Next Article in Journal
Fractional-Order Tabu Learning Neuron Models and Their Dynamics
Next Article in Special Issue
Analytic Solution of the Time-Fractional Partial Differential Equation Using a Multi-G-Laplace Transform Method
Previous Article in Journal
Multiple Solutions to the Fractional p-Laplacian Equations of Schrödinger–Hardy-Type Involving Concave–Convex Nonlinearities
Previous Article in Special Issue
The Existence and Ulam Stability Analysis of a Multi-Term Implicit Fractional Differential Equation with Boundary Conditions
 
 
Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
Journals
Fractal Fract
Volume 8
Issue 7
10.3390/fractalfract8070427
fractalfract-logo
Submit to this Journal Review for this Journal Propose a Special Issue
► Article Menu

Article Menu

share Share announcement Help format_quote Cite question_answer Discuss in SciProfiles
first_page
settings
Order Article Reprints
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation

by
Mohammed H. Alharbi
1,
Abdullah F. Abu Sunayh
1,
Ahmed Gamal Atta
2 and
Waleed Mohamed Abd-Elhameed
1,3,*
1
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23218, Saudi Arabia
2
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt
3
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
*
Author to whom correspondence should be addressed.
Fractal Fract. 2024, 8(7), 427; https://doi.org/10.3390/fractalfract8070427
Submission received: 18 June 2024 / Revised: 18 July 2024 / Accepted: 18 July 2024 / Published: 20 July 2024
(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)
Browse Figures
Versions Notes

Abstract

This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs.
Keywords: Fibonacci polynomials; golden ratio; shifted polynomials; collocation procedure; operational matrices; error analysis Fibonacci polynomials; golden ratio; shifted polynomials; collocation procedure; operational matrices; error analysis

Share and Cite

MDPI and ACS Style

Alharbi, M.H.; Abu Sunayh, A.F.; Atta, A.G.; Abd-Elhameed, W.M. Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation. Fractal Fract. 2024, 8, 427. https://doi.org/10.3390/fractalfract8070427

AMA Style

Alharbi MH, Abu Sunayh AF, Atta AG, Abd-Elhameed WM. Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation. Fractal and Fractional. 2024; 8(7):427. https://doi.org/10.3390/fractalfract8070427

Chicago/Turabian Style

Alharbi, Mohammed H., Abdullah F. Abu Sunayh, Ahmed Gamal Atta, and Waleed Mohamed Abd-Elhameed. 2024. "Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation" Fractal and Fractional 8, no. 7: 427. https://doi.org/10.3390/fractalfract8070427

APA Style

Alharbi, M. H., Abu Sunayh, A. F., Atta, A. G., & Abd-Elhameed, W. M. (2024). Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation. Fractal and Fractional, 8(7), 427. https://doi.org/10.3390/fractalfract8070427

Article Metrics

Back to TopTop