Jacobi spectral method for variable-order fractional Benney-Lin equation arising in falling film problems. (English) Zbl 1524.65654
Summary: In this paper, a variable-order fractional version of the Benney-Lin equation is defined using the variable-order fractional derivative in the Caputo type. A collocation method based on the shifted Jacobi polynomials is applied to deal with this problem. Some matrix relationships related to these polynomials are extracted and used in constructing the established method. The obtained relations cause the method calculations to be significantly reduced, which reduce the method execution time. The established technique converts solving the problem under study into solving an algebraic system of equations. The high accuracy and low computations of the presented scheme are investigated by solving some numerical examples.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
| 34A08 | Fractional ordinary differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65D32 | Numerical quadrature and cubature formulas |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 76A20 | Thin fluid films |
| 76B45 | Capillarity (surface tension) for incompressible inviscid fluids |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35B36 | Pattern formations in context of PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Keywords:
Benney-Lin equation; Jacobi polynomials; Jacobi-Gauss-Lobatto quadrature technique; variable-order fractional derivative matrixReferences:
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