Fractional convolution quadrature based on generalized Adams methods
The authors obtain numerical solutions of fractional differential equations of the form \[ D^{\alpha}_{t_{0}}y(t)=f(t,y(t)),\, t_{0}<t\leq T,\, 0<\alpha<1,\tag{1} \] \(y(t_{0})=y_{0}\), where \( D^{\alpha}_{t_{0}}y(t)\) denotes the Caputo fractional derivative defined by \[ D^{\alpha}_{t_{0}}y(t)=\frac{1}{\Gamma(1-\alpha)}\int^{t}_{t_{0}}\frac{y'(u)}{(t-u)^{\alpha}}du. \tag{2} \] Since the solution of (1) can be written as \[ y(t)=y_{0}+\frac{1}{\Gamma(\alpha)}\int^{t}_{t_{0}}(t-u)^{\alpha-1}f(u,y(u))du,\tag{3} \] each quadrature scheme for (3) leads to a numerical solver for (2). The generalized Adam's method for ordinary differential equations is extended to define product quadrature rules for the solution of (3). Numerical examples are given to illustrate the results obtained.
scientific article; zbMATH DE number 6418679
scientific article