A numerical approach for solving Volterra type functional integral equations with variable bounds and mixed delays

Gökmen, Elçin; Yüksel, Gamze; Sezer, Mehmet

doi:10.1016/j.cam.2016.08.004

Cited by 24 publications

(19 citation statements)

References 12 publications

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“…The classical concepts related to Chelyshkov polynomials (a new class of orthogonal polynomials) are presented. In 2006, Chelyshkov [34] introduced this class of polynomials while Gokmen et al [35] (in 2017) redefined said polynomials explicitly as stated in (3) and (4).…”

Section: Classical Concepts and Propertiesmentioning

confidence: 99%

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A New Operational Matrices-Based Spectral Method for Multi-Order Fractional Problems

et al. 2020

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The operational matrices-based computational algorithms are the promising tools to tackle the problems of non-integer derivatives and gained a substantial devotion among the scientific community. Here, an accurate and efficient computational scheme based on another new type of polynomial with the help of collocation method (CM) is presented for different nonlinear multi-order fractional differentials (NMOFDEs) and Bagley–Torvik (BT) equations. The methods are proposed utilizing some new operational matrices of derivatives using Chelyshkov polynomials (CPs) through Caputo’s sense. Two different ways are adopted to construct the approximated (AOM) and exact (EOM) operational matrices of derivatives for integer and non-integer orders and used to propose an algorithm. The understudy problems have been transformed to an equivalent nonlinear algebraic equations system and solved by means of collocation method. The proposed computational method is authenticated through convergence and error-bound analysis. The exactness and effectiveness of said method are shown on some fractional order physical problems. The attained outcomes are endorsing that the recommended method is really accurate, reliable and efficient and could be used as suitable tool to attain the solutions for a variety of the non-integer order differential equations arising in applied sciences.

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Section: Classical Concepts and Propertiesmentioning

confidence: 99%

“…A Chelyshkov wavelet-based technique is developed to compute the control theory problems by Moradi et al [33]. Readers are suggested to find some associated research to such polynomials in [34][35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

A New Operational Matrices-Based Spectral Method for Multi-Order Fractional Problems

et al. 2020

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“…The Chelyshkov polynomials are one of the latest classes of orthogonal polynomials that introduced by Chelyshkov [4,5]. In this section, we aim to introduce the Chelyshkov orthogonal polynomials briefly.…”

Section: Basic Definition Of Chelyshkov Polynomialsmentioning

confidence: 99%

“…Integro-differential equations plays an important role in modeling of various problems arise in scientific fields such as biology, ecology, medicine and physics [5,11,15,17]. Particularly, population dynamics of two separate species in biology, which the second species will feed on the first, can be described by a fractional-order nonlinear system of delay integro-differential equations as [1,2,16,18,19]…”

Section: Introductionmentioning

confidence: 99%

Numerical treatment of fractional-order nonlinear system of delay integro-differential equations arising in biology

Mohammadi

Moradi

2019

Asian-European J. Math.

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This paper presents a fractional-order nonlinear system of delay integro-differential equations which is used to model biological species living together. A numerical scheme based on the Chelyshkov polynomials is implemented to solve this fractional order system of integro-differential equations. The main advantages of the presented method is that it reduces under consideration problem to a system of nonlinear algebraic equations. Some test problems are considered to confirm validity and accuracy of the presented method. Moreover, the obtained numerical results are compared with those existing in the literature.Numerical treatment of fractional-order nonlinear system are derived in Sec. 4. In Sec. 5, an efficient numerical method is introduced to solve fractional-order nonlinear system of delay integro-differential equations (3). To confirm efficiency and accuracy of the presented method, we give some examples and numerical results in Sec. 6. Finally, a conclusion is drawn in Sec. 7. Preliminary Remarks on Fractional CalculusFractional order calculus is a branch of calculus which deals with integration and differentiation operators of non-integer order. Although the fractional order operators enable to model a wider class of problems, there not exists a unique definition of fractional derivative. Among the several formulations of the generalized derivative, the Riemann-Liouville and Liouville-Caputo definition is the most commonly used, which can be described as follows (see e.g. [7,13]).

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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 approach for solving Volterra type functional integral equations with variable bounds and mixed delays

Cited by 24 publications

References 12 publications

A New Operational Matrices-Based Spectral Method for Multi-Order Fractional Problems

A New Operational Matrices-Based Spectral Method for Multi-Order Fractional Problems

Numerical treatment of fractional-order nonlinear system of delay integro-differential equations arising in biology

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

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