Third-Kind Chebyshev Wavelet Method for the Solution of Fractional Order Riccati Differential Equations

Polat, S. Nergis Tural

doi:10.1142/s0218126619502475

Cited by 7 publications

(3 citation statements)

References 39 publications

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“…In Tural-Polat and Dincel (2022) and Dincel and Tural-Polat (2021), authors utilized the third and fourth kind Chebyshev wavelet methods for solving multi-term variable order fractional differential equations. Third-kind Chebyshev wavelet method and Euler wavelet method are applied on fractional order Riccati differential equations in Tural-Polat (2019) and Dincel (2019).…”

Section: Introductionmentioning

confidence: 99%

Vieta–Lucas wavelets method for fractional linear and nonlinear delay differential equations

Idrees

Saeed

2022

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PurposeIn this article, the authors aims to introduce a novel Vieta–Lucas wavelets method by generalizing the Vieta–Lucas polynomials for the numerical solutions of fractional linear and non-linear delay differential equations on semi-infinite interval.Design/methodology/approachThe authors have worked on the development of the operational matrices for the Vieta–Lucas wavelets and their Riemann–Liouville fractional integral, and these matrices are successfully utilized for the solution of fractional linear and non-linear delay differential equations on semi-infinite interval. The method which authors have introduced in the current paper utilizes the operational matrices of Vieta–Lucas wavelets to converts the fractional delay differential equations (FDDEs) into a system of algebraic equations. For non-linear FDDE, the authors utilize the quasilinearization technique in conjunction with the Vieta–Lucas wavelets method.FindingsThe purpose of utilizing the new operational matrices is to make the method more efficient, because the operational matrices contains many zero entries. Authors have worked out on both error and convergence analysis of the present method. Procedure of implementation for FDDE is also provided. Furthermore, numerical simulations are provided to illustrate the reliability and accuracy of the method.Originality/valueMany engineers or scientist can utilize the present method for solving their ordinary or Caputo–fractional differential models. To the best of authors’ knowledge, the present work has not been used or introduced for the considered type of differential equations.

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Section: Introductionmentioning

confidence: 99%

Vieta–Lucas wavelets method for fractional linear and nonlinear delay differential equations

Idrees

Saeed

2022

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“…Constant-order FDEs have been solved using a variety of wavelet basis functions. Legendre, Bernoulli, Euler, CAS, Chebyshev wavelets of first and second kind attracted a great deal of attention (Li, 2010; Rehman and Khan, 2011; Zhu and Wang, 2013; Razzaghi and Yousefi, 2002; Keshavarz et al , 2019; Turan Dincel, 2019; Tural-Polat, 2019).…”

Section: Introductionmentioning

confidence: 99%

Fourth kind Chebyshev Wavelet Method for the solution of multi-term variable order fractional differential equations

Ocak

Polat

2021

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PurposeMulti-term variable-order fractional differential equations (VO-FDEs) are powerful tools in accurate modeling of transient-regime real-life problems such as diffusion phenomena and nonlinear viscoelasticity. In this paper the Chebyshev polynomials of the fourth kind is employed to obtain a numerical solution for those multi-term VO-FDEs.Design/methodology/approachTo this end, operational matrices for the approximation of the VO-FDEs are obtained using the Fourth kind Chebyshev Wavelets (FKCW). Thus, the VO-FDE is condensed into an algebraic equation system. The solution of the system of those equations yields a coefficient vector, the coefficient vector in turn yields the approximate solution.FindingsSeveral examples that we present at the end of the paper emphasize the efficacy and preciseness of the proposed method.Originality/valueThe value of the paper stems from the exploitation of FKCWs for the numerical solution of multi-term VO-FDEs. The method produces accurate results even for relatively small collocation points. What is more, FKCW method provides a compact mapping between multi-term VO-FDEs and a system of algebraic equations given in vector-matrix form.

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“…e authors of [32] utilized the finite difference scheme for the approximation of RDE. Other works on the numerical approximation of the solution of fractional-order RDE can be found in [33][34][35][36][37][38][39][40][41][42] and references therein. In this work, we approximate the solution of fractional-order RDE with ABC derivative of the following form:…”

Section: Introductionmentioning

confidence: 99%

Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional Derivative

Liu

Kamran

Yao

2020

Journal of Mathematics

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The Riccati differential equation is a well-known nonlinear differential equation and has different applications in engineering and science domains, such as robust stabilization, stochastic realization theory, network synthesis, and optimal control, and in financial mathematics. In this study, we aim to approximate the solution of a fractional Riccati equation of order 0<β<1 with Atangana–Baleanu derivative (ABC). Our numerical scheme is based on Laplace transform (LT) and quadrature rule. We apply LT to the given fractional differential equation, which reduces it to an algebraic equation. The reduced equation is solved for the unknown in LT space. The solution of the original problem is retrieved by representing it as a Bromwich integral in the complex plane along a smooth curve. The Bromwich integral is approximated using the trapezoidal rule. Some numerical experiments are performed to validate our numerical scheme.

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Third-Kind Chebyshev Wavelet Method for the Solution of Fractional Order Riccati Differential Equations

Cited by 7 publications

References 39 publications

Vieta–Lucas wavelets method for fractional linear and nonlinear delay differential equations

Vieta–Lucas wavelets method for fractional linear and nonlinear delay differential equations

Fourth kind Chebyshev Wavelet Method for the solution of multi-term variable order fractional differential equations

Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional Derivative

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