A quadrature method for numerical solutions of fractional differential equations

Rehman, Mujeeb ur; Idrees, Amna; Saeed, Umer

doi:10.1016/j.amc.2017.02.053

Cited by 18 publications

(11 citation statements)

References 22 publications

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“…Jüttler, B. et al presented a detailed case study of different quadrature schemes for isogeometric discretizations of partial differential equations on closed surfaces with Loop's subdivision scheme [8]. ur Rehman, M. et al developed a numerical method to obtain approximate solutions for a certain class of fractional differential equations [9]. However, Thiagarajan, V. et al demonstrated the application of AW integration scheme in the context of the Finite Cell Method which must perform numerical integration over arbitrary domains without meshing [10].…”

Section: Introductionmentioning

confidence: 99%

A New Analysis of Approximate Solutions for Numerical Integration Problems with Quadrature-based Methods

Moheuddin¹,

Titu²,

Hossain³

2020

PAMJ

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In this paper, we mainly propose the approximate solutions to solve the integration problems numerically using the quadrature method including the Trapezoidal method, Simpson's 1/3 method, and Simpson's 3/8 method. The three proposed methods are quite workable and practically well suitable for solving integration problems. Through the MATLAB program, our numerical solutions are determined as well as compared with the exact values to verify the higher accuracy of the proposed methods. Some numerical examples have been utilized to give the accuracy rate and simple implementation of our methods. In this study, we have compared the performance of our solutions and the computational attempt of our proposed methods. Moreover, we explore and calculate the errors of the three proposed methods for the sake of showing our approximate solution's superiority. Then, among these three methods, we analyzed the approximate errors to prove which method shows more appropriate results. We also demonstrated the approximate results and observed errors to give clear idea graphically. Therefore, from the analysis, we can point out that only the minimum error is in Simpson's 1/3 method which will beneficial for the readers to understand the effectiveness in solving the several numerical integration problems.

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Section: Introductionmentioning

confidence: 99%

A New Analysis of Approximate Solutions for Numerical Integration Problems with Quadrature-based Methods

Moheuddin¹,

Titu²,

Hossain³

2020

PAMJ

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“…, so that m 1 , m 2 , and m 3 are respectively N , 0, 2N for odd N and 0, 2N , 0 for even N . Then, we substitute equation 10into (9) to get…”

Section: The Operational Matrix Of Derivativementioning

confidence: 99%

Numerical solution of the Bagley–Torvik equation using shifted Chebyshev operational matrix

Tian-fu¹,

Hou²,

Yang³

2020

Adv Differ Equ

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In this study, an efficient numerical scheme based on shifted Chebyshev polynomials is established to obtain numerical solutions of the Bagley–Torvik equation. We first derive the shifted Chebyshev operational matrix of fractional derivative. Then, by the use of these operational matrices, we reduce the corresponding fractional order differential equation to a system of algebraic equations, which can be solved numerically by Newton’s method. Furthermore, the maximum absolute error is obtained through error analysis. Finally, numerical examples are presented to validate our theoretical analysis.

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“…Efficient numerical methods are developed in order to eliminate this hardship. Some of them are the Taylor matrix [23,24], Bernstein matrix collocation [51], quadrature [44], differential transform [47], Fermat tau [50], multiscale collocation [40], and homotopy perturbation methods [3].…”

Section: Ams Mathematics Subject Classification: 34k37 65l60 68r10mentioning

confidence: 99%

A novel graph-operational matrix method for solving multidelay fractional differential equations with variable coefficients and a numerical comparative survey of fractional derivative types

Kürkçü¹,

Aslan²,

Sezer³

2019

Turk J Math

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In this study, we introduce multidelay fractional differential equations with variable coefficients in a unique formula. A novel graph-operational matrix method based on the fractional Caputo, Riemann-Liouville, Caputo-Fabrizio, and Jumarie derivative types is developed to efficiently solve them. We also make use of the collocation points and matrix relations of the matching polynomial of the complete graph in the method. We determine which of the fractional derivative types is more appropriate for the method. The solutions of model problems are improved via a new residual error analysis technique. We design a general computer program module. Thus, we can explicitly monitor the usefulness of the method. All results are scrutinized in tables and figures. Finally, an illustrative algorithm is presented.

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A quadrature method for numerical solutions of fractional differential equations

Cited by 18 publications

References 22 publications

A New Analysis of Approximate Solutions for Numerical Integration Problems with Quadrature-based Methods

A New Analysis of Approximate Solutions for Numerical Integration Problems with Quadrature-based Methods

Numerical solution of the Bagley–Torvik equation using shifted Chebyshev operational matrix

A novel graph-operational matrix method for solving multidelay fractional differential equations with variable coefficients and a numerical comparative survey of fractional derivative types

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