A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro‐differential equations with weakly singular kernels

Keshavarz, Elham; Ordokhani, Yadollah

doi:10.1002/mma.5663

Cited by 34 publications

(14 citation statements)

References 35 publications

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“…Convergence analysis of approximation techniques for integral equations can be also observed in few articles, say previous studies 20–22 . Recently, spline collocation 23 and Taylor wavelets 24 are proposed to solve fractional integro differential equations with weakly singular kernel for the particular case 0 < α < 1. Numerical methods based on cubic spline approximations 25 are also popular on adaptive meshes 26–29 which do not assume semi analytical homotopy construction.…”

Section: Introductionmentioning

confidence: 91%

Theoretical prospects of fractional order weakly singular Volterra Integro differential equations and their approximations with convergence analysis

Das

Rana

2021

Math Methods in App Sciences

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In this research, we study a weakly singular Volterra integro differential equation with Caputo‐type fractional derivative. First, we derive a sufficient condition for the existence and uniqueness of the solution of this problem based on the maximum norm. It is observed that the condition depends on the domain of definition of the problem. Thereafter, we show that this condition will be independent of the domain of definition based on an equivalent weighted maximum norm. In addition, we have also provided a procedure to extend the existence and uniqueness of the solution in its domain of definition by partitioning it. We also derive a sufficient condition under which the model problem will provide an analytic solution. Next, we introduce an operator‐based parameterized method to generate an approximate solution of this problem. Convergence analysis of this approach is established here. Next, we have optimized this solution based on least square method. For this, residual minimization is used to obtain the optimal values of the auxiliary parameter. In addition, we have also provided an error bound based on this technique. Several numerical examples are produced to clarify the effective behavior of the convergence of the present method. Comparison of the standard method and optimized method based on residual minimization signify the better accuracy of modified one. In addition, we also consider an equivalent form of weakly singular integro differential equation of Heat transfer problem to show the high effectiveness of the present approach.

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Section: Introductionmentioning

confidence: 91%

Theoretical prospects of fractional order weakly singular Volterra Integro differential equations and their approximations with convergence analysis

Das

Rana

2021

Math Methods in App Sciences

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“…The two scale relations for nth order derivative of the scale function (x) and wavelet function (x) are obtained from (11) and (12) followed by differentiating n times…”

Section: Multiscale (Wavelet) Basis Of Daubechies Familymentioning

confidence: 99%

“…Solving of FIDE with weakly singular kernel might be difficult analytically, so many researchers have tried to propose different accurate and efficient numerical methods. In very short time a lot of different numerical methods have been proposed such as spline collocation, 5 second kind Chebyshev polynomial method, 6 Legendre wavelets, 7 an operational Jacobi Tau method, 8 Hat function, 9 Shifted Jacobi polynomials, 10 Block pulse, 11 Taylor wavelets, 12 and so on. All these methods provide adequate results.…”

Section: Introductionmentioning

confidence: 99%

Wavelet‐based collocation technique for fractional integro‐differential equation with weakly singular kernel

Mouley

Mandal

2021

Comp and Math Methods

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Fractional integro-differential equation (FIDE) with weakly singular kernel is an important topic in mathematics and engineering dealing with mathematical modeling and simulation of numerous systems and processes. A wavelet-based collocation technique has been developed here to obtain approximate numerical solution of a FIDE with weakly singular kernel. The present method avoids complicated integrations and elaborate numerical calculations. The multiscale error approximation associated with this method has also been explained. The efficiency of the proposed method has been demonstrated by including some illustrative examples. K E Y W O R D Scollocation method, fractional derivative, multiscale basis of Daubechies family, weakly singular fractional integro-differential equation Here D y(x) represents the fractional derivative of order of the unknown function y(x) and the fractional derivative is understood in Caputo sense in this paper. The explicit forms of V(x, t) and f (x) are known and they are considered sufficiently smooth to confirm the existence and uniqueness of solution. 4

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“…Wavelet functions have been previously applied for obtaining approximate solutions for some of these problems. Authors in (8) have constructed a fast algorithm for some linear and nonlinear wave equations using Taylor wavelets. In the two papers (9,10), the authors treated weakly kernel integral equation of the second kind and fractional delay differential equation respectively using Hermit wavelets.…”

Section: Introductionmentioning

confidence: 99%

Boubaker Wavelets Functions: Properties and Applications

2021

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This paper is concerned with introducing an explicit expression for orthogonal Boubaker polynomial functions with some important properties. Taking advantage of the interesting properties of Boubaker polynomials, the definition of Boubaker wavelets on interval [0,1) is achieved. These basic functions are orthonormal and have compact support. Wavelets have many advantages and applications in the theoretical and applied fields, and they are applied with the orthogonal polynomials to propose a new method for treating several problems in sciences, and engineering that is wavelet method, which is computationally more attractive in the various fields. A novel property of Boubaker wavelet function derivative in terms of Boubaker wavelet themselves is also obtained. This Boubaker wavelet is utilized along with a collocation method to obtain an approximate numerical solution of singular linear type of Lane-Emden equations. Lane-Emden equations describe several important phenomena in mathematical science and astrophysics such as thermal explosions and stellar structure. It is one of the cases of singular initial value problem in the form of second order nonlinear ordinary differential equation. The suggested method converts Lane-Emden equation into a system of linear differential equations, which can be performed easily on computer. Consequently, the numerical solution concurs with the exact solution even with a small number of Boubaker wavelets used in estimation. An estimation of error bound for the present method is also proved in this work. Three examples of Lane-Emden type equations are included to demonstrate the applicability of the proposed method. The exact known solutions against the obtained approximate results are illustrated in figures for comparison

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A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro‐differential equations with weakly singular kernels

Cited by 34 publications

References 35 publications

Theoretical prospects of fractional order weakly singular Volterra Integro differential equations and their approximations with convergence analysis

Theoretical prospects of fractional order weakly singular Volterra Integro differential equations and their approximations with convergence analysis

Wavelet‐based collocation technique for fractional integro‐differential equation with weakly singular kernel

Boubaker Wavelets Functions: Properties and Applications

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