An approach based on fractional-order Lagrange polynomials for the numerical approximation of fractional order non-linear Volterra-Fredholm integro-differential equations

Kumar, Saurabh; Gupta, Vikas

doi:10.1007/s12190-022-01743-w

Cited by 10 publications

(7 citation statements)

References 25 publications

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“…Recently, researchers have focused on finding the numerical solution to FDEs due to its increasing applications in several domains. Various researchers have discussed several mathematical techniques for solving FDEs, including integer-order differential equations technique [6], homotopy perturbation method [7], method of extended Jacobi elliptic function expansion [8], homotopy analysis method [9], variational iteration method [10][11][12], predictorcorrector method [13], Adams-Bashforth-Moulton (ABM) method [14], finite difference method [15], differential transform method [16], Adomian decomposition method [17], fractional-order polynomial's operational matrix method [18][19][20], Quasi-Wavelet numerical method [21], fifth kind orthonormal Chebsychev polynomial method [22], Haar wavelet method [23] and Legendre wavelets method [24], etc.…”

Section: Introductionmentioning

confidence: 99%

“…In general, the Lagrange polynomial provides many advantages, including a clear structure, simple calculation, and ease of integral and derivative calculation. The use of operational matrix approaches based on Lagrange polynomials has proven effective in [19,20] also. As a result, we are inspired to develop an effective numerical technique for solving the pantograph delay and Riccati differential equations with fractional order based on fractional-order Lagrange polynomials (FOLPs).…”

Section: Introductionmentioning

confidence: 99%

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An accurate operational matrix method based on Lagrange polynomials for solving fractional-order pantograph delay and Riccati differential equations

et al. 2023

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This paper introduces the fractional-order Lagrange polynomials approach to solve initial value problems for pantograph delay and Riccati differential equations involving fractional-order derivatives. The fractional derivative is determined as per the idea of Caputo. First, operational matrices of fractional integration with fractional-order Lagrange polynomials have been constructed using the Laplace transform. Then, we use these operational matrices and the collocation method to convert the given initial value problem to a system of algebraic equations. Subsequently, we use Newton’s iterative approach to solve the resultant system of algebraic equations. Error estimates for the function approximation also have been discussed. Finally, some numerical examples supported the theoretical findings by demonstrating the applicability and accuracy of the proposed strategy.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An accurate operational matrix method based on Lagrange polynomials for solving fractional-order pantograph delay and Riccati differential equations

et al. 2023

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“…However, one challenging aspect of FDEs is that most of the problems cannot be solved analytically, necessitating the employment of numerical methods to solve them approximately. Numerous semi-analytical and numerical techniques have been explored in recent years to solve different classes of FDEs (see [1][2][3][4][5][6][7][8][9][10][11] and references therein).…”

Section: Introductionmentioning

confidence: 99%

“…The authors in [35] derived the general VO Riemann-Liouville pseudo-operational matrix and a general VO fractional derivative pseudo-operational matrix for the general Lagrange scaling functions and used them to solve VO partial differential equations. Also, readers can see previous studies [10,11,[36][37][38] for applications of Lagrange functions in solving fractional-order problems that arise in various areas of science and engineering.…”

Section: Introductionmentioning

confidence: 99%

Collocation method with Lagrange polynomials for variable‐order time‐fractional advection–diffusion problems

Kumar,

Gupta

2023

Math Methods in App Sciences

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In this study, we proposed a numerical technique to solve a class of variable‐order time‐fractional advection–diffusion equations (VOTFADEs) by applying an operational matrix of differentiation based on fractional‐order Lagrange polynomials (FOLPs). The variable‐order fractional derivative is assumed to be Caputo's derivative. Using the operational matrix and collocation method, the advection–diffusion equation can be reduced to an algebraic system of equations that can be solved using Newton's iterative method. Error analysis also has been carried out for the proposed method. The current approach is simple to use and computer oriented and provides highly accurate approximate solutions. The effectiveness and accuracy of the proposed method are demonstrated using a few numerical examples.

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“…Maleknejad and Hoseingholipour [23] implemented Laguerre functions for singular integral equation in unbounded domain. Kumar and Gupta [24] analyzed an operational matrix based 2D fractional-order Lagrange polynomials for approximating nonlinear 2DFIDEs. Mirzaee and Samadyar [25] obtained an operational matrix based on 2D hat basis functions for stochastic 2DFIEs.…”

Section: Introductionmentioning

confidence: 99%

Hybridization of Block-Pulse and Taylor Polynomials for Approximating 2D Fractional Volterra Integral Equations

et al. 2022

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This paper proposes an accurate numerical approach for computing the solution of two-dimensional fractional Volterra integral equations. The operational matrices of fractional integration based on the Hybridization of block-pulse and Taylor polynomials are implemented to transform these equations into a system of linear algebraic equations. The error analysis of the proposed method is examined in detail. Numerical results highlight the robustness and accuracy of the proposed strategy.

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An approach based on fractional-order Lagrange polynomials for the numerical approximation of fractional order non-linear Volterra-Fredholm integro-differential equations

Cited by 10 publications

References 25 publications

An accurate operational matrix method based on Lagrange polynomials for solving fractional-order pantograph delay and Riccati differential equations

An accurate operational matrix method based on Lagrange polynomials for solving fractional-order pantograph delay and Riccati differential equations

Collocation method with Lagrange polynomials for variable‐order time‐fractional advection–diffusion problems

Hybridization of Block-Pulse and Taylor Polynomials for Approximating 2D Fractional Volterra Integral Equations

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