Semi-Analytical Solutions for Some Types of Nonlinear Fractional-Order Differential Equations Based on Third-Kind Chebyshev Polynomials
Abstract
:1. Introduction
2. Important Preliminaries
2.1. Derivatives in Fractions
2.2. Third-Kind Chebyshev Polynomials: Definition and Properties
3. Operational Matrices of Derivatives for Third-Kind Chebyshev Polynomials
3.1. Third-Kind Chebyshev Polynomials: Operational Matrix of Integer-Order Derivatives IC3OM
3.2. Third-Kind Chebyshev Polynomials: Operational Matrix of Fractional-Order Derivatives FC3OM
4. Nonlinear Multi-Term Fractional-Order Differential Equations (NMFDEs)
4.1. Initial-Value Problem (INMFDEs)
4.2. Boundary Value Problem (BNMFDEs)
5. Error Bound
6. Numerical Applications
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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m | -Error | -Error | ||||
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In [38] | In [28] | Our Method | In [38] | In [28] | Our Method | |
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In [38] | In [28] | Our Method | In [38] | In [28] | Our Method | |
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t | In [29] | Proposed Method |
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0.7 | 0.0 | |
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1.0 | 0.0 |
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El-Sayed, A.A.E.; Boulaaras, S.; AbaOud, M. Semi-Analytical Solutions for Some Types of Nonlinear Fractional-Order Differential Equations Based on Third-Kind Chebyshev Polynomials. Fractal Fract. 2023, 7, 784. https://doi.org/10.3390/fractalfract7110784
El-Sayed AAE, Boulaaras S, AbaOud M. Semi-Analytical Solutions for Some Types of Nonlinear Fractional-Order Differential Equations Based on Third-Kind Chebyshev Polynomials. Fractal and Fractional. 2023; 7(11):784. https://doi.org/10.3390/fractalfract7110784
Chicago/Turabian StyleEl-Sayed, Adel Abd Elaziz, Salah Boulaaras, and Mohammed AbaOud. 2023. "Semi-Analytical Solutions for Some Types of Nonlinear Fractional-Order Differential Equations Based on Third-Kind Chebyshev Polynomials" Fractal and Fractional 7, no. 11: 784. https://doi.org/10.3390/fractalfract7110784