A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order
Abstract
:1. Introduction
- Collocation-type methods include the generalized Taylor collocation method [16], a wavelet based collocation method [17], the Legendre collocation method for the nonlinear spacetime fractional partial differential equations [18] and the Sinc collocation method for variable-order fractional integro-partial differential equations [19].
- Other methods such as a neural network method based on the sine and the cosine functions [1].
2. The Least Squares Differential Quadrature Method
- From the first initial condition we have:By imposing that the relation is satisfied in the nodes we obtain a linear system of equations in unknowns which has the unique solution .
- From the second initial condition we have:Again, by imposing that the relation is satisfied in the corresponding nodes, we obtain a linear system which has the unique solution .
3. Numerical Examples
3.1. Nonlinear Time-Fractional Advection Partial Differential Equation
3.1.1. The Case
3.1.2. The Case
3.1.3. The Case
3.2. Nonlinear Time-Fractional Hyperbolic Equation
3.2.1. The Case
3.2.2. The Case
3.2.3. The case
3.3. The Fractional BBM-Burgers’ Equation
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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x | t | ADM | HPM | LSDQM |
---|---|---|---|---|
0 | 0 | 0 | 0 | 0 |
0.25 | 0.25 | |||
0.5 | 0.5 | |||
0.75 | 0.75 | |||
1 | 1 | |||
1.25 | 1.25 | |||
1.5 | 1.5 |
x | t | ADM | HPM | LSDQM |
---|---|---|---|---|
0 | 0 | 0 | 0 | 0 |
0.25 | 0.25 | |||
0.5 | 0.5 | |||
0.75 | 0.75 | |||
1 | 1 | |||
1.25 | 1.25 | |||
1.5 | 1.5 |
x | t | ADM | HPM | LSDQM |
---|---|---|---|---|
0 | 0 | 0 | 0 | 0 |
0.25 | 0.25 | |||
0.5 | 0.5 | |||
0.75 | 0.75 | |||
1 | 1 | |||
1.25 | 1.25 |
x | t | ADM | HPM | LSDQM |
---|---|---|---|---|
0 | 0 | 0 | 0 | 0 |
0.25 | 0.25 | |||
0.5 | 0.5 | |||
0.75 | 0.75 | |||
1 | 1 |
x | t | ADM | HPM | LSDQM |
---|---|---|---|---|
0 | 0 | 0 | 0 | 0 |
0.25 | 0.25 | |||
0.5 | 0.5 | |||
0.75 | 0.75 | |||
1 | 1 |
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Bota, C.; Căruntu, B.; Ţucu, D.; Lăpădat, M.; Paşca, M.S. A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order. Mathematics 2020, 8, 1336. https://doi.org/10.3390/math8081336
Bota C, Căruntu B, Ţucu D, Lăpădat M, Paşca MS. A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order. Mathematics. 2020; 8(8):1336. https://doi.org/10.3390/math8081336
Chicago/Turabian StyleBota, Constantin, Bogdan Căruntu, Dumitru Ţucu, Marioara Lăpădat, and Mădălina Sofia Paşca. 2020. "A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order" Mathematics 8, no. 8: 1336. https://doi.org/10.3390/math8081336