Abstract
A radial basis function method for solving time-fractional KdV equation is presented. The Caputo derivative is approximated by the high order formulas introduced in Buhman (Proc. Edinb. Math. Soc. 36:319–333, 1993). By choosing the centers of radial basis functions as collocation points, in each time step a nonlinear system of algebraic equations is obtained. A fixed point predictor–corrector method for solving the system is introduced. The efficiency and accuracy of our method are demonstrated through several illustrative examples. By the examples, the experimental convergence order is approximately 4-alpha , where alpha is the order of time derivative.
Highlights
In this paper, we consider the time-fractional KdV equation ∂αu(x, t) ∂tα +ε u(x, t) m ∂u(x, t) ∂x∂3u(x, t) + ν ∂x3 = f (x, t), x∈ = [a, b], t ≥ 0, (1)with the following initial condition: u(x, 0) = g(x), x ∈, (2)
There is a sufficient condition for the convergence of the iteration formula (11) [31]
∂αu(X,tn ) ∂tα for n = 1, n = 2 and n ≥3. These formulas are of orders 2 − α, 3 − α and 4 − α, for, respectively, n=1, n=2 and n ≥3, where α is the order of derivative
Summary
Name of function Gaussian (GA) Hardy multiquadric (MQ) Inverse multiquadric (IMQ) Inverse quadric (IQ). Several numerical and analytical methods for solving fractional KdV equations have been introduced. In [29], the variational iteration method for the space- and time-fractional KdV equation was applied. To see the applications of RBFs in the numerical solution of partial differential equations (PDEs) and fractional PDEs, for example, see [7,11,15,24,27,36]. (1)–(4) is approximated by a linear combination of RBFs with unknown coefficients For finding these coefficients, we choose the centers of RBFs as collocation points, and a nonlinear system of algebraic equations is obtained. By the fixed point method, the computations of the nonlinear system are reduced to some linear systems of equations. 2, a fixed point iteration method for solving the systems of nonlinear algebraic equations is introduced.
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