The combined reproducing kernel method and Taylor series for handling nonlinear Volterra integro-differential equations with derivative type kernel

Alvandi, Azizallah; Paripour, Mahmoud

doi:10.1016/j.amc.2019.02.023

Cited by 9 publications

(3 citation statements)

References 15 publications

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“…In particular, the first two elements of the Taylor expansion are used as a linearization of a function behaviour at the given point and its surroundings [20]. The Taylor expansion is used in solutions of partial differential equations (PDE) [1,7,17], ordinary differential equations (ODE) [2,6,21] , integral equations (IE) integro-differential equations (IDE) [1,11], approximation of inverse functions (AIF) [8], control theory [4], fluid flow visualization of 3D flow using radial basis functions [12,13,15], computer vision [5,18], in statistical mechanics [10], antenna design [6,9], operator theory [3], etc.…”

Section: Introductionmentioning

confidence: 99%

Efficient Taylor expansion computation of multidimensional vector functions on GPU

Skala¹

2021

AMI

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The Taylor expansion [19] is used in many applications for a value estimation of scalar functions of one or two variables in the neighbour point. Usually, only the first two elements of the Taylor expansion are used, i.e. a value in the given point and derivatives estimation. The Taylor expansion can be also used for vector functions, too. The usual formulae are well known, but if the second element of the expansion, i.e. with the second derivatives are to be used, mathematical formulations are getting too complex for efficient programming, as it leads to the use of multi-dimensional matrices.This contribution describes a new form of the Taylor expansion for multidimensional vector functions. The proposed approach uses "standard" formalism of linear algebra, i.e. using vectors and matrices, which is simple, easy to implement. It leads to efficient computation on the GPU in the three dimensional case, as the GPU offers fast vector-vector computation and many parts can be done in parallel.

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Section: Introductionmentioning

confidence: 99%

Efficient Taylor expansion computation of multidimensional vector functions on GPU

Skala¹

2021

AMI

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“…Reproducing kernel theory has important applications in numerical analysis, differential equations, probability and statistics, learning theory and so on (Arqub et al 2012;Allahviranloo et al 2015;FarzanehJavan et al 2017;Xueqin and Sixing 1965;Du and JiHong 2014;Du and Cui 2008;Genga and b Minggen Cui, Bo Z, 2010;Cui and Geng 2007;Arqub and Maayah 2018;Mohammadi and Mokhtari 2011;Du and Zhao 2014). Also the idea of reproducing kernel theory has been studied by scientists and engineers (Alvandi and Paripour 2016;Sahihi et al 2018;FarzanehJavan et al 2017). They considered a new method for solving initial value problems, singular integral equations, nonlinear partial differential equations and operator equations with the help of the concept of reproducing kernel Hilbert space method.…”

Section: Introductionmentioning

confidence: 99%

Combining fractional differential transform method and reproducing kernel Hilbert space method to solve fuzzy impulsive fractional differential equations

Najafi

Allahviranloo

2020

Comp. Appl. Math.

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The aim of this paper is to use the combination of Reproducing kernel Hilbert space method (RKHSM) and fractional differential transform method (FDTM) to solve the linear and nonlinear fuzzy impulsive fractional differential equations. Finding the numerical solutions of this class of equations are a difficult topic to analyze. In this study, convergence analysis, estimations error and bounds errors are discussed in detail under some hypotheses which provide the theoretical basis of the proposed algorithm. Some numerical examples indicate that this method is an efficient one to solve the mentioned equations.

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“…Alvandi and Paripour solved nonlinear Abel's integral equations with weakly singular kernel in the reproducing kernel space and removed the singularity of the equation considered [13]. The same authors presented a simple and efficient method to solve linear Volterra integro-differential equations [14], nonlinear Volterra-Fredholm integro-differential equations [15], and see [16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Reproducing kernel method for a class of weakly singular Fredholm integral equations

Alvandi

Paripour

2018

Journal of Taibah University for Science

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Numerical methods for solving integral equations have been the focus of much research, including reproducing kernel methods. We present a new algorithm to solve weakly singular Fredholm integral equations (WSFIEs). The advantage of this method is that it is possible to pick any point in the interval of integration and also the approximate solution. The advantage of this method is used to remove singularity and reproducing kernel functions are used as a basis. The convergence of approximation solution to the exact solution is also proved. Some examples are displayed to demonstrate that the method is accurate and efficient for WSFIEs.

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The combined reproducing kernel method and Taylor series for handling nonlinear Volterra integro-differential equations with derivative type kernel

Cited by 9 publications

References 15 publications

Efficient Taylor expansion computation of multidimensional vector functions on GPU

Efficient Taylor expansion computation of multidimensional vector functions on GPU

Combining fractional differential transform method and reproducing kernel Hilbert space method to solve fuzzy impulsive fractional differential equations

Reproducing kernel method for a class of weakly singular Fredholm integral equations

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