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A high order numerical method for solving nonlinear fractional differential equation with non-uniform meshes
Fan, Lili Yan, Yubin
Fan, Lili
Yan, Yubin
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2019-01-18
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Abstract
We introduce a high-order numerical method for solving nonlinear fractional differential equation with non-uniform meshes. We first transform the fractional nonlinear differential equation into the equivalent Volterra integral equation. Then we approximate the integral by using the quadratic interpolation polynomials. On the first subinterval $[t_{0}, t_{1}]$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{0}, t_{1}, t_{2}$ and in the other subinterval $[t_{j}, t_{j+1}], j=1, 2, \dots N-1$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{j-1}, t_{j}, t_{j+1}$. A high-order numerical method is obtained. Then we apply this numerical method with the non-uniform meshes with the step size $\tau_{j}= t_{j+1}- t_{j}= (j+1) \mu$ where $\mu= \frac{2T}{N (N+1)}$. Numerical results show that this method with the non-uniform meshes has the higher convergence order than the standard numerical methods obtained by using the rectangle and the trapzoid rules with the same non-uniform meshes.
Citation
Fan, L., & Yan Y. (2019). A high order numerical method for solving nonlinear fractional differential equation with non-uniform meshes. In G. Nikolov, N. Kolkovska, & K. Georgiev (Eds.) Numerical methods and applications: 9th International Conference, Borovets, Bulgaria, August 20-24, 2018. Springer.
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Springer
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10.1007/978-3-030-10692-8_23
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Book chapter
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en
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Lecture Notes in Computer Science, volume 11189
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9783030106911