Numerical treatment for the solution of fractional fifth-order Sawada–Kotera equation using second kind Chebyshev wavelet method

Gupta, A. K.; Ray, S. Saha

doi:10.1016/j.apm.2015.04.003

Cited by 62 publications

(16 citation statements)

References 16 publications

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“…where the Chebyshev wavelet ψ n,m (·) in (13). In the other hand, the function ω(x, t) in (14) can be rewritten a finite sum of entries of the spatial matrix as…”

Section: Chebyshev Wavelets Approximationmentioning

confidence: 99%

“…Analytical and approximate series solutions for the nonlinear fractional differential equations are fundamental importance for seeking solutions of the most complex phenomena that are modeled. There are many methods that have also been proposing for solving analytical and approximate series solutions: the transform methods, including Laplace, Fourier, and Mellin transforms [5]; the Tau method [6]; the Adomian decomposition method [7]; the variational iteration method [7,8]; the Sumudu decomposition method [9]; the blockpulse functions [10]; shifted Chebyshev polynomials [11]; shifted Legendre polynomials [12]; Chebyshev wavelets [13,14]; and Legendre wavelets [15].…”

Section: Introductionmentioning

confidence: 99%

“…A.K. Gupta and S.S. Ray studied the solution of fractional fifth-order Sawada-Kotera equation using second kind Chebyshev wavelet method [14] and many others [16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

New Analytical Solutions for Time-Fractional Kolmogorov-Petrovsky-Piskunov Equation with Variety of Initial Boundary Conditions

2019

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The generalized time fractional Kolmogorov-Petrovsky-Piskunov equation (FKPP),, which plays an important role in engineering, chemical reaction problem is proposed by Caputo fractional order derivative sense. In this paper, we develop a framework wavelet, including shift Chebyshev polynomial of the first kind as a mother wavelet, and also construct some operational matrices that represent Caputo fractional derivative to obtain analytical solutions for FKPP equation with three different types of Initial Boundary conditions (Dirichlet, Dirichlet-Neumann, and Neumann-Robin). Our results shown that the Chebyshev wavelet is a powerful method, due to its simplicity, efficiency in analytical approximations, and its fast convergence. The comparison of the Chebyshev wavelet results indicates that the proposed method not only gives satisfactory results but also do not need large amount of CPU times.

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“…where the Chebyshev wavelet ψ n,m (·) in (13). In the other hand, the function ω(x, t) in (14) can be rewritten a finite sum of entries of the spatial matrix as…”

Section: Chebyshev Wavelets Approximationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

New Analytical Solutions for Time-Fractional Kolmogorov-Petrovsky-Piskunov Equation with Variety of Initial Boundary Conditions

2019

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“…Exact traveling wave solution for nonlinear differential equations (Sayevand et al, 2014) has also been investigated using a general functional transformation. Two-dimensional Chebyshev wavelet method (Gupta and Saha Ray, 2015) has been used to obtain numerical approximations to the solution of fractional fifth-order Sawada-Kotera equation. Sayevand et al (2015) approximated the solutions of some important integral equations using a numerical technique based on homotopy analysis method.…”

Section: Introductionmentioning

confidence: 99%

Application of homotopy analysis method to the solution of ninth order boundary value problems in AFTI-F16 fighters

Akram

Sadaf

2017

Journal of the Association of Arab Universities for Basic and A

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This paper is devoted to the study of ninth order boundary value problems that emerge in the mathematical modeling of AFTI-F16 fighter. An augmented longitudinal and lateral dynamics of the AFTI-F16 fighter is described by ninth order differential equations, containing unknown parameters which can be determined using automated system identification algorithms. The solution of the boundary value problem is obtained in terms of a convergent series using homotopy analysis method (HAM). The method is effectively applied on numerical examples and the results are compared with those given in the literature, revealing that the presented method gives better approximations to the exact solution.

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“…Particularly, orthogonal wavelets are widely used in approximating numerical solutions of various types of fractional order differential equations in the relevant literatures; see [18][19][20][21][22]. Among them, the second-kind Chebyshev wavelets have gained much attention due to their useful properties ( [23][24][25][26]) and can handle different types of differential problems. It is observed that most papers using these wavelets methods to approximate numerical solutions of fractional order differential equations are based on the operational matrix of fractional integral or fractional derivatives.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method

Zhou

2017

Advances in Mathematical Physics

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The second-kind Chebyshev wavelets collocation method is applied for solving a class of time-fractional diffusion-wave equation. Fractional integral formula of a single Chebyshev wavelet in the Riemann-Liouville sense is derived by means of shifted Chebyshev polynomials of the second kind. Moreover, convergence and accuracy estimation of the second-kind Chebyshev wavelets expansion of two dimensions are given. During the process of establishing the expression of the solution, all the initial and boundary conditions are taken into account automatically, which is very convenient for solving the problem under consideration. Based on the collocation technique, the second-kind Chebyshev wavelets are used to reduce the problem to the solution of a system of linear algebraic equations. Several examples are provided to confirm the reliability and effectiveness of the proposed method.

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Numerical treatment for the solution of fractional fifth-order Sawada–Kotera equation using second kind Chebyshev wavelet method

Cited by 62 publications

References 16 publications

New Analytical Solutions for Time-Fractional Kolmogorov-Petrovsky-Piskunov Equation with Variety of Initial Boundary Conditions

New Analytical Solutions for Time-Fractional Kolmogorov-Petrovsky-Piskunov Equation with Variety of Initial Boundary Conditions

Application of homotopy analysis method to the solution of ninth order boundary value problems in AFTI-F16 fighters

Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method

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