involving variable delays

Gümgüm

Savaşaneril

Kürkçü

et al. 2018

Sakarya University Journal of Science

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In this paper, a new numerical matrix-collocation technique is considered to solve functional integrodifferential equations involving variable delays under the initial conditions. This technique is based essentially on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points. Some descriptive examples are performed to observe the practicability of the technique and the residual error analysis is employed to improve the obtained solutions. Also, the numerical results obtained by using these collocation points are compared in tables and figures.

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“…Q pqr (t) y (p) (t) y (q) (t) y (r) is considered, where P k (t) , Q pq (t) , Q pqr (t) and g (t) are the given analytic functions defined on the interval a ≤ t ≤ b; λ j , a kj and b kj are the known rael coefficients. In order to solve the nonlinear problem (1.1)-(1.2), we utilize the matrix-collocation method, which have been developed by Sezer and Coworkers [7,10,[12][13][14], and research the numerical solution in the truncated Morgan-Voyce series form…”

Section: Introductionmentioning

confidence: 99%

Solution of nonlinear ordinary differential equations with quadratic and cubic terms by Morgan-Voyce matrix-collocation method

Tarakçı¹,

Özel²,

Sezer³

2020

Turk J Math

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Nonlinear differential equations have many applications in different science and engineering disciplines. However, a nonlinear differential equation cannot be solved analytically and so must be solved numerically. Thus, we aim to develop a novel numerical algorithm based on Morgan-Voyce polynomials with collocation points and operational matrix method to solve nonlinear differential equations. In the our proposed method, the nonlinear differential equations including quadratic and cubic terms having the initial conditions are converted to a matrix equation. In order to obtain the matrix equations and solutions for the selected problems, code was developed in MATLAB. The solution of this method for the convergence and efficiency was compared with the equations such as Van der Pol differential equation calculated by different methods.

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A numerical method for solving some model problems arising in science and convergence analysis based on residual function

Cited by 32 publications

References 32 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

A numerical technique based on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points for solving functional integro-differential equations involving variable delays

Solution of nonlinear ordinary differential equations with quadratic and cubic terms by Morgan-Voyce matrix-collocation method

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