A comparative study on interior penalty discontinuous Galerkin and enriched Galerkin methods for time-fractional Sobolev equation

Mohammadi-Firouzjaei, Hadi; Adibi, Hojatollah; Dehghan, Mehdi

doi:10.1007/s00366-022-01624-7

Cited by 5 publications

(1 citation statement)

References 29 publications

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“…According to earlier studies [10,11], the fractional differential equation (FDE) has numerous applications in engineering, mathematical physics, and fluid flow. In the literature, the direct and indirect methods are frequently used to solve differential equations (DEs) and FDEs [12][13][14][15][16][17]. The indirect approach has the benefit of guaranteeing the results' existence and originality.…”

Section: Introductionmentioning

confidence: 99%

The general Bernstein function: Application to χ‐fractional differential equations

Sadek,

Sami Bataineh

2024

Math Methods in App Sciences

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In this paper, we present the general Bernstein functions for the first time. The properties of generalized Bernstein basis functions are given and demonstrated. The classical Bernstein polynomial bases are merely a subset of the general Bernstein functions. Based on the new Bernstein base functions and the collocation method, we present a numerical method for solving linear and nonlinear ‐fractional differential equations ( ‐FDEs) with variable coefficients. The fractional derivative used in this work is the ‐Caputo fractional derivative sense ( ‐CFD). Combining the Bernstein functions basis and the collocation methods yields the approximation solution of nonlinear differential equations. These base functions can be used to solve many problems in applied mathematics, including calculus of variations, differential equations, optimal control, and integral equations. Furthermore, the convergence of the method is rigorously justified and supported by numerical experiments.

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