Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein–Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz–Galerkin Method

Kanwal, Afshan; Phang, Chang; Iqbal, Umer

doi:10.3390/computation6030040

Cited by 22 publications

(16 citation statements)

References 21 publications

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“…Numerous researchers have developed various methods for solving a time-fractional DWE; see [14,[25][26][27]. Recently, Kanwal et al [28] presented a Ritz-Galerkin method together with two-dimensional Genocchi polynomials to establish the numerical solutions of a time-fractional DWE and a time-fractional Klein-Gordon equation.…”

Section: Introductionmentioning

confidence: 99%

A numerical algorithm based on modified extended B-spline functions for solving time-fractional diffusion wave equation involving reaction and damping terms

et al. 2019

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In this study, we have proposed an efficient numerical algorithm based on third degree modified extended B-spline (EBS) functions for solving time-fractional diffusion wave equation with reaction and damping terms. The Caputo time-fractional derivative has been approximated by means of usual finite difference scheme and the modified EBS functions are used for spatial discretization. The stability analysis and derivation of theoretical convergence validates the authenticity and effectiveness of the proposed algorithm. The numerical experiments show that the computational outcomes are in line with the theoretical expectations. Moreover, the numerical results are proved to be better than other methods on the topic.

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Section: Introductionmentioning

confidence: 99%

A numerical algorithm based on modified extended B-spline functions for solving time-fractional diffusion wave equation involving reaction and damping terms

et al. 2019

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“…Bernoulli polynomials have been used in solving various type fractional calculus problems, such as in [21,22]. These kinds of semi-orthogonal polynomials, including Genocchi polynomials and Euler polynomials, are widely used in solving differential equations up to arbitrary orders [23][24][25]. More specifically, we derive the new operational matrix based on poly-Bernoulli polynomials for solving the fractional differential equation in an Atangana-Baleanu sense.…”

Section: Introductionmentioning

confidence: 99%

An Operational Matrix Method Based on Poly-Bernoulli Polynomials for Solving Fractional Delay Differential Equations

2020

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In this work, we derive the operational matrix using poly-Bernoulli polynomials. These polynomials generalize the Bernoulli polynomials using a generating function involving a polylogarithm function. We first show some new properties for these poly-Bernoulli polynomials; then we derive new operational matrix based on poly-Bernoulli polynomials for the Atangana–Baleanu derivative. A delay operational matrix based on poly-Bernoulli polynomials is derived. The error bound of this new method is shown. We applied this poly-Bernoulli operational matrix for solving fractional delay differential equations with variable coefficients. The numerical examples show that this method is easy to use and yet able to give accurate results.

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“…A sinc-Chebyshev collocation method (SCCM) was developed in Nagy [42] for numerical treatment of the time-fractional nonlinear KGE. Recently, in Kanwal et al [43], Genocchi polynomials were employed together with the Ritz-Galerkin scheme to solve fractional KGEs and diffusion wave equations. A linearized second-order scheme was introduced in Lyu and Vong [44] to solve non-linear time-fractional Klein-Gordon-type equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical Treatment of Time-Fractional Klein–Gordon Equation Using Redefined Extended Cubic B-Spline Functions

et al. 2020

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In this article we develop a numerical algorithm based on redefined extended cubic B-spline functions to explore the approximate solution of the time-fractional Klein-Gordon equation. The proposed technique employs the finite difference formulation to discretize the Caputo fractional time derivative of order α ∈ (1, 2] and uses redefined extended cubic B-spline functions to interpolate the solution curve over a spatial grid. A stability analysis of the scheme is conducted, which confirms that the errors do not amplify during execution of the numerical procedure. The derivation of a uniform convergence result reveals that the scheme is O(h 2 + t 2−α) accurate. Some computational experiments are carried out to verify the theoretical results. Numerical simulations comparing the proposed method with existing techniques demonstrate that our scheme yields superior outcomes.

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Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein–Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz–Galerkin Method

Cited by 22 publications

References 21 publications

A numerical algorithm based on modified extended B-spline functions for solving time-fractional diffusion wave equation involving reaction and damping terms

A numerical algorithm based on modified extended B-spline functions for solving time-fractional diffusion wave equation involving reaction and damping terms

An Operational Matrix Method Based on Poly-Bernoulli Polynomials for Solving Fractional Delay Differential Equations

Numerical Treatment of Time-Fractional Klein–Gordon Equation Using Redefined Extended Cubic B-Spline Functions

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