Application of Fractional Residual Power Series Algorithm to Solve Newell–Whitehead–Segel Equation of Fractional Order
Abstract
:1. Introduction
2. Preliminaries and Notations
- , .
- , .
- , that is, is a linear, .
3. Solution Methodology of the FRPS Algorithm
- Step 1: Assume that the solution of FNWSEs (1) and (2) has the MFPS about :
- Step 2: Define the th truncated series of such that
- Step 3: Consider the initial condition , then the zeroth MFPS approximate solution of is .
- Step 4: Define the th residual function such that
- Step 5: Substitute the th truncated series into the th residual function such that
- Step 6: Set in Step 5, then by using , the first unknown coefficient is obtained. Therefore, the first approximate PS solution is also obtained.
- Step 7: For , do the following subroutine:
- (A)
- Apply the operator , () times, on both sides of the th residual function in Step 4 such that .
- (B)
- Compute the resulting equation at with equality to zero such that , with the help of for at .
- (C)
- Find the th unknown coefficient and do Step 7 for until the arbitrary .
- Step 8: Collect the obtained coefficients for each in terms of expanded MFPS and try to find a general pattern with the term of infinite series so that the exact solution of FNWSEs (1) and (2) is obtained; otherwise, the pattern obtained in the sense of the series coefficients will be the th approximate MFPS solution of FNWSEs (1) and (2).
- Step 9: Stop.
4. Numerical Results and Discussion
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Exact Solution | Approximation | Absolute Error | ||
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Iteration | Errors | ||||
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Absolute | |||||
Relative | |||||
Absolute | |||||
Relative | |||||
Absolute | |||||
Relative | |||||
Absolute | |||||
Relative |
Exact | FPS Method | FCT-HP Method [12] | |||
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Fifth Appr. Sol. | Absolute Error | Fifth Appr. Sol. | Absolute Error | ||
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Saadeh, R.; Alaroud, M.; Al-Smadi, M.; Ahmad, R.R.; Salma Din, U.K. Application of Fractional Residual Power Series Algorithm to Solve Newell–Whitehead–Segel Equation of Fractional Order. Symmetry 2019, 11, 1431. https://doi.org/10.3390/sym11121431
Saadeh R, Alaroud M, Al-Smadi M, Ahmad RR, Salma Din UK. Application of Fractional Residual Power Series Algorithm to Solve Newell–Whitehead–Segel Equation of Fractional Order. Symmetry. 2019; 11(12):1431. https://doi.org/10.3390/sym11121431
Chicago/Turabian StyleSaadeh, Rania, Mohammad Alaroud, Mohammed Al-Smadi, Rokiah Rozita Ahmad, and Ummul Khair Salma Din. 2019. "Application of Fractional Residual Power Series Algorithm to Solve Newell–Whitehead–Segel Equation of Fractional Order" Symmetry 11, no. 12: 1431. https://doi.org/10.3390/sym11121431
APA StyleSaadeh, R., Alaroud, M., Al-Smadi, M., Ahmad, R. R., & Salma Din, U. K. (2019). Application of Fractional Residual Power Series Algorithm to Solve Newell–Whitehead–Segel Equation of Fractional Order. Symmetry, 11(12), 1431. https://doi.org/10.3390/sym11121431