Chebyshev wavelets operational matrices for solving nonlinear variable-order fractional integral equations

Yang, Yin; Heydari, Mohammad Hossein; Avazzadeh, Z.; Atangana, Abdon

doi:10.1186/s13662-020-03047-4

Cited by 12 publications

(4 citation statements)

References 38 publications

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“…Since the variable-order operators can more accurately describe properties of systems with spatial and temporal behavior, many problems in physics and other disciplines [1,2] can be modeled as variable-order fractional equations. Recently, many efficient numerical schemes have emerged to solve these equations, for example, the Chebyshev wavelets Galerkin method in [3] and shifted Legendre Gauss-Lobatto collocation method in [4].…”

Section: Introductionmentioning

confidence: 99%

Bernoulli polynomial approximation method for solving the multi-dimensional Volterra integral equations with variable-order weakly singular kernels

Wang¹,

Huang²,

Hu³

2023

3rd International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC 2023)

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Section: Introductionmentioning

confidence: 99%

Bernoulli polynomial approximation method for solving the multi-dimensional Volterra integral equations with variable-order weakly singular kernels

Wang¹,

Huang²,

Hu³

2023

3rd International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC 2023)

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“…In recent years, different kinds of orthogonal wavelets have been employed for solving divers problems. For instance, the interested reader can be refer to previous works [28][29][30][31][32] to see some well-known wavelets utilized for solving fractional problems. Recently, the Müntz-Legender wavelets (MLWs) as a specific family of orthogonal wavelets have been employed for fractional Volterra-Fredholm integro-differential equations, 33 fractional integro-differential equation, 34 fractional pantograph differential equation, 35 and fractional optimal control problem.…”

Section: Introductionmentioning

confidence: 99%

A hybrid approach established upon the Müntz‐Legender functions and 2D Müntz‐Legender wavelets for fractional Sobolev equation

Hosseininia

Heydari

Avazzadeh

2022

Math Methods in App Sciences

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This article proposes a hybrid technique for finding approximation solutions of the fractional 2D Sobolev equation. In the proposed approach, the Müntz-Legender functions and Müntz-Legender wavelets are, respectively, utilized to approximate the solution of the problem under consideration in the time and spatial directions. By implementing the presented technique, solving the 2D fractional Sobolev equation is converted into solving a system of algebraic equations. Three examples are solved to examine the validity of the proposed method.

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“…There are many researchers in literature, who utilized the different types of wavelets for the solution of fractional mathematical differential models. For example, Euler wavelets, 23,24 CAS wavelets, 19,25,26 Legendre wavelets, [27][28][29] Chebyshev wavelets, [30][31][32] Daubechies wavelets, 16,33 Taylor wavelets, 34,35 Gegenbauer wavelets, 20,36 Sine-Cosine wavelets, 37 and B-Spline wavelets 38 methods.…”

Section: Introductionmentioning

confidence: 99%

Krawtchouk wavelets method for solving Caputo and Caputo–Hadamard fractional differential equations

Saeed

Idrees

Javid

et al. 2022

Math Methods in App Sciences

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The purpose of the present work is to develop a new wavelet method, named as Krawtchouk wavelets method, for solving both Caputo fractional and Caputo-Hadamard fractional differential equations on a semi-infinite domain.Design/methodology/approach: We have utilized the discrete Krawtchouk orthogonal polynomial for the construction of Krawtchouk wavelets method. The supporting analysis of the method such as construction of operational matrices, procedure of implementation, and convergence analysis of the method are being provided. We have also proposed a method by combining the Krawtchouk wavelets method with the method of step for the solution of Caputo-Hadamard fractional delay differential equations. Findings:We have provided the orthogonality condition for the Krawtchouk wavelets. We have derived and constructed the Krawtchouk wavelets matrix, Krawtchouk wavelets operational matrix of Riemann-Liouville and Hadamard-type fractional-order integration, and Krawtchouk wavelets operational matrix of Riemann-Liouville and Hadamard-type fractional-order integration for boundary value problems. These matrices are successfully utilized for the solution of Caputo and Caputo-Hadamard fractional differential equations. Operational matrices contains many zero entries, which makes the present method more efficient. Furthermore, we workout on the procedure of implementation of the method for the Caputo fractional differential equations as well as for the Caputo-Hadamard fractional differential equations. We also derived the convergence analysis of the Krawtchouk wavelets method, which completes the theoretical analysis of the proposed method.We have applied the Krawtchouk wavelets method for the numerical solutions of several Caputo fractional differential equations and Caputo-Hadamard fractional differential equations and compare the obtained results with the analytical solutions. The comparison shows the effectiveness of the present numerical method. In this paper, we have considered initial value problem, boundary value problem, and delay problem. According to the numerical results, the present method is more efficient and accurate.Originality/value: Since fractional differential equation is a latest and emerging field, many engineers and scientists can utilize the present method for solving their fractional models. To the best of the author's knowledge, the present

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Chebyshev wavelets operational matrices for solving nonlinear variable-order fractional integral equations

Cited by 12 publications

References 38 publications

Bernoulli polynomial approximation method for solving the multi-dimensional Volterra integral equations with variable-order weakly singular kernels

Bernoulli polynomial approximation method for solving the multi-dimensional Volterra integral equations with variable-order weakly singular kernels

A hybrid approach established upon the Müntz‐Legender functions and 2D Müntz‐Legender wavelets for fractional Sobolev equation

Krawtchouk wavelets method for solving Caputo and Caputo–Hadamard fractional differential equations

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