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Numerical Approximations for the Solutions of Fourth Order Time Fractional Evolution Problems Using a Novel Spline Technique

by Ghazala Akram, Muhammad Abbas, Hira Tariq, Maasoomah Sadaf, Thabet Abdeljawad and Manar A. Alqudah

Fractal Fract. 2022, 6(3), 170; https://doi.org/10.3390/fractalfract6030170 - 19 Mar 2022

Cited by 11 | Viewed by 3327

Abstract

Developing mathematical models of fractional order for physical phenomena and constructing numerical solutions for these models are crucial issues in mathematics, physics, and engineering. Higher order temporal fractional evolution problems (EPs) with Caputo’s derivative (CD) are numerically solved using a sextic polynomial spline technique (SPST). These equations are frequently applied in a wide variety of real-world applications, such as strain gradient elasticity, phase separation in binary mixtures, and modelling of thin beams and plates, all of which are key parts of mechanical engineering. The SPST can be used for space discretization, whereas the backward Euler formula can be used for time discretization. For the temporal discretization, the method’s convergence and stability are assessed. To show the accuracy and applicability of the proposed technique, numerical simulations are employed. Full article

(This article belongs to the Special Issue Advances in Fractional and Fractal Boundary Value Problems in Applied Sciences)

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22 pages, 1850 KB

Open AccessArticle

Quasi-Interpolation in a Space of C² Sextic Splines over Powell–Sabin Triangulations

by Salah Eddargani, María José Ibáñez, Abdellah Lamnii, Mohamed Lamnii and Domingo Barrera

Mathematics 2021, 9(18), 2276; https://doi.org/10.3390/math9182276 - 16 Sep 2021

Cited by 7 | Viewed by 2435

Abstract

In this work, we study quasi-interpolation in a space of sextic splines defined over Powell–Sabin triangulations. These spline functions are of class

C^{2}

on the whole domain but fourth-order regularity is required at vertices and

C^{3}

regularity is imposed across the [...] Read more.

In this work, we study quasi-interpolation in a space of sextic splines defined over Powell–Sabin triangulations. These spline functions are of class

C^{2}

on the whole domain but fourth-order regularity is required at vertices and

C^{3}

regularity is imposed across the edges of the refined triangulation and also at the interior point chosen to define the refinement. An algorithm is proposed to define the Powell–Sabin triangles with a small area and diameter needed to construct a normalized basis. Quasi-interpolation operators which reproduce sextic polynomials are constructed after deriving Marsden’s identity from a more explicit version of the control polynomials introduced some years ago in the literature. Finally, some tests show the good performance of these operators. Full article

(This article belongs to the Section E: Applied Mathematics)

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