A numerical method based on the Chebyshev cardinal functions for variable‐order fractional version of the fourth‐order 2D Kuramoto‐Sivashinsky equation

An innovative numerical procedure for solving the viscoelastic arch problem based on variable fractional rheological models, directly in time domain, is proposed and investigated. First, the nonlinear integral‐differential governing equation is established according to the variable fractional constitutive relation and geometrical relationship. Second, the nonlinear integral‐differential governing equation is transformed into algebraic equations and solved by using the shifted Legendre polynomials. Furthermore, the accuracy and effectiveness of the algorithm are verified according to the mathematical example. A small value of the absolute error between numerical and accurate solution is obtained. Finally, the dynamic analysis of viscoelastic arch is investigated to determine the displacement at different times and positions. The displacement of the viscoelastic arch is compared under various loading (uniformly distributed load and linear load). The displacement of the viscoelastic arch of different materials under the same load conditions is also investigated. The results in the paper show the efficiency of the proposed numerical algorithm in the dynamical analysis of the viscoelastic arch.

Section: Introductionmentioning

confidence: 99%

Numerical analysis of nonlinear variable fractional viscoelastic arch based on shifted Legendre polynomials

Cao

Chen

Wang

et al. 2021

“…To solve this drawback, in recent years, numerical methods have been used as a suitable alternative. A semi‐discrete approach is presented in Hosseininia et al 6 through 2D Chebyshev cardinal functions for solving variable‐order fractional version of the fourth‐order 2D Kuramoto‐Sivashinsky equation. A meshless approach is applied in Heydari et al 7 for solving nonlinear variable‐order time fractional 2D Ginzburg‐Landau equation.…”

Section: Introductionmentioning

confidence: 99%

A hybrid approach established upon the Müntz‐Legender functions and 2D Müntz‐Legender wavelets for fractional Sobolev equation

Hosseininia

Heydari

Avazzadeh

2022

Self Cite

This article proposes a hybrid technique for finding approximation solutions of the fractional 2D Sobolev equation. In the proposed approach, the Müntz-Legender functions and Müntz-Legender wavelets are, respectively, utilized to approximate the solution of the problem under consideration in the time and spatial directions. By implementing the presented technique, solving the 2D fractional Sobolev equation is converted into solving a system of algebraic equations. Three examples are solved to examine the validity of the proposed method.

“…The cardinality enables us to compute exactly the CCFs expansion coefficients of a function. Recently, these functions have been utilized to solve various equations, including a category of systems of VO fractional quadratic integral equations, 51 VO fractional optimal control problems, 52 VO fractional 2D Kuramoto–Sivashinsky equation, 53 VO fractional pantograph equation, 54 coupled fractal‐fractional Schrödinger equations, 55 and coupled VO fractional sine‐Gordon equations 7 …”

Section: Introductionmentioning

confidence: 99%

Numerical investigation of variable‐order fractional Benjamin–Bona–Mahony–Burgers equation using a pseudo‐spectral method

Heydari

Razzaghi

Avazzadeh

2021

Self Cite

This article introduces a new local variable‐order (VO) fractional differentiation called the VO conformable fractional derivative. This derivative is employed to define the VO time fractional Benjamin–Bona–Mahony–Burgers (BBMB) equation. A pseudo‐spectral algorithm using the Chebyshev cardinal functions (CCFs) is adopted to find a numerical solution for this equation. The presented method with the help of the CCFs derivative matrices (which are obtained in this research) turns problem solving into solving an algebraic system of equations. The proposed approach is applied on some test problems, and its accuracy is examined in terms of L∞ and L2 error norms. A numerical comparison is performed between the achieved results of the method with the results obtained from the cubic B‐spline method and the hybrid method generated using the quintic Hermite approach and the finite difference technique.