Spline collocation method for integro-differential equations with weakly singular kernels

Pedas, Arvet; Tamme, Enn

doi:10.1016/j.cam.2005.07.035

Cited by 43 publications

(34 citation statements)

References 16 publications

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“…In the same way as in the proof of Theorem 4.1 of [8] we obtain that there exists an integer N 0 such that for every N ≥ N 0 equation (4.14) possesses a unique solution v N ∈ S (−1) m−1 (Π N ) and…”

Section: Convergence Of the Fully Discrete Collocation Methodsmentioning

confidence: 54%

“…The convergence of such collocation method for solving (1.1), (1.2) is investigated in [8]. In order to obtain a high-order convergence a special graded grid reflecting the possible singular behavior of the solution is used.…”

Section: Collocation Methodsmentioning

confidence: 99%

“…A popular class of methods for solution of weakly singular integro-differential equations is the class of piecewise polynomial collocation methods using nonuniform grids (see, e.g., [1,2,3,7,8,12]). In order to apply these methods it is necessary to compute certain integrals that determine the linear systems to be solved.…”

Section: Introductionmentioning

confidence: 99%

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On Fully Discrete Collocation Methods for Solving Weakly Singular Integro‐differential Equations

Kangro

Tamme

2010

Mathematical Modelling and Analysis

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Abstract. In order to find approximate solutions of Volterra and Fredholm integrodifferential equations by collocation methods it is necessary to compute certain integrals that determine the required algebraic systems. Those integrals usually can not be computed exactly and if the kernels of the integral operators are not smooth, simple quadrature formula approximations of the integrals do not preserve the convergence rate of the collocation method. In the present paper fully discrete analogs of collocation methods where non-smooth integrals are replaced by appropriate quadrature formulas approximations, are considered and corresponding error estimates are derived. Presented numerical examples display that theoretical results are in a good accordance with the actual convergence rates of the proposed algorithms.

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Section: Convergence Of the Fully Discrete Collocation Methodsmentioning

confidence: 54%

Section: Collocation Methodsmentioning

confidence: 99%

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On Fully Discrete Collocation Methods for Solving Weakly Singular Integro‐differential Equations

Kangro

Tamme

2010

Mathematical Modelling and Analysis

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“…Therefore, many researchers have tried their best to use different techniques to find the analytical and numerical solutions of these problems, for example, Adomian decomposition method (ADM), 5 spline collocation method (SCM), 6 fractional transform method (FTM), 7 homotopy perturbation method (HPM), 8 operational tau method (OTM), 9 shifted Chebyshev polynomial method (SCPM), 10 rationalized Haar function method (RHFM), 11 exp-function method, 12 traveling wave transformation method, 13 and Cole-Hopf transformation method, 14 and also see the work in Yang et al, 15 Sayevand and Pichaghchi, 16 and Wang and Liu. 17 Recently, Yang et al 18 did a comprehensive study of the methods which have been used for the solutions of the problems containing fractional derivatives and integral operators.…”

Section: Introductionmentioning

confidence: 99%

Chebyshev wavelet method to nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions

Mohyud‐Din

Khan

Arif

et al. 2017

Advances in Mechanical Engineering

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This research work addresses the numerical solutions of nonlinear fractional integro-differential equations with mixed boundary conditions, using Chebyshev wavelet method. The basic idea of this work started from the Caputo definition of fractional differential operator. The fractional derivatives are replaced by Caputo operator, and the solution is approximated by wavelet family of functions. The numerical scheme by Chebyshev wavelet method is constructed through a very simple and straightforward way. The numerical results of the current method are compared with the exact solutions of the problems, which show that the proposed method has a strong agreement with the exact solutions of the problems. The numerical solutions of the present method are also compared with steepest decent method and Adomian decomposition method. The comparison with other methods reveals that this method has the highest degree of accuracy than those methods.

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“…We can point out to the collocation method [6], Adomian decomposition method (ADM) [7]- [9], Spline collocation method [10], fractional differential transform method [11] and the method of combination of forward and central differences [12]. Out of the aforesaid methods, we desire to consider OTM for solving FIDEs.…”

Section: Introductionmentioning

confidence: 99%

Operational Tau approximation for a general class of fractional integro-differential equations

Vanani

Aminataei

2011

Comput. Appl. Math.

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Abstract. In this work, an extension of the algebraic formulation of the operational Tau method (OTM) for the numerical solution of the linear and nonlinear fractional integro-differential equations (FIDEs) is proposed. The main idea behind the OTM is to convert the fractional differential and integral parts of the desired FIDE to some operational matrices. Then the FIDE reduces to a set of algebraic equations. We demonstrate the Tau matrix representation for solving FIDEs based on arbitrary orthogonal polynomials. Some advantages of using the method, error estimation and computer algorithm are also presented. Illustrative linear and nonlinear experiments are included to show the validity and applicability of the presented method.Mathematical subject classification: 65M70, 34A25, 26A33, 47Gxx.

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Spline collocation method for integro-differential equations with weakly singular kernels

Cited by 43 publications

References 16 publications

On Fully Discrete Collocation Methods for Solving Weakly Singular Integro‐differential Equations

On Fully Discrete Collocation Methods for Solving Weakly Singular Integro‐differential Equations

Chebyshev wavelet method to nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions

Operational Tau approximation for a general class of fractional integro-differential equations

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