Two‐dimensional shifted Legendre polynomial collocation method for electromagnetic waves in dielectric media via almost operational matrices

Patel, Vijay Kumar; Singh, Shatrughan; Singh, Vineet Kumar

doi:10.1002/mma.4257

Cited by 32 publications

(10 citation statements)

References 31 publications

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“…) and F n (t) is the polygamma function which is given in Singh et al 68 Theorem 5.4 (Patel et al 67 ). Let…”

Section: Error Estimatementioning

confidence: 99%

“…By using Theorem 5.3 for a nm in Patel et al 67 of Equation ( 53), we have Thus, ||u N (x, t)−u N (x, t)|| 2 2 → 0 as N, N → ∞, that this shows the sequence of partial sums u N (x, t) is a Cauchy sequence in Hilbert space L 2 ([0, 1] × [0, 1]) and it converges to say Ξ. To complete the proof, we show that u(t) = Ξ, and then we have…”

Section: Convergence Analysismentioning

confidence: 99%

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A numerical method for finding solution of the distributed‐order time‐fractional forced Korteweg–de Vries equation including the Caputo fractional derivative

Derakhshan

Aminataei

2021

Math Methods in App Sciences

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In this paper, for the first time, the distributed-order time-fractional forced Korteweg-de Vries equation is studied. We use a numerical method based on the shifted Legendre operational matrix of distributed-order fractional derivative with Tau method to find approximate solution of distributed-order forced Korteweg-de Vries equation. This shifted Legendre operational matrix of distributed-order fractional derivative with Tau method is used to reduce the solution of the distributed-order time-fractional forced Korteweg-de Vries equations to a system of algebraic equations. An error analysis and convergence are obtained. Finally, to display the applicability and validity of the numerical method, some examples are implemented.

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“…) and F n (t) is the polygamma function which is given in Singh et al 68 Theorem 5.4 (Patel et al 67 ). Let…”

Section: Error Estimatementioning

confidence: 99%

Section: Convergence Analysismentioning

confidence: 99%

A numerical method for finding solution of the distributed‐order time‐fractional forced Korteweg–de Vries equation including the Caputo fractional derivative

Derakhshan

Aminataei

2021

Math Methods in App Sciences

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“…In this article, we develop a stable and effective collocation method to solve (3). The collocation method is one of the most efficient methods for obtaining accurate numerical solutions of differential equations including variable coefficients and nonlinear differential equations [11][12][13][14]. The stability of collocation methods has always been an important topic.…”

Section: Introductionmentioning

confidence: 99%

A new stable collocation method for solving a class of nonlinear fractional delay differential equations

et al. 2020

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In this paper, a stable collocation method for solving the nonlinear fractional delay differential equations is proposed by constructing a new set of multiscale orthonormal bases of W 1 2,0. Error estimations of approximate solutions are given and the highest convergence order can reach four in the sense of the norm of W 1 2,0. To overcome the nonlinear condition, we make use of Newton's method to transform the nonlinear equation into a sequence of linear equations. For the linear equations, a rigorous theory is given for obtaining their ε-approximate solutions by solving a system of equations or searching the minimum value. Stability analysis is also obtained. Some examples are discussed to illustrate the efficiency of the proposed method.

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“…They applied spectral tau method by using fractional-order shifted Jacobi orthogonal function along with the operational matrix. Moreover, Patel et al [16] used the collocation method which is based on 2D shifted Legendre polynomials for the solution of FPDEs. Yi et al [25] applied the two-dimensional Block pulse operational matrix for FPDEs.…”

Section: Introductionmentioning

confidence: 99%

New collocation scheme for solving fractional partial differential equations

Phang

Kanwal

Loh

2020

Hacettepe Journal of Mathematics and Statistics

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This article concerned about the numerical solution of time fractional partial differential equations (FPDEs). The proposed technique is using shifted Chebyshev-Gauss-Lobatto (CGL) collocation points in conjunction with an operational matrix of Caputo sense derivatives via Genocchi polynomials. The system of linear algebraic equations is obtained when the main equation along with the initial as well as boundary conditions is collocated by using shifted CGL collocation points. The main approach to this method is to transform the FPDEs to system of algebraic equations, hence, greatly simplify the numerical scheme. Comparison of the obtained results with the existing methods depicts that the suggested method is highly effect, more efficient and have less computational work. Some examples are given to illustrate the effectiveness and applicability of the proposed technique.

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Two‐dimensional shifted Legendre polynomial collocation method for electromagnetic waves in dielectric media via almost operational matrices

Cited by 32 publications

References 31 publications

A numerical method for finding solution of the distributed‐order time‐fractional forced Korteweg–de Vries equation including the Caputo fractional derivative

A numerical method for finding solution of the distributed‐order time‐fractional forced Korteweg–de Vries equation including the Caputo fractional derivative

A new stable collocation method for solving a class of nonlinear fractional delay differential equations

New collocation scheme for solving fractional partial differential equations

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