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Waveform relaxation methods for fractional differential equations with the Caputo derivatives. (English) Zbl 1259.65113

In waveform relaxation methods for the numerical solution of linear initial value problems (IVPs) in differential equations:

$\stackrel{˙}{x}\left(t\right)+Ax\left(t\right)=b\left(t\right),\phantom{\rule{1.em}{0ex}}t\in \left[0,T\right],\phantom{\rule{3.33333pt}{0ex}}x\left(0\right)={x}^{0}\in {ℝ}^{d},\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $A\in {ℝ}^{d×d}$, $b\left(t\right)\in {ℝ}^{d}$, the solution $x\left(t\right)$ of the IVP (1) is approximated by some ${x}_{k}\left(t\right)$ obtained from the iterative process

${\stackrel{˙}{x}}_{i+1}\left(t\right)+{A}_{1}{x}_{i+1}\left(t\right)=-{A}_{2}{x}_{i}\left(t\right)+b\left(t\right),\phantom{\rule{1.em}{0ex}}i=0,1,\cdots ,\phantom{\rule{3.33333pt}{0ex}}{x}_{i+1}\left(0\right)={x}^{0},\phantom{\rule{2.em}{0ex}}\left(2\right)$

with a suitable splitting $A={A}_{1}+{A}_{2}$ so that (2) is easily solved in ${x}_{i+1}$ and $\left({x}_{i}\right)$ converges as fast as possible to the exact solution (see e.g. [U. Miekkala and O. Nevanlinna, SIAM J. Sci. Stat. Comput. 8, 459–482 (1987; Zbl 0625.65063)]).

In the paper under consideration, the standard derivative $\stackrel{˙}{x}=Dx$ in (1) is substituted by the Caputo fractional derivative ${}^{C}{D}_{t}^{\alpha }$ of order $\alpha \in \left(0,1\right)$ studying the application of waveform relaxation methods in this context of fractional differential equations. The contents of the paper is as follows: In Section 2, an introduction of the basic notations, definitions and main results is briefly outlined. Then in the first part of Section 3, several theorems on convergence results for linear problems and finite time intervals with several standard splittings are given. Further, some examples in which the linear term of (1) $Ax$ comes from spatial discretization of diffusion and convection operators are presented to illustrate the application of this theory. In the second part of Section 3, nonlinear IVPs for fractional differential equations: ${}^{C}{D}_{t}^{\alpha }x\left(t\right)=f\left(t,x\left(t\right)\right)$, $x\left(0\right)={x}^{0}$ on finite intervals are considered. Now, the nonlinear splitting of $f\left(t,x\right)$ is a suitable function $F\left(t,u,v\right)$ such that $F\left(t,x,x\right)\equiv f\left(t,x\right)$ and under suitable conditions on $F$ sufficient conditions are given for the waveform relation iteration ${}^{C}{D}_{t}^{\alpha }{x}_{i+1}\left(t\right)=F\left(t,{x}_{i}\left(t\right),{x}_{i+1}\left(t\right)\right)$, ${x}_{i+1}\left(0\right)={x}^{0}$.

##### MSC:
 65L05 Initial value problems for ODE (numerical methods) 34A08 Fractional differential equations 65L20 Stability and convergence of numerical methods for ODE 34A30 Linear ODE and systems, general 34A34 Nonlinear ODE and systems, general