| Citation: |
Amjid Ali, Teruya Minamoto. A NEW NUMERICAL TECHNIQUE FOR INVESTIGATING BOUNDARY VALUE PROBLEMS WITH Ψ-CAPUTO FRACTIONAL OPERATOR[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 275-297. doi: 10.11948/20220062
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A NEW NUMERICAL TECHNIQUE FOR INVESTIGATING BOUNDARY VALUE PROBLEMS WITH Ψ-CAPUTO FRACTIONAL OPERATOR
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Abstract
This article introduces a new numerical approach for solving linear and non-linear boundary value problems for Ψ-fractional differential equations (Ψ-FDEs). This approach relies on the Ψ-Haar wavelet operational integration matrices. The Ψ-operational matrices (Ψ-OMs) are used to convert the Ψ-FDE to an algebraic system of equations. The non-linear fractional boundary value problems are first linearized using the quasi-linearization technique, and then the Ψ-Haar wavelet technique is applied to the linearized problem. The solution is updated by the Ψ-Haar wavelet method in each iteration of the quasi-linearization technique. The proposed method is a good and simple mathematical technique for numerically solving non-linear Ψ-FDEs. The operational matrix (OM) method is computationally more efficient. Several linear and non-linear boundary value problems are discussed to demonstrate the applicability, efficiency, and simplicity of the method. Moreover, the error analysis is carried out resulting a rigorous error bound for the proposed method.
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References
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Amjid Ali, Teruya Minamoto. A NEW NUMERICAL TECHNIQUE FOR INVESTIGATING BOUNDARY VALUE PROBLEMS WITH Ψ-CAPUTO FRACTIONAL OPERATOR[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 275-297. doi: 10.11948/20220062
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Figure 1.
exact and numerical integration of
$ h({t})=\frac{{t}(e^{{t}}+2)}{e+2} $ -
Figure 2.
Exact and Approximate solutions and the corresponding max absolute error of equation (5.1) for
$ \Psi({t})={t} $ $ \Psi({t})=({t}^2+t)/2 $ -
Figure 3.
For equation (5.7). Exact and approximate solutions for
$ {\Psi}({t})={t} $ $ {\Psi}({t})={{t}}^3 $ $ \max $ -
Figure 4.
For equation (5.11): Approximate solutions for different choices of
$ {{\alpha}} $ -
Figure 5.
exact and approximate solutions of (5.18) for
$ {\Psi}({t})={t} $ $ {\Psi}({t})=\tan(\frac{{t}}{2}) $ $ \max $